1,1,112,132,0.1240034,"\int x^2 (d+e x) \sqrt{d^2-e^2 x^2} \, dx","Integrate[x^2*(d + e*x)*Sqrt[d^2 - e^2*x^2],x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 d^4 \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-16 d^4-15 d^3 e x-8 d^2 e^2 x^2+30 d e^3 x^3+24 e^4 x^4\right)\right)}{120 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{d x \left(d^2-e^2 x^2\right)^{3/2}}{4 e^2}-\frac{d^2 \left(d^2-e^2 x^2\right)^{3/2}}{3 e^3}+\frac{\left(d^2-e^2 x^2\right)^{5/2}}{5 e^3}+\frac{d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}+\frac{d^3 x \sqrt{d^2-e^2 x^2}}{8 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-16*d^4 - 15*d^3*e*x - 8*d^2*e^2*x^2 + 30*d*e^3*x^3 + 24*e^4*x^4) + 15*d^4*ArcSin[(e*x)/d]))/(120*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
2,1,157,201,0.2135955,"\int x^4 (d+e x) \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(945 d^8 \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(1024 d^8+945 d^7 e x+512 d^6 e^2 x^2+630 d^5 e^3 x^3+384 d^4 e^4 x^4-7560 d^3 e^5 x^5-6400 d^2 e^6 x^6+5040 d e^7 x^7+4480 e^8 x^8\right)\right)}{40320 e^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{x^4 \left(d^2-e^2 x^2\right)^{5/2}}{9 e}-\frac{d x^3 \left(d^2-e^2 x^2\right)^{5/2}}{8 e^2}-\frac{4 d^2 x^2 \left(d^2-e^2 x^2\right)^{5/2}}{63 e^3}+\frac{3 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e^5}+\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left(d^2-e^2 x^2\right)^{3/2}}{64 e^4}-\frac{d^3 (128 d+315 e x) \left(d^2-e^2 x^2\right)^{5/2}}{5040 e^5}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(1024*d^8 + 945*d^7*e*x + 512*d^6*e^2*x^2 + 630*d^5*e^3*x^3 + 384*d^4*e^4*x^4 - 7560*d^3*e^5*x^5 - 6400*d^2*e^6*x^6 + 5040*d*e^7*x^7 + 4480*e^8*x^8)) + 945*d^8*ArcSin[(e*x)/d]))/(40320*e^5*Sqrt[1 - (e^2*x^2)/d^2])","A",1
3,1,146,172,0.1907723,"\int x^3 (d+e x) \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(105 d^7 \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(256 d^7+105 d^6 e x+128 d^5 e^2 x^2+70 d^4 e^3 x^3-1024 d^3 e^4 x^4-840 d^2 e^5 x^5+640 d e^6 x^6+560 e^7 x^7\right)\right)}{4480 e^4 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{d x^2 \left(d^2-e^2 x^2\right)^{5/2}}{7 e^2}-\frac{x^3 \left(d^2-e^2 x^2\right)^{5/2}}{8 e}-\frac{d^2 (32 d+35 e x) \left(d^2-e^2 x^2\right)^{5/2}}{560 e^4}+\frac{3 d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e^4}+\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}+\frac{d^4 x \left(d^2-e^2 x^2\right)^{3/2}}{64 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(256*d^7 + 105*d^6*e*x + 128*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 1024*d^3*e^4*x^4 - 840*d^2*e^5*x^5 + 640*d*e^6*x^6 + 560*e^7*x^7)) + 105*d^7*ArcSin[(e*x)/d]))/(4480*e^4*Sqrt[1 - (e^2*x^2)/d^2])","A",1
4,1,135,159,0.1747033,"\int x^2 (d+e x) \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(105 d^6 \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(96 d^6+105 d^5 e x+48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+280 d e^5 x^5+240 e^6 x^6\right)\right)}{1680 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{d x \left(d^2-e^2 x^2\right)^{5/2}}{6 e^2}-\frac{d^2 \left(d^2-e^2 x^2\right)^{5/2}}{5 e^3}+\frac{\left(d^2-e^2 x^2\right)^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^3}+\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left(d^2-e^2 x^2\right)^{3/2}}{24 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(96*d^6 + 105*d^5*e*x + 48*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 384*d^2*e^4*x^4 + 280*d*e^5*x^5 + 240*e^6*x^6)) + 105*d^6*ArcSin[(e*x)/d]))/(1680*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
5,1,124,116,0.1418541,"\int x (d+e x) \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[x*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 d^5 \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(48 d^5+15 d^4 e x-96 d^3 e^2 x^2-70 d^2 e^3 x^3+48 d e^4 x^4+40 e^5 x^5\right)\right)}{240 e^2 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^2 x \left(d^2-e^2 x^2\right)^{3/2}}{24 e}-\frac{(6 d+5 e x) \left(d^2-e^2 x^2\right)^{5/2}}{30 e^2}+\frac{d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^2}+\frac{d^4 x \sqrt{d^2-e^2 x^2}}{16 e}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(48*d^5 + 15*d^4*e*x - 96*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 48*d*e^4*x^4 + 40*e^5*x^5)) + 15*d^5*ArcSin[(e*x)/d]))/(240*e^2*Sqrt[1 - (e^2*x^2)/d^2])","A",1
6,1,124,116,0.0421149,"\int x (d+e x) \left(d^2-e^2 x^2\right)^{3/2} \, dx","Integrate[x*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 d^5 \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(48 d^5+15 d^4 e x-96 d^3 e^2 x^2-70 d^2 e^3 x^3+48 d e^4 x^4+40 e^5 x^5\right)\right)}{240 e^2 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^2 x \left(d^2-e^2 x^2\right)^{3/2}}{24 e}-\frac{(6 d+5 e x) \left(d^2-e^2 x^2\right)^{5/2}}{30 e^2}+\frac{d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^2}+\frac{d^4 x \sqrt{d^2-e^2 x^2}}{16 e}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(48*d^5 + 15*d^4*e*x - 96*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 48*d*e^4*x^4 + 40*e^5*x^5)) + 15*d^5*ArcSin[(e*x)/d]))/(240*e^2*Sqrt[1 - (e^2*x^2)/d^2])","A",1
7,1,124,113,0.1835719,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x,x]","d^4 \left(-\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)\right)+\frac{3 d^3 \sqrt{d^2-e^2 x^2} \sin ^{-1}\left(\frac{e x}{d}\right)}{8 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{1}{24} \sqrt{d^2-e^2 x^2} \left(32 d^3+15 d^2 e x-8 d e^2 x^2-6 e^3 x^3\right)","\frac{1}{8} d^2 (8 d+3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d+3 e x) \left(d^2-e^2 x^2\right)^{3/2}+\frac{3}{8} d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(Sqrt[d^2 - e^2*x^2]*(32*d^3 + 15*d^2*e*x - 8*d*e^2*x^2 - 6*e^3*x^3))/24 + (3*d^3*Sqrt[d^2 - e^2*x^2]*ArcSin[(e*x)/d])/(8*Sqrt[1 - (e^2*x^2)/d^2]) - d^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]","A",1
8,1,124,117,0.17446,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^2} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^2,x]","-\frac{d^5 \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x \sqrt{d^2-e^2 x^2}}-\frac{1}{3} e \left(\sqrt{d^2-e^2 x^2} \left(e^2 x^2-4 d^2\right)+3 d^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)\right)","\frac{1}{2} d e (2 d-3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d^3 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-1/3*(e*(Sqrt[d^2 - e^2*x^2]*(-4*d^2 + e^2*x^2) + 3*d^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])) - (d^5*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, (e^2*x^2)/d^2])/(x*Sqrt[d^2 - e^2*x^2])","C",1
9,1,110,121,0.0766066,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^3} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^3,x]","-\frac{d^2 e \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{e^2 \left(d^2-e^2 x^2\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right)}{5 d^3}","-\frac{3 d e (d+e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{2 x^2}-\frac{3}{2} d^2 e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-((d^2*e*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, (e^2*x^2)/d^2])/(x*Sqrt[1 - (e^2*x^2)/d^2])) - (e^2*(d^2 - e^2*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 - (e^2*x^2)/d^2])/(5*d^3)","C",1
10,1,111,120,0.0603263,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^4} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^4,x]","-\frac{e^3 \left(d^2-e^2 x^2\right)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right)}{5 d^4}-\frac{d^3 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{e^2 (2 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(2 d+3 e x) \left(d^2-e^2 x^2\right)^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{3}{2} d e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-1/3*(d^3*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (e^2*x^2)/d^2])/(x^3*Sqrt[1 - (e^2*x^2)/d^2]) - (e^3*(d^2 - e^2*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 - (e^2*x^2)/d^2])/(5*d^4)","C",1
11,1,133,118,0.0894158,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^5} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^5,x]","-\frac{\sqrt{d^2-e^2 x^2} \left(3 d^2 \left(2 d^2-5 e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}}+9 e^4 x^4 \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+8 d^3 e x \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)\right)}{24 d x^4 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{e^2 (3 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{(3 d+4 e x) \left(d^2-e^2 x^2\right)^{3/2}}{12 x^4}+e^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{3}{8} e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-1/24*(Sqrt[d^2 - e^2*x^2]*(3*d^2*(2*d^2 - 5*e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2] + 9*e^4*x^4*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]] + 8*d^3*e*x*Hypergeometric2F1[-3/2, -3/2, -1/2, (e^2*x^2)/d^2]))/(d*x^4*Sqrt[1 - (e^2*x^2)/d^2])","C",1
12,1,133,108,0.0612908,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^6} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^6,x]","-\frac{8 d^6+10 d^5 e x-24 d^4 e^2 x^2-35 d^3 e^3 x^3+24 d^2 e^4 x^4+15 d e^5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+25 d e^5 x^5-8 e^6 x^6}{40 d x^5 \sqrt{d^2-e^2 x^2}}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{5 d x^5}-\frac{e \left(d^2-e^2 x^2\right)^{3/2}}{4 x^4}-\frac{3 e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d}+\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{8 x^2}",1,"-1/40*(8*d^6 + 10*d^5*e*x - 24*d^4*e^2*x^2 - 35*d^3*e^3*x^3 + 24*d^2*e^4*x^4 + 25*d*e^5*x^5 - 8*e^6*x^6 + 15*d*e^5*x^5*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(d*x^5*Sqrt[d^2 - e^2*x^2])","A",1
13,1,59,143,0.0208528,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^7} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^7,x]","-\frac{e \left(d^2-e^2 x^2\right)^{5/2} \left(d^5+e^5 x^5 \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right)\right)}{5 d^7 x^5}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{6 d x^6}-\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{5 d^2 x^5}-\frac{e^2 \left(d^2-e^2 x^2\right)^{3/2}}{24 d x^4}-\frac{e^6 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{16 d^2}+\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}",1,"-1/5*(e*(d^2 - e^2*x^2)^(5/2)*(d^5 + e^5*x^5*Hypergeometric2F1[5/2, 4, 7/2, 1 - (e^2*x^2)/d^2]))/(d^7*x^5)","C",1
14,1,72,172,0.0200961,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^8} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^8,x]","-\frac{\left(d^2-e^2 x^2\right)^{5/2} \left(5 d^7+2 d^5 e^2 x^2+7 e^7 x^7 \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right)\right)}{35 d^8 x^7}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{7 d x^7}-\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{6 d^2 x^6}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left(d^2-e^2 x^2\right)^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left(d^2-e^2 x^2\right)^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{16 d^3}",1,"-1/35*((d^2 - e^2*x^2)^(5/2)*(5*d^7 + 2*d^5*e^2*x^2 + 7*e^7*x^7*Hypergeometric2F1[5/2, 4, 7/2, 1 - (e^2*x^2)/d^2]))/(d^8*x^7)","C",1
15,1,73,201,0.0224789,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{x^9} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,x]","-\frac{e \left(d^2-e^2 x^2\right)^{5/2} \left(5 d^7+2 d^5 e^2 x^2+7 e^7 x^7 \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right)\right)}{35 d^9 x^7}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{8 d x^8}-\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{128 d^4}-\frac{2 e^3 \left(d^2-e^2 x^2\right)^{5/2}}{35 d^4 x^5}-\frac{e^2 \left(d^2-e^2 x^2\right)^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left(d^2-e^2 x^2\right)^{3/2}}{64 d^3 x^4}",1,"-1/35*(e*(d^2 - e^2*x^2)^(5/2)*(5*d^7 + 2*d^5*e^2*x^2 + 7*e^7*x^7*Hypergeometric2F1[5/2, 5, 7/2, 1 - (e^2*x^2)/d^2]))/(d^9*x^7)","C",1
16,1,70,103,0.0381266,"\int \frac{x^2 (d+e x)}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(x^2*(d + e*x))/Sqrt[d^2 - e^2*x^2],x]","\frac{3 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(4 d^2+3 d e x+2 e^2 x^2\right)}{6 e^3}","-\frac{d x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2}}{e^3}+\frac{\left(d^2-e^2 x^2\right)^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^3}",1,"(-(Sqrt[d^2 - e^2*x^2]*(4*d^2 + 3*d*e*x + 2*e^2*x^2)) + 3*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^3)","A",1
17,1,77,73,0.0306355,"\int \frac{x^2 (d+e x)}{\left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(3/2),x]","\frac{-d \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+2 d^2+d e x-e^2 x^2}{e^3 \sqrt{d^2-e^2 x^2}}","\frac{d (d+e x)}{e^3 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}",1,"(2*d^2 + d*e*x - e^2*x^2 - d*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(e^3*Sqrt[d^2 - e^2*x^2])","A",1
18,1,52,58,0.0211391,"\int \frac{x^2 (d+e x)}{\left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(5/2),x]","\frac{-2 d^2+2 d e x+e^2 x^2}{3 d e^3 (d-e x) \sqrt{d^2-e^2 x^2}}","\frac{x^2 (d+e x)}{3 d e \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}}",1,"(-2*d^2 + 2*d*e*x + e^2*x^2)/(3*d*e^3*(d - e*x)*Sqrt[d^2 - e^2*x^2])","A",1
19,1,155,161,0.0967275,"\int \frac{x^7 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^7*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-105 d^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-15 d e^5 x^5-15 e^6 x^6}{30 e^8 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^6 (d+e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^8}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(96*d^6 + 9*d^5*e*x - 249*d^4*e^2*x^2 + 4*d^3*e^3*x^3 + 176*d^2*e^4*x^4 - 15*d*e^5*x^5 - 15*e^6*x^6 - 105*d^2*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
20,1,142,147,0.0873683,"\int \frac{x^6 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^6*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{48 d^5-33 d^4 e x-87 d^3 e^2 x^2+52 d^2 e^3 x^3-15 d (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+38 d e^4 x^4-15 e^5 x^5}{15 e^7 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^5 (d+e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^7}+\frac{x (5 d+8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d+6 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(48*d^5 - 33*d^4*e*x - 87*d^3*e^2*x^2 + 52*d^2*e^3*x^3 + 38*d*e^4*x^4 - 15*e^5*x^5 - 15*d*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^7*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
21,1,130,122,0.0814741,"\int \frac{x^5 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^5*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{8 d^4+7 d^3 e x-27 d^2 e^2 x^2-15 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-8 d e^3 x^3+23 e^4 x^4}{15 e^6 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^4 (d+e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^6}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(8*d^4 + 7*d^3*e*x - 27*d^2*e^2*x^2 - 8*d*e^3*x^3 + 23*e^4*x^4 - 15*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
22,1,82,84,0.0251341,"\int \frac{x^4 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{8 d^4-8 d^3 e x-12 d^2 e^2 x^2+12 d e^3 x^3+3 e^4 x^4}{15 d e^5 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^4 (d+e x)}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}+\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{4 d^2}{15 e^5 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(8*d^4 - 8*d^3*e*x - 12*d^2*e^2*x^2 + 12*d*e^3*x^3 + 3*e^4*x^4)/(15*d*e^5*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
23,1,82,90,0.0227354,"\int \frac{x^3 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^3*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{-2 d^4+2 d^3 e x+3 d^2 e^2 x^2-3 d e^3 x^3+3 e^4 x^4}{15 d^2 e^4 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^2 (d+e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}",1,"(-2*d^4 + 2*d^3*e*x + 3*d^2*e^2*x^2 - 3*d*e^3*x^3 + 3*e^4*x^4)/(15*d^2*e^4*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
24,1,82,94,0.024772,"\int \frac{x^2 (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{-2 d^4+2 d^3 e x+3 d^2 e^2 x^2+2 d e^3 x^3-2 e^4 x^4}{15 d^3 e^3 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{x^2 (d+e x)}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 (d-e x)}{15 d e^3 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}",1,"(-2*d^4 + 2*d^3*e*x + 3*d^2*e^2*x^2 + 2*d*e^3*x^3 - 2*e^4*x^4)/(15*d^3*e^3*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
25,1,82,83,0.0348932,"\int \frac{x (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{3 d^4-3 d^3 e x+3 d^2 e^2 x^2+2 d e^3 x^3-2 e^4 x^4}{15 d^4 e^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","-\frac{x}{15 d^2 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{d+e x}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}}",1,"(3*d^4 - 3*d^3*e*x + 3*d^2*e^2*x^2 + 2*d*e^3*x^3 - 2*e^4*x^4)/(15*d^4*e^2*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
26,1,82,80,0.0286511,"\int \frac{d+e x}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(d^2 - e^2*x^2)^(7/2),x]","\frac{3 d^4+12 d^3 e x-12 d^2 e^2 x^2-8 d e^3 x^3+8 e^4 x^4}{15 d^5 e (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{d+e x}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(3*d^4 + 12*d^3*e*x - 12*d^2*e^2*x^2 - 8*d*e^3*x^3 + 8*e^4*x^4)/(15*d^5*e*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
27,1,131,117,0.062932,"\int \frac{d+e x}{x \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(x*(d^2 - e^2*x^2)^(7/2)),x]","\frac{23 d^4-8 d^3 e x-27 d^2 e^2 x^2-15 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)+7 d e^3 x^3+8 e^4 x^4}{15 d^6 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{d+e x}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{15 d+8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^6}+\frac{5 d+4 e x}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(23*d^4 - 8*d^3*e*x - 27*d^2*e^2*x^2 + 7*d*e^3*x^3 + 8*e^4*x^4 - 15*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(15*d^6*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
28,1,147,153,0.0727441,"\int \frac{d+e x}{x^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(x^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{-15 d^5+38 d^4 e x+52 d^3 e^2 x^2-87 d^2 e^3 x^3-15 e x (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-33 d e^4 x^4+48 e^5 x^5}{15 d^7 x (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}}","\frac{d+e x}{5 d^2 x \left(d^2-e^2 x^2\right)^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^7}+\frac{8 d+5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d+5 e x}{15 d^4 x \left(d^2-e^2 x^2\right)^{3/2}}",1,"(-15*d^5 + 38*d^4*e*x + 52*d^3*e^2*x^2 - 87*d^2*e^3*x^3 - 33*d*e^4*x^4 + 48*e^5*x^5 - 15*e*x*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(15*d^7*x*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])","A",1
29,1,183,184,0.1345271,"\int \frac{d+e x}{x^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{105 e^2 x^2 (d+e x)^2 (e x-d)^3 \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+d \sqrt{1-\frac{e^2 x^2}{d^2}} \left(-15 d^6-15 d^5 e x+176 d^4 e^2 x^2+4 d^3 e^3 x^3-249 d^2 e^4 x^4+9 d e^5 x^5+96 e^6 x^6\right)}{30 d^9 x^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d+e x}{5 d^2 x^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(d*Sqrt[1 - (e^2*x^2)/d^2]*(-15*d^6 - 15*d^5*e*x + 176*d^4*e^2*x^2 + 4*d^3*e^3*x^3 - 249*d^2*e^4*x^4 + 9*d*e^5*x^5 + 96*e^6*x^6) + 105*e^2*x^2*(-d + e*x)^3*(d + e*x)^2*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(30*d^9*x^2*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
30,1,104,121,0.0404613,"\int \frac{x^2 (d+e x)}{\left(d^2-e^2 x^2\right)^{9/2}} \, dx","Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(9/2),x]","\frac{-6 d^6+6 d^5 e x+15 d^4 e^2 x^2+20 d^3 e^3 x^3-20 d^2 e^4 x^4-8 d e^5 x^5+8 e^6 x^6}{105 d^5 e^3 (d-e x)^3 (d+e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{x^2 (d+e x)}{7 d e \left(d^2-e^2 x^2\right)^{7/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(-6*d^6 + 6*d^5*e*x + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 - 20*d^2*e^4*x^4 - 8*d*e^5*x^5 + 8*e^6*x^6)/(105*d^5*e^3*(d - e*x)^3*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
31,1,126,148,0.0494978,"\int \frac{x^2 (d+e x)}{\left(d^2-e^2 x^2\right)^{11/2}} \, dx","Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x]","\frac{-10 d^8+10 d^7 e x+35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+16 d e^7 x^7-16 e^8 x^8}{315 d^7 e^3 (d-e x)^4 (d+e x)^3 \sqrt{d^2-e^2 x^2}}","\frac{x^2 (d+e x)}{9 d e \left(d^2-e^2 x^2\right)^{9/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left(d^2-e^2 x^2\right)^{7/2}}-\frac{16 x}{315 d^7 e^2 \sqrt{d^2-e^2 x^2}}-\frac{8 x}{315 d^5 e^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 x}{105 d^3 e^2 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(-10*d^8 + 10*d^7*e*x + 35*d^6*e^2*x^2 + 70*d^5*e^3*x^3 - 70*d^4*e^4*x^4 - 56*d^3*e^5*x^5 + 56*d^2*e^6*x^6 + 16*d*e^7*x^7 - 16*e^8*x^8)/(315*d^7*e^3*(d - e*x)^4*(d + e*x)^3*Sqrt[d^2 - e^2*x^2])","A",1
32,1,50,54,0.0292201,"\int \frac{x^2 (1-a x)}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(x^2*(1 - a*x))/(1 - a^2*x^2)^(3/2),x]","\frac{a^2 x^2-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)+a x-2}{a^3 \sqrt{1-a^2 x^2}}","-\frac{\sin ^{-1}(a x)}{a^3}-\frac{1-a x}{a^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{a^3}",1,"(-2 + a*x + a^2*x^2 - Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(a^3*Sqrt[1 - a^2*x^2])","A",1
33,1,103,173,0.0993749,"\int \frac{x^4 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(x^4*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]","\frac{165 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(256 d^5+165 d^4 e x+128 d^3 e^2 x^2+110 d^2 e^3 x^3+96 d e^4 x^4+40 e^5 x^5\right)}{240 e^5}","-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}+\frac{11 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^5}-\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}",1,"(-(Sqrt[d^2 - e^2*x^2]*(256*d^5 + 165*d^4*e*x + 128*d^3*e^2*x^2 + 110*d^2*e^3*x^3 + 96*d*e^4*x^4 + 40*e^5*x^5)) + 165*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^5)","A",1
34,1,92,144,0.089817,"\int \frac{x^3 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(x^3*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]","\frac{15 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(24 d^4+15 d^3 e x+12 d^2 e^2 x^2+10 d e^3 x^3+4 e^4 x^4\right)}{20 e^4}","-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{4 e^4}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}",1,"(-(Sqrt[d^2 - e^2*x^2]*(24*d^4 + 15*d^3*e*x + 12*d^2*e^2*x^2 + 10*d*e^3*x^3 + 4*e^4*x^4)) + 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(20*e^4)","A",1
35,1,81,115,0.0702554,"\int \frac{x^2 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(x^2*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]","\frac{21 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(32 d^3+21 d^2 e x+16 d e^2 x^2+6 e^3 x^3\right)}{24 e^3}","-\frac{2 d x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} x^3 \sqrt{d^2-e^2 x^2}-\frac{d^2 (32 d+21 e x) \sqrt{d^2-e^2 x^2}}{24 e^3}+\frac{7 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}",1,"(-(Sqrt[d^2 - e^2*x^2]*(32*d^3 + 21*d^2*e*x + 16*d*e^2*x^2 + 6*e^3*x^3)) + 21*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e^3)","A",1
36,1,69,83,0.0541128,"\int \frac{x (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(x*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]","\frac{3 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\sqrt{d^2-e^2 x^2} \left(5 d^2+3 d e x+e^2 x^2\right)}{3 e^2}","-\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}+\frac{d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^2}",1,"(-(Sqrt[d^2 - e^2*x^2]*(5*d^2 + 3*d*e*x + e^2*x^2)) + 3*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e^2)","A",1
37,1,58,83,0.0359105,"\int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/Sqrt[d^2 - e^2*x^2],x]","\frac{3 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-(4 d+e x) \sqrt{d^2-e^2 x^2}}{2 e}","-\frac{3 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{(d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"(-((4*d + e*x)*Sqrt[d^2 - e^2*x^2]) + 3*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)","A",1
38,1,66,66,0.0268173,"\int \frac{(d+e x)^2}{x \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x*Sqrt[d^2 - e^2*x^2]),x]","-\sqrt{d^2-e^2 x^2}+2 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)","-\sqrt{d^2-e^2 x^2}+2 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-Sqrt[d^2 - e^2*x^2] + 2*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - d*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]","A",1
39,1,68,68,0.0295974,"\int \frac{(d+e x)^2}{x^2 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x^2*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2}}{x}+e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-2 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)","-\frac{\sqrt{d^2-e^2 x^2}}{x}+e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-2 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-(Sqrt[d^2 - e^2*x^2]/x) + e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 2*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]","A",1
40,1,122,80,0.2434847,"\int \frac{(d+e x)^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x^3*Sqrt[d^2 - e^2*x^2]),x]","\frac{e \left(-\frac{4 d \sqrt{d^2-e^2 x^2}}{x}-2 d e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-e \sqrt{d^2-e^2 x^2} \left(\frac{d^2}{e^2 x^2}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)}{\sqrt{1-\frac{e^2 x^2}{d^2}}}\right)\right)}{2 d^2}","-\frac{2 e \sqrt{d^2-e^2 x^2}}{d x}-\frac{\sqrt{d^2-e^2 x^2}}{2 x^2}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d}",1,"(e*((-4*d*Sqrt[d^2 - e^2*x^2])/x - 2*d*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d] - e*Sqrt[d^2 - e^2*x^2]*(d^2/(e^2*x^2) + ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]/Sqrt[1 - (e^2*x^2)/d^2])))/(2*d^2)","A",1
41,1,87,107,0.1497255,"\int \frac{(d+e x)^2}{x^4 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x^4*Sqrt[d^2 - e^2*x^2]),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-\frac{d \left(d^2+3 d e x+5 e^2 x^2\right)}{x^3}-\frac{3 e^3 \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)}{\sqrt{1-\frac{e^2 x^2}{d^2}}}\right)}{3 d^3}","-\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}-\frac{e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-((d*(d^2 + 3*d*e*x + 5*e^2*x^2))/x^3) - (3*e^3*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/Sqrt[1 - (e^2*x^2)/d^2]))/(3*d^3)","A",1
42,1,155,140,0.1477384,"\int \frac{(d+e x)^2}{x^5 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x^5*Sqrt[d^2 - e^2*x^2]),x]","-\frac{e \sqrt{d^2-e^2 x^2} \left(d \left(4 d^2+3 d e x+8 e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}}+6 e^3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+3 e^3 x^3 \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)\right)}{6 d^4 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{7 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}-\frac{2 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{7 e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d^3}-\frac{4 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}",1,"-1/6*(e*Sqrt[d^2 - e^2*x^2]*(d*(4*d^2 + 3*d*e*x + 8*e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2] + 3*e^3*x^3*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]] + 6*e^3*x^3*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (e^2*x^2)/d^2]))/(d^4*x^3*Sqrt[1 - (e^2*x^2)/d^2])","C",1
43,1,79,169,0.0399284,"\int \frac{(d+e x)^2}{x^6 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(d + e*x)^2/(x^6*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(d^5+3 d^3 e^2 x^2+10 e^5 x^5 \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+6 d e^4 x^4\right)}{5 d^5 x^5}","-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{4 d^4}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}",1,"-1/5*(Sqrt[d^2 - e^2*x^2]*(d^5 + 3*d^3*e^2*x^2 + 6*d*e^4*x^4 + 10*e^5*x^5*Hypergeometric2F1[1/2, 3, 3/2, 1 - (e^2*x^2)/d^2]))/(d^5*x^5)","C",1
44,1,111,143,0.2119517,"\int \frac{x^5 (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^5*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{56 d^4-82 d^3 e x-32 d^2 e^2 x^2-\frac{30 (d-e x)^3 (d+e x) \sin ^{-1}\left(\frac{e x}{d}\right)}{\sqrt{1-\frac{e^2 x^2}{d^2}}}+76 d e^3 x^3-15 e^4 x^4}{15 e^6 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^6}+\frac{d^4 (d+e x)^2}{5 e^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(56*d^4 - 82*d^3*e*x - 32*d^2*e^2*x^2 + 76*d*e^3*x^3 - 15*e^4*x^4 - (30*(d - e*x)^3*(d + e*x)*ArcSin[(e*x)/d])/Sqrt[1 - (e^2*x^2)/d^2])/(15*e^6*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
45,1,96,121,0.2021593,"\int \frac{x^4 (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^4*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{16 d^3-15 d (d-e x)^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left(\frac{e x}{d}\right)-17 d^2 e x-22 d e^2 x^2+26 e^3 x^3}{15 e^5 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","-\frac{17 d^2 (d+e x)}{15 e^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 (15 d+13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}+\frac{d^3 (d+e x)^2}{5 e^5 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(16*d^3 - 17*d^2*e*x - 22*d*e^2*x^2 + 26*e^3*x^3 - 15*d*(d - e*x)^2*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d])/(15*e^5*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
46,1,63,97,0.0599194,"\int \frac{x^3 (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^3*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{2 d^3-4 d^2 e x+d e^2 x^2+2 e^3 x^3}{5 d e^4 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{d^2 (d+e x)^2}{5 e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{4 d (d+e x)}{5 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{5 d+2 e x}{5 d e^4 \sqrt{d^2-e^2 x^2}}",1,"(2*d^3 - 4*d^2*e*x + d*e^2*x^2 + 2*e^3*x^3)/(5*d*e^4*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
47,1,63,87,0.0552182,"\int \frac{x^2 (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^2*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{-4 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{15 d^2 e^3 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}+\frac{d (d+e x)^2}{5 e^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{7 (d+e x)}{15 e^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(-4*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)/(15*d^2*e^3*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
48,1,62,89,0.0505543,"\int \frac{x (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{d^3-2 d^2 e x+8 d e^2 x^2-4 e^3 x^3}{15 d^3 e^2 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^2}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}",1,"(d^3 - 2*d^2*e*x + 8*d*e^2*x^2 - 4*e^3*x^3)/(15*d^3*e^2*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
49,1,63,77,0.0423809,"\int \frac{(d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(d^2 - e^2*x^2)^(7/2),x]","\frac{2 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3}{5 d^4 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{x}{5 d^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 (d+e x)}{5 e \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}",1,"(2*d^3 + d^2*e*x - 4*d*e^2*x^2 + 2*e^3*x^3)/(5*d^4*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
50,1,81,117,0.042251,"\int \frac{(d+e x)^2}{x \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(x*(d^2 - e^2*x^2)^(7/2)),x]","\frac{3 d^5+30 d^4 e x-40 d^2 e^3 x^3+3 d^5 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+16 e^5 x^5}{15 d^5 \left(d^2-e^2 x^2\right)^{5/2}}","\frac{2 (d+e x)}{5 d \left(d^2-e^2 x^2\right)^{5/2}}+\frac{15 d+16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^5}+\frac{5 d+8 e x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(3*d^5 + 30*d^4*e*x - 40*d^2*e^3*x^3 + 16*e^5*x^5 + 3*d^5*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2])/(15*d^5*(d^2 - e^2*x^2)^(5/2))","C",1
51,1,90,145,0.0500649,"\int \frac{(d+e x)^2}{x^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(x^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{-15 d^6+105 d^4 e^2 x^2-140 d^2 e^4 x^4+6 d^5 e x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+56 e^6 x^6}{15 d^6 x \left(d^2-e^2 x^2\right)^{5/2}}","\frac{2 e (d+e x)}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{e (30 d+41 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^6}+\frac{e (10 d+13 e x)}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(-15*d^6 + 105*d^4*e^2*x^2 - 140*d^2*e^4*x^4 + 56*e^6*x^6 + 6*d^5*e*x*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2])/(15*d^6*x*(d^2 - e^2*x^2)^(5/2))","C",1
52,1,117,182,0.0564285,"\int \frac{(d+e x)^2}{x^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(x^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{e \left(-10 d^6+60 d^4 e^2 x^2-80 d^2 e^4 x^4+d^5 e x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+d^5 e x \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+32 e^6 x^6\right)}{5 d^7 x \left(d^2-e^2 x^2\right)^{5/2}}","\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^7}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 e^2 (d+e x)}{5 d^3 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(e*(-10*d^6 + 60*d^4*e^2*x^2 - 80*d^2*e^4*x^4 + 32*e^6*x^6 + d^5*e*x*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2] + d^5*e*x*Hypergeometric2F1[-5/2, 2, -3/2, 1 - (e^2*x^2)/d^2]))/(5*d^7*x*(d^2 - e^2*x^2)^(5/2))","C",1
53,1,105,209,0.0535345,"\int \frac{(d+e x)^2}{x^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{-5 d^8-55 d^6 e^2 x^2+330 d^4 e^4 x^4-440 d^2 e^6 x^6+6 d^5 e^3 x^3 \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+176 e^8 x^8}{15 d^8 x^3 \left(d^2-e^2 x^2\right)^{5/2}}","-\frac{14 e^2 \sqrt{d^2-e^2 x^2}}{3 d^8 x}+\frac{2 e^3 (45 d+53 e x)}{15 d^8 \sqrt{d^2-e^2 x^2}}-\frac{7 e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^8}-\frac{e \sqrt{d^2-e^2 x^2}}{d^7 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^6 x^3}+\frac{e^3 (20 d+23 e x)}{15 d^6 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 e^3 (d+e x)}{5 d^4 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(-5*d^8 - 55*d^6*e^2*x^2 + 330*d^4*e^4*x^4 - 440*d^2*e^6*x^6 + 176*e^8*x^8 + 6*d^5*e^3*x^3*Hypergeometric2F1[-5/2, 2, -3/2, 1 - (e^2*x^2)/d^2])/(15*d^8*x^3*(d^2 - e^2*x^2)^(5/2))","C",1
54,1,42,81,0.0359204,"\int \frac{x^3 (1+x)^2}{\sqrt{1-x^2}} \, dx","Integrate[(x^3*(1 + x)^2)/Sqrt[1 - x^2],x]","\frac{3}{4} \sin ^{-1}(x)-\frac{1}{20} \sqrt{1-x^2} \left(4 x^4+10 x^3+12 x^2+15 x+24\right)","-\frac{3}{5} \sqrt{1-x^2} x^2-\frac{3}{20} (5 x+8) \sqrt{1-x^2}-\frac{1}{5} \sqrt{1-x^2} x^4-\frac{1}{2} \sqrt{1-x^2} x^3+\frac{3}{4} \sin ^{-1}(x)",1,"-1/20*(Sqrt[1 - x^2]*(24 + 15*x + 12*x^2 + 10*x^3 + 4*x^4)) + (3*ArcSin[x])/4","A",1
55,1,37,63,0.0278391,"\int \frac{x^2 (1+x)^2}{\sqrt{1-x^2}} \, dx","Integrate[(x^2*(1 + x)^2)/Sqrt[1 - x^2],x]","\frac{7}{8} \sin ^{-1}(x)-\frac{1}{24} \sqrt{1-x^2} \left(6 x^3+16 x^2+21 x+32\right)","-\frac{2}{3} \sqrt{1-x^2} x^2-\frac{1}{24} (21 x+32) \sqrt{1-x^2}-\frac{1}{4} \sqrt{1-x^2} x^3+\frac{7}{8} \sin ^{-1}(x)",1,"-1/24*(Sqrt[1 - x^2]*(32 + 21*x + 16*x^2 + 6*x^3)) + (7*ArcSin[x])/8","A",1
56,1,26,41,0.0173176,"\int \frac{x (1+x)^2}{\sqrt{1-x^2}} \, dx","Integrate[(x*(1 + x)^2)/Sqrt[1 - x^2],x]","\sin ^{-1}(x)-\frac{1}{3} \sqrt{1-x^2} \left(x^2+3 x+5\right)","-\frac{1}{3} \sqrt{1-x^2} x^2-\frac{1}{3} (3 x+5) \sqrt{1-x^2}+\sin ^{-1}(x)",1,"-1/3*(Sqrt[1 - x^2]*(5 + 3*x + x^2)) + ArcSin[x]","A",1
57,1,25,40,0.0152267,"\int \frac{(1+x)^2}{\sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/Sqrt[1 - x^2],x]","\frac{1}{2} \left(3 \sin ^{-1}(x)-(x+4) \sqrt{1-x^2}\right)","-\frac{1}{2} \sqrt{1-x^2} (x+1)-\frac{3 \sqrt{1-x^2}}{2}+\frac{3}{2} \sin ^{-1}(x)",1,"(-((4 + x)*Sqrt[1 - x^2]) + 3*ArcSin[x])/2","A",1
58,1,32,32,0.0104568,"\int \frac{(1+x)^2}{x \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x*Sqrt[1 - x^2]),x]","-\sqrt{1-x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)+2 \sin ^{-1}(x)","-\sqrt{1-x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)+2 \sin ^{-1}(x)",1,"-Sqrt[1 - x^2] + 2*ArcSin[x] - ArcTanh[Sqrt[1 - x^2]]","A",1
59,1,33,33,0.0158001,"\int \frac{(1+x)^2}{x^2 \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x^2*Sqrt[1 - x^2]),x]","-\frac{\sqrt{1-x^2}}{x}-2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)+\sin ^{-1}(x)","-\frac{\sqrt{1-x^2}}{x}-2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)+\sin ^{-1}(x)",1,"-(Sqrt[1 - x^2]/x) + ArcSin[x] - 2*ArcTanh[Sqrt[1 - x^2]]","A",1
60,1,40,51,0.0199121,"\int \frac{(1+x)^2}{x^3 \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x^3*Sqrt[1 - x^2]),x]","-\frac{\sqrt{1-x^2} (4 x+1)}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left(\sqrt{1-x^2}\right)","-\frac{2 \sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2}}{2 x^2}-\frac{3}{2} \tanh ^{-1}\left(\sqrt{1-x^2}\right)",1,"-1/2*((1 + 4*x)*Sqrt[1 - x^2])/x^2 - (3*ArcTanh[Sqrt[1 - x^2]])/2","A",1
61,1,43,67,0.023974,"\int \frac{(1+x)^2}{x^4 \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x^4*Sqrt[1 - x^2]),x]","-\tanh ^{-1}\left(\sqrt{1-x^2}\right)-\frac{\sqrt{1-x^2} \left(5 x^2+3 x+1\right)}{3 x^3}","-\frac{5 \sqrt{1-x^2}}{3 x}-\frac{\sqrt{1-x^2}}{x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)-\frac{\sqrt{1-x^2}}{3 x^3}",1,"-1/3*(Sqrt[1 - x^2]*(1 + 3*x + 5*x^2))/x^3 - ArcTanh[Sqrt[1 - x^2]]","A",1
62,1,73,89,0.0382421,"\int \frac{(1+x)^2}{x^5 \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x^5*Sqrt[1 - x^2]),x]","-\sqrt{1-x^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-x^2\right)-\frac{1}{2} \tanh ^{-1}\left(\sqrt{1-x^2}\right)-\frac{\sqrt{1-x^2} \left(8 x^2+3 x+4\right)}{6 x^3}","-\frac{4 \sqrt{1-x^2}}{3 x}-\frac{7 \sqrt{1-x^2}}{8 x^2}-\frac{7}{8} \tanh ^{-1}\left(\sqrt{1-x^2}\right)-\frac{\sqrt{1-x^2}}{4 x^4}-\frac{2 \sqrt{1-x^2}}{3 x^3}",1,"-1/6*(Sqrt[1 - x^2]*(4 + 3*x + 8*x^2))/x^3 - ArcTanh[Sqrt[1 - x^2]]/2 - Sqrt[1 - x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - x^2]","C",1
63,1,50,107,0.017458,"\int \frac{(1+x)^2}{x^6 \sqrt{1-x^2}} \, dx","Integrate[(1 + x)^2/(x^6*Sqrt[1 - x^2]),x]","-\frac{\sqrt{1-x^2} \left(10 x^5 \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-x^2\right)+6 x^4+3 x^2+1\right)}{5 x^5}","-\frac{6 \sqrt{1-x^2}}{5 x}-\frac{3 \sqrt{1-x^2}}{4 x^2}-\frac{3}{4} \tanh ^{-1}\left(\sqrt{1-x^2}\right)-\frac{\sqrt{1-x^2}}{5 x^5}-\frac{\sqrt{1-x^2}}{2 x^4}-\frac{3 \sqrt{1-x^2}}{5 x^3}",1,"-1/5*(Sqrt[1 - x^2]*(1 + 3*x^2 + 6*x^4 + 10*x^5*Hypergeometric2F1[1/2, 3, 3/2, 1 - x^2]))/x^5","C",1
64,1,196,134,0.2433323,"\int \frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{x^5} \, dx","Integrate[((d + e*x)^3*Sqrt[d^2 - e^2*x^2])/x^5,x]","-\frac{e \sqrt{d^2-e^2 x^2} \left(6 d^2 e^3 x^3 \sin ^{-1}\left(\frac{e x}{d}\right)+2 e^3 x^3 \left(d^2-e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};1-\frac{e^2 x^2}{d^2}\right)-9 d^2 e^3 x^3 \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+6 d^5 \sqrt{1-\frac{e^2 x^2}{d^2}}+9 d^4 e x \sqrt{1-\frac{e^2 x^2}{d^2}}\right)}{6 d^3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left(d^2-e^2 x^2\right)^{3/2}}{4 x^4}-\frac{e \left(d^2-e^2 x^2\right)^{3/2}}{x^3}+e^4 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)+\frac{13}{8} e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-1/6*(e*Sqrt[d^2 - e^2*x^2]*(6*d^5*Sqrt[1 - (e^2*x^2)/d^2] + 9*d^4*e*x*Sqrt[1 - (e^2*x^2)/d^2] + 6*d^2*e^3*x^3*ArcSin[(e*x)/d] - 9*d^2*e^3*x^3*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]] + 2*e^3*x^3*(d^2 - e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[3/2, 3, 5/2, 1 - (e^2*x^2)/d^2]))/(d^3*x^3*Sqrt[1 - (e^2*x^2)/d^2])","C",1
65,1,212,310,0.3580881,"\int x^5 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[x^5*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(315315 d^{13} \sin ^{-1}\left(\frac{e x}{d}\right)-\sqrt{1-\frac{e^2 x^2}{d^2}} \left(507904 d^{13}+315315 d^{12} e x+253952 d^{11} e^2 x^2+210210 d^{10} e^3 x^3+190464 d^9 e^4 x^4+168168 d^8 e^5 x^5-2916352 d^7 e^6 x^6-7763184 d^6 e^7 x^7-2551808 d^5 e^8 x^8+9499776 d^4 e^9 x^9+8773632 d^3 e^{10} x^{10}-1427712 d^2 e^{11} x^{11}-4257792 d e^{12} x^{12}-1317888 e^{13} x^{13}\right)\right)}{18450432 e^6 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{1}{14} e x^7 \left(d^2-e^2 x^2\right)^{7/2}-\frac{3}{13} d x^6 \left(d^2-e^2 x^2\right)^{7/2}-\frac{7 d^2 x^5 \left(d^2-e^2 x^2\right)^{7/2}}{24 e}+\frac{35 d^{14} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2048 e^6}+\frac{35 d^{12} x \sqrt{d^2-e^2 x^2}}{2048 e^5}+\frac{35 d^{10} x \left(d^2-e^2 x^2\right)^{3/2}}{3072 e^5}+\frac{7 d^8 x \left(d^2-e^2 x^2\right)^{5/2}}{768 e^5}-\frac{d^6 (31744 d+63063 e x) \left(d^2-e^2 x^2\right)^{7/2}}{1153152 e^6}-\frac{124 d^5 x^2 \left(d^2-e^2 x^2\right)^{7/2}}{1287 e^4}-\frac{7 d^4 x^3 \left(d^2-e^2 x^2\right)^{7/2}}{48 e^3}-\frac{31 d^3 x^4 \left(d^2-e^2 x^2\right)^{7/2}}{143 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(507904*d^13 + 315315*d^12*e*x + 253952*d^11*e^2*x^2 + 210210*d^10*e^3*x^3 + 190464*d^9*e^4*x^4 + 168168*d^8*e^5*x^5 - 2916352*d^7*e^6*x^6 - 7763184*d^6*e^7*x^7 - 2551808*d^5*e^8*x^8 + 9499776*d^4*e^9*x^9 + 8773632*d^3*e^10*x^10 - 1427712*d^2*e^11*x^11 - 4257792*d*e^12*x^12 - 1317888*e^13*x^13)) + 315315*d^13*ArcSin[(e*x)/d]))/(18450432*e^6*Sqrt[1 - (e^2*x^2)/d^2])","A",1
66,1,200,281,0.3298664,"\int x^4 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(135135 d^{12} \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-204800 d^{12}-135135 d^{11} e x-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right)\right)}{5125120 e^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{1}{13} e x^6 \left(d^2-e^2 x^2\right)^{7/2}-\frac{1}{4} d x^5 \left(d^2-e^2 x^2\right)^{7/2}-\frac{45 d^2 x^4 \left(d^2-e^2 x^2\right)^{7/2}}{143 e}+\frac{27 d^{13} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{1024 e^5}+\frac{27 d^{11} x \sqrt{d^2-e^2 x^2}}{1024 e^4}+\frac{9 d^9 x \left(d^2-e^2 x^2\right)^{3/2}}{512 e^4}+\frac{9 d^7 x \left(d^2-e^2 x^2\right)^{5/2}}{640 e^4}-\frac{d^5 (12800 d+27027 e x) \left(d^2-e^2 x^2\right)^{7/2}}{320320 e^5}-\frac{20 d^4 x^2 \left(d^2-e^2 x^2\right)^{7/2}}{143 e^3}-\frac{9 d^3 x^3 \left(d^2-e^2 x^2\right)^{7/2}}{40 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-204800*d^12 - 135135*d^11*e*x - 102400*d^10*e^2*x^2 - 90090*d^9*e^3*x^3 - 76800*d^8*e^4*x^4 + 952952*d^7*e^5*x^5 + 2498560*d^6*e^6*x^6 + 816816*d^5*e^7*x^7 - 2938880*d^4*e^8*x^8 - 2690688*d^3*e^9*x^9 + 430080*d^2*e^10*x^10 + 1281280*d*e^11*x^11 + 394240*e^12*x^12) + 135135*d^12*ArcSin[(e*x)/d]))/(5125120*e^5*Sqrt[1 - (e^2*x^2)/d^2])","A",1
67,1,189,252,0.2965026,"\int x^3 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[x^3*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(142065 d^{11} \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-235520 d^{11}-142065 d^{10} e x-117760 d^9 e^2 x^2-94710 d^8 e^3 x^3+798720 d^7 e^4 x^4+2053128 d^6 e^5 x^5+665600 d^5 e^6 x^6-2295216 d^4 e^7 x^7-2078720 d^3 e^8 x^8+325248 d^2 e^9 x^9+967680 d e^{10} x^{10}+295680 e^{11} x^{11}\right)\right)}{3548160 e^4 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{1}{12} e x^5 \left(d^2-e^2 x^2\right)^{7/2}-\frac{3}{11} d x^4 \left(d^2-e^2 x^2\right)^{7/2}-\frac{41 d^2 x^3 \left(d^2-e^2 x^2\right)^{7/2}}{120 e}+\frac{41 d^{12} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{1024 e^4}+\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left(d^2-e^2 x^2\right)^{3/2}}{1536 e^3}+\frac{41 d^6 x \left(d^2-e^2 x^2\right)^{5/2}}{1920 e^3}-\frac{d^4 (14720 d+28413 e x) \left(d^2-e^2 x^2\right)^{7/2}}{221760 e^4}-\frac{23 d^3 x^2 \left(d^2-e^2 x^2\right)^{7/2}}{99 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-235520*d^11 - 142065*d^10*e*x - 117760*d^9*e^2*x^2 - 94710*d^8*e^3*x^3 + 798720*d^7*e^4*x^4 + 2053128*d^6*e^5*x^5 + 665600*d^5*e^6*x^6 - 2295216*d^4*e^7*x^7 - 2078720*d^3*e^8*x^8 + 325248*d^2*e^9*x^9 + 967680*d*e^10*x^10 + 295680*e^11*x^11) + 142065*d^11*ArcSin[(e*x)/d]))/(3548160*e^4*Sqrt[1 - (e^2*x^2)/d^2])","A",1
68,1,178,223,0.2673583,"\int x^2 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[x^2*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(65835 d^{10} \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-94720 d^{10}-65835 d^9 e x-47360 d^8 e^2 x^2+251790 d^7 e^3 x^3+629760 d^6 e^4 x^4+201432 d^5 e^5 x^5-657920 d^4 e^6 x^6-587664 d^3 e^7 x^7+89600 d^2 e^8 x^8+266112 d e^9 x^9+80640 e^{10} x^{10}\right)\right)}{887040 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{37 d^2 x^2 \left(d^2-e^2 x^2\right)^{7/2}}{99 e}-\frac{1}{11} e x^4 \left(d^2-e^2 x^2\right)^{7/2}-\frac{3}{10} d x^3 \left(d^2-e^2 x^2\right)^{7/2}+\frac{19 d^{11} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{256 e^3}+\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left(d^2-e^2 x^2\right)^{3/2}}{384 e^2}+\frac{19 d^5 x \left(d^2-e^2 x^2\right)^{5/2}}{480 e^2}-\frac{d^3 (5920 d+13167 e x) \left(d^2-e^2 x^2\right)^{7/2}}{55440 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-94720*d^10 - 65835*d^9*e*x - 47360*d^8*e^2*x^2 + 251790*d^7*e^3*x^3 + 629760*d^6*e^4*x^4 + 201432*d^5*e^5*x^5 - 657920*d^4*e^6*x^6 - 587664*d^3*e^7*x^7 + 89600*d^2*e^8*x^8 + 266112*d*e^9*x^9 + 80640*e^10*x^10) + 65835*d^10*ArcSin[(e*x)/d]))/(887040*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
69,1,167,230,0.3789995,"\int x (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[x*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(3465 d^9 \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-6400 d^9-3465 d^8 e x+10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7+8960 d e^8 x^8+2688 e^9 x^9\right)\right)}{26880 e^2 \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{11 d^2 (d+e x) \left(d^2-e^2 x^2\right)^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left(d^2-e^2 x^2\right)^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{7/2}}{10 e^2}+\frac{33 d^{10} \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{256 e^2}+\frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left(d^2-e^2 x^2\right)^{3/2}}{128 e}+\frac{11 d^4 x \left(d^2-e^2 x^2\right)^{5/2}}{160 e}-\frac{33 d^3 \left(d^2-e^2 x^2\right)^{7/2}}{560 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-6400*d^9 - 3465*d^8*e*x + 10240*d^7*e^2*x^2 + 24570*d^6*e^3*x^3 + 7680*d^5*e^4*x^4 - 23352*d^4*e^5*x^5 - 20480*d^3*e^6*x^6 + 3024*d^2*e^7*x^7 + 8960*d*e^8*x^8 + 2688*e^9*x^9) + 3465*d^9*ArcSin[(e*x)/d]))/(26880*e^2*Sqrt[1 - (e^2*x^2)/d^2])","A",1
70,1,156,188,0.3264071,"\int (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(3465 d^8 \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(-3712 d^8+4599 d^7 e x+10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+3024 d e^7 x^7+896 e^8 x^8\right)\right)}{8064 e \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{11 d^2 \left(d^2-e^2 x^2\right)^{7/2}}{56 e}-\frac{11 d (d+e x) \left(d^2-e^2 x^2\right)^{7/2}}{72 e}-\frac{(d+e x)^2 \left(d^2-e^2 x^2\right)^{7/2}}{9 e}+\frac{55 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e}+\frac{55}{128} d^7 x \sqrt{d^2-e^2 x^2}+\frac{55}{192} d^5 x \left(d^2-e^2 x^2\right)^{3/2}+\frac{11}{48} d^3 x \left(d^2-e^2 x^2\right)^{5/2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-3712*d^8 + 4599*d^7*e*x + 10240*d^6*e^2*x^2 + 3066*d^5*e^3*x^3 - 8448*d^4*e^4*x^4 - 7224*d^3*e^5*x^5 + 1024*d^2*e^6*x^6 + 3024*d*e^7*x^7 + 896*e^8*x^8) + 3465*d^8*ArcSin[(e*x)/d]))/(8064*e*Sqrt[1 - (e^2*x^2)/d^2])","A",1
71,1,168,190,0.3552087,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x,x]","d^8 \left(-\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)\right)+\frac{125 d^7 \sqrt{d^2-e^2 x^2} \sin ^{-1}\left(\frac{e x}{d}\right)}{128 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{\sqrt{d^2-e^2 x^2} \left(14848 d^7+27195 d^6 e x+7424 d^5 e^2 x^2-17710 d^4 e^3 x^3-14592 d^3 e^4 x^4+1960 d^2 e^5 x^5+5760 d e^6 x^6+1680 e^7 x^7\right)}{13440}","\frac{1}{240} d^2 (48 d+125 e x) \left(d^2-e^2 x^2\right)^{5/2}-\frac{3}{7} d \left(d^2-e^2 x^2\right)^{7/2}-\frac{1}{8} e x \left(d^2-e^2 x^2\right)^{7/2}+\frac{125}{128} d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d^8 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)+\frac{1}{128} d^6 (128 d+125 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{192} d^4 (64 d+125 e x) \left(d^2-e^2 x^2\right)^{3/2}",1,"(Sqrt[d^2 - e^2*x^2]*(14848*d^7 + 27195*d^6*e*x + 7424*d^5*e^2*x^2 - 17710*d^4*e^3*x^3 - 14592*d^3*e^4*x^4 + 1960*d^2*e^5*x^5 + 5760*d*e^6*x^6 + 1680*e^7*x^7))/13440 + (125*d^7*Sqrt[d^2 - e^2*x^2]*ArcSin[(e*x)/d])/(128*Sqrt[1 - (e^2*x^2)/d^2]) - d^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]","A",1
72,1,221,193,0.5252404,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^2} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^2,x]","-\frac{d^7 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x \sqrt{1-\frac{e^2 x^2}{d^2}}}-3 d^7 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)+\frac{15 d^6 e \sqrt{d^2-e^2 x^2} \sin ^{-1}\left(\frac{e x}{d}\right)}{16 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{1}{560} e \sqrt{d^2-e^2 x^2} \left(2496 d^6+1155 d^5 e x-992 d^4 e^2 x^2-910 d^3 e^3 x^3+96 d^2 e^4 x^4+280 d e^5 x^5+80 e^6 x^6\right)","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{x}+\frac{1}{10} d e (6 d-5 e x) \left(d^2-e^2 x^2\right)^{5/2}-\frac{1}{7} e \left(d^2-e^2 x^2\right)^{7/2}-\frac{15}{16} d^7 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-3 d^7 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)+\frac{3}{16} d^5 e (16 d-5 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{8} d^3 e (8 d-5 e x) \left(d^2-e^2 x^2\right)^{3/2}",1,"(e*Sqrt[d^2 - e^2*x^2]*(2496*d^6 + 1155*d^5*e*x - 992*d^4*e^2*x^2 - 910*d^3*e^3*x^3 + 96*d^2*e^4*x^4 + 280*d*e^5*x^5 + 80*e^6*x^6))/560 + (15*d^6*e*Sqrt[d^2 - e^2*x^2]*ArcSin[(e*x)/d])/(16*Sqrt[1 - (e^2*x^2)/d^2]) - 3*d^7*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d] - (d^7*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, (e^2*x^2)/d^2])/(x*Sqrt[1 - (e^2*x^2)/d^2])","C",1
73,1,259,207,0.6401226,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^3} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^3,x]","-\frac{e \left(5040 d^9 \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)+e x \left(240 \left(d^2-e^2 x^2\right)^4 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)-7 d \left(1104 d^7+165 d^6 e x-1632 d^5 e^2 x^2-295 d^4 e^3 x^3+672 d^3 e^4 x^4+170 d^2 e^5 x^5+75 d^7 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left(\frac{e x}{d}\right)-720 d^6 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-144 d e^6 x^6-40 e^7 x^7\right)\right)\right)}{1680 d x \sqrt{d^2-e^2 x^2}}","\frac{1}{24} d^2 e^2 (4 d-85 e x) \left(d^2-e^2 x^2\right)^{3/2}-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{2 x^2}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{x}+\frac{1}{30} e^2 (3 d-85 e x) \left(d^2-e^2 x^2\right)^{5/2}-\frac{85}{16} d^6 e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{1}{2} d^6 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)+\frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}",1,"-1/1680*(e*(5040*d^9*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, (e^2*x^2)/d^2] + e*x*(-7*d*(1104*d^7 + 165*d^6*e*x - 1632*d^5*e^2*x^2 - 295*d^4*e^3*x^3 + 672*d^3*e^4*x^4 + 170*d^2*e^5*x^5 - 144*d*e^6*x^6 - 40*e^7*x^7 + 75*d^7*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d] - 720*d^6*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]) + 240*(d^2 - e^2*x^2)^4*Hypergeometric2F1[2, 7/2, 9/2, 1 - (e^2*x^2)/d^2])))/(d*x*Sqrt[d^2 - e^2*x^2])","C",1
74,1,251,210,0.2722985,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^4} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^4,x]","-\frac{3 e^3 \left(d^2-e^2 x^2\right)^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)}{7 d^2}-\frac{d^7 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 d^5 e^2 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{1}{15} e^3 \left(\sqrt{d^2-e^2 x^2} \left(23 d^4-11 d^2 e^2 x^2+3 e^4 x^4\right)-15 d^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)\right)","-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{2 x^2}-\frac{e^2 (50 d+39 e x) \left(d^2-e^2 x^2\right)^{5/2}}{30 x}-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{3 x^3}-\frac{1}{12} d e^3 (26 d+25 e x) \left(d^2-e^2 x^2\right)^{3/2}-\frac{25}{8} d^5 e^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{13}{2} d^5 e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}",1,"(e^3*(Sqrt[d^2 - e^2*x^2]*(23*d^4 - 11*d^2*e^2*x^2 + 3*e^4*x^4) - 15*d^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]))/15 - (d^7*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -3/2, -1/2, (e^2*x^2)/d^2])/(3*x^3*Sqrt[1 - (e^2*x^2)/d^2]) - (3*d^5*e^2*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -1/2, 1/2, (e^2*x^2)/d^2])/(x*Sqrt[1 - (e^2*x^2)/d^2]) - (3*e^3*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^2)","C",1
75,1,195,209,0.099812,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^5} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^5,x]","\frac{e \sqrt{d^2-e^2 x^2} \left(3 \left(e^3 x^2-d^2 e\right)^3 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)+\left(e^3 x^2-d^2 e\right)^3 \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)-\frac{7 d^9 \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{7 d^7 e^2 \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x \sqrt{1-\frac{e^2 x^2}{d^2}}}\right)}{7 d^3}","-\frac{3 e^2 (3 d+2 e x) \left(d^2-e^2 x^2\right)^{5/2}}{8 x^2}-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{4 x^4}-\frac{e \left(d^2-e^2 x^2\right)^{7/2}}{x^3}-\frac{45}{8} d^2 e^4 (d-e x) \sqrt{d^2-e^2 x^2}+\frac{15 d e^3 (2 d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{8 x}+\frac{45}{8} d^4 e^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{45}{8} d^4 e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(e*Sqrt[d^2 - e^2*x^2]*((-7*d^9*Hypergeometric2F1[-5/2, -3/2, -1/2, (e^2*x^2)/d^2])/(x^3*Sqrt[1 - (e^2*x^2)/d^2]) - (7*d^7*e^2*Hypergeometric2F1[-5/2, -1/2, 1/2, (e^2*x^2)/d^2])/(x*Sqrt[1 - (e^2*x^2)/d^2]) + 3*(-(d^2*e) + e^3*x^2)^3*Hypergeometric2F1[2, 7/2, 9/2, 1 - (e^2*x^2)/d^2] + (-(d^2*e) + e^3*x^2)^3*Hypergeometric2F1[3, 7/2, 9/2, 1 - (e^2*x^2)/d^2]))/(7*d^3)","C",1
76,1,199,216,0.0935767,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^6} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^6,x]","\frac{\sqrt{d^2-e^2 x^2} \left(5 e^5 \left(e^2 x^2-d^2\right)^3 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)+15 e^5 \left(e^2 x^2-d^2\right)^3 \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)-\frac{7 d^{11} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{35 d^9 e^2 \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}\right)}{35 d^4}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{5 x^5}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{4 x^4}-\frac{e^2 (52 d+25 e x) \left(d^2-e^2 x^2\right)^{5/2}}{60 x^3}+\frac{d^2 e^4 (52 d+25 e x) \sqrt{d^2-e^2 x^2}}{8 x}+\frac{d e^3 (25 d-52 e x) \left(d^2-e^2 x^2\right)^{3/2}}{24 x^2}+\frac{13}{2} d^3 e^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{25}{8} d^3 e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(Sqrt[d^2 - e^2*x^2]*((-7*d^11*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (35*d^9*e^2*Hypergeometric2F1[-5/2, -3/2, -1/2, (e^2*x^2)/d^2])/(x^3*Sqrt[1 - (e^2*x^2)/d^2]) + 5*e^5*(-d^2 + e^2*x^2)^3*Hypergeometric2F1[2, 7/2, 9/2, 1 - (e^2*x^2)/d^2] + 15*e^5*(-d^2 + e^2*x^2)^3*Hypergeometric2F1[3, 7/2, 9/2, 1 - (e^2*x^2)/d^2]))/(35*d^4)","C",1
77,1,286,214,0.2225464,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^7} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^7,x]","-\frac{3 d^6 e \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 e^6 \left(d^2-e^2 x^2\right)^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)}{7 d^5}-\frac{d^4 e^3 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{-8 d^9+34 d^7 e^2 x^2-59 d^5 e^4 x^4+33 d^3 e^6 x^6+15 d^3 e^6 x^6 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)}{48 x^6 \sqrt{d^2-e^2 x^2}}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{6 x^6}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{5 x^5}-\frac{e^2 (85 d+12 e x) \left(d^2-e^2 x^2\right)^{5/2}}{120 x^4}-\frac{1}{2} d^2 e^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{85}{16} d^2 e^6 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left(d^2-e^2 x^2\right)^{3/2}}{48 x^3}",1,"(-8*d^9 + 34*d^7*e^2*x^2 - 59*d^5*e^4*x^4 + 33*d^3*e^6*x^6 + 15*d^3*e^6*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(48*x^6*Sqrt[d^2 - e^2*x^2]) - (3*d^6*e*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (d^4*e^3*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -3/2, -1/2, (e^2*x^2)/d^2])/(3*x^3*Sqrt[1 - (e^2*x^2)/d^2]) - (3*e^6*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^5)","C",1
78,1,247,206,0.1519579,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^8} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{7 x^7}-\frac{e^7 \left(d^2-e^2 x^2\right)^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)}{7 d^6}-\frac{3 d^5 e^2 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{-8 d^8 e+34 d^6 e^3 x^2-59 d^4 e^5 x^4+33 d^2 e^7 x^6+15 d^2 e^7 x^6 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)}{16 x^6 \sqrt{d^2-e^2 x^2}}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{7 x^7}-\frac{e \left(d^2-e^2 x^2\right)^{7/2}}{2 x^6}-\frac{e^2 (24 d+5 e x) \left(d^2-e^2 x^2\right)^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{15}{16} d e^7 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-\frac{3 e^6 (16 d-5 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{e^4 (16 d+5 e x) \left(d^2-e^2 x^2\right)^{3/2}}{16 x^3}",1,"-1/7*(d*(d^2 - e^2*x^2)^(7/2))/x^7 + (-8*d^8*e + 34*d^6*e^3*x^2 - 59*d^4*e^5*x^4 + 33*d^2*e^7*x^6 + 15*d^2*e^7*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(16*x^6*Sqrt[d^2 - e^2*x^2]) - (3*d^5*e^2*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (e^7*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^6)","C",1
79,1,245,204,0.149187,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^9} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]","-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{7 x^7}-\frac{e^8 \left(d^2-e^2 x^2\right)^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)}{7 d^7}-\frac{d^4 e^3 \sqrt{d^2-e^2 x^2} \, _2F_1\left(-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{-8 d^7 e^2+34 d^5 e^4 x^2-59 d^3 e^6 x^4+15 d e^8 x^6 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+33 d e^8 x^6}{16 x^6 \sqrt{d^2-e^2 x^2}}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{8 x^8}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{7 x^7}-\frac{e^2 (125 d+48 e x) \left(d^2-e^2 x^2\right)^{5/2}}{240 x^6}+e^8 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)+\frac{125}{128} e^8 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left(d^2-e^2 x^2\right)^{3/2}}{192 x^4}",1,"(-3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^7) + (-8*d^7*e^2 + 34*d^5*e^4*x^2 - 59*d^3*e^6*x^4 + 33*d*e^8*x^6 + 15*d*e^8*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(16*x^6*Sqrt[d^2 - e^2*x^2]) - (d^4*e^3*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (e^8*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^7)","C",1
80,1,218,187,0.1654304,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^{10}} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^10,x]","\frac{-112 d^{10}-16 d^8 e^2 x^2-168 d^7 e^3 x^3+1184 d^6 e^4 x^4+714 d^5 e^5 x^5-2336 d^4 e^6 x^6-1239 d^3 e^7 x^7+1744 d^2 e^8 x^8+315 d e^9 x^9 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)+693 d e^9 x^9-464 e^{10} x^{10}}{1008 d x^9 \sqrt{d^2-e^2 x^2}}-\frac{3 e^9 \left(d^2-e^2 x^2\right)^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)}{7 d^8}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{9 x^9}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{8 x^8}-\frac{29 e^2 \left(d^2-e^2 x^2\right)^{7/2}}{63 d x^7}+\frac{55 e^9 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{128 d}-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left(d^2-e^2 x^2\right)^{3/2}}{192 x^4}-\frac{11 e^3 \left(d^2-e^2 x^2\right)^{5/2}}{48 x^6}",1,"(-112*d^10 - 16*d^8*e^2*x^2 - 168*d^7*e^3*x^3 + 1184*d^6*e^4*x^4 + 714*d^5*e^5*x^5 - 2336*d^4*e^6*x^6 - 1239*d^3*e^7*x^7 + 1744*d^2*e^8*x^8 + 693*d*e^9*x^9 - 464*e^10*x^10 + 315*d*e^9*x^9*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(1008*d*x^9*Sqrt[d^2 - e^2*x^2]) - (3*e^9*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^8)","C",1
81,1,102,225,0.0644799,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^{11}} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]","-\frac{e \left(d^2-e^2 x^2\right)^{7/2} \left(7 d^9+5 d^7 e^2 x^2+9 e^9 x^9 \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)+3 e^9 x^9 \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)\right)}{21 d^9 x^9}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{10 x^{10}}-\frac{e \left(d^2-e^2 x^2\right)^{7/2}}{3 x^9}-\frac{33 e^2 \left(d^2-e^2 x^2\right)^{7/2}}{80 d x^8}+\frac{33 e^{10} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{256 d^2}-\frac{33 e^8 \sqrt{d^2-e^2 x^2}}{256 d x^2}+\frac{11 e^6 \left(d^2-e^2 x^2\right)^{3/2}}{128 d x^4}-\frac{11 e^4 \left(d^2-e^2 x^2\right)^{5/2}}{160 d x^6}-\frac{5 e^3 \left(d^2-e^2 x^2\right)^{7/2}}{21 d^2 x^7}",1,"-1/21*(e*(d^2 - e^2*x^2)^(7/2)*(7*d^9 + 5*d^7*e^2*x^2 + 9*e^9*x^9*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2] + 3*e^9*x^9*Hypergeometric2F1[7/2, 6, 9/2, 1 - (e^2*x^2)/d^2]))/(d^9*x^9)","C",1
82,1,112,254,0.0610064,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}{x^{12}} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]","-\frac{\left(d^2-e^2 x^2\right)^{7/2} \left(63 d^{11}+259 d^9 e^2 x^2+74 d^7 e^4 x^4+99 e^{11} x^{11} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)+297 e^{11} x^{11} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right)\right)}{693 d^{10} x^{11}}","-\frac{d \left(d^2-e^2 x^2\right)^{7/2}}{11 x^{11}}-\frac{3 e \left(d^2-e^2 x^2\right)^{7/2}}{10 x^{10}}-\frac{37 e^2 \left(d^2-e^2 x^2\right)^{7/2}}{99 d x^9}-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left(d^2-e^2 x^2\right)^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left(d^2-e^2 x^2\right)^{5/2}}{480 d^2 x^6}-\frac{19 e^3 \left(d^2-e^2 x^2\right)^{7/2}}{80 d^2 x^8}+\frac{19 e^{11} \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{256 d^3}-\frac{74 e^4 \left(d^2-e^2 x^2\right)^{7/2}}{693 d^3 x^7}",1,"-1/693*((d^2 - e^2*x^2)^(7/2)*(63*d^11 + 259*d^9*e^2*x^2 + 74*d^7*e^4*x^4 + 99*e^11*x^11*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2] + 297*e^11*x^11*Hypergeometric2F1[7/2, 6, 9/2, 1 - (e^2*x^2)/d^2]))/(d^10*x^11)","C",1
83,1,131,174,0.2367906,"\int \frac{x^5 (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(304 d^4-717 d^3 e x+479 d^2 e^2 x^2-45 d e^3 x^3-15 e^4 x^4\right)-195 d (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{30 e^6 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}-\frac{13 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d+e x)^3}{5 e^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(304*d^4 - 717*d^3*e*x + 479*d^2*e^2*x^2 - 45*d*e^3*x^3 - 15*e^4*x^4) - 195*d*(d - e*x)^3*ArcSin[(e*x)/d]))/(30*e^6*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
84,1,119,142,0.1986129,"\int \frac{x^4 (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^4*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} \left(24 d^3-57 d^2 e x+39 d e^2 x^2-5 e^3 x^3\right)-15 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{5 e^5 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{6 d^2 (d+e x)^2}{5 e^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{24 d (d+e x)}{5 e^5 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^5}-\frac{3 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}+\frac{d^3 (d+e x)^3}{5 e^5 \left(d^2-e^2 x^2\right)^{5/2}}",1,"((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(24*d^3 - 57*d^2*e*x + 39*d*e^2*x^2 - 5*e^3*x^3) - 15*(d - e*x)^3*ArcSin[(e*x)/d]))/(5*e^5*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
85,1,112,118,0.1509344,"\int \frac{x^3 (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^3*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(d \left(22 d^2-51 d e x+32 e^2 x^2\right) \sqrt{1-\frac{e^2 x^2}{d^2}}-15 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{15 d e^4 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^2 (d+e x)^3}{5 e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{13 d (d+e x)^2}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{32 (d+e x)}{15 e^4 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^4}",1,"((d + e*x)*(d*(22*d^2 - 51*d*e*x + 32*e^2*x^2)*Sqrt[1 - (e^2*x^2)/d^2] - 15*(d - e*x)^3*ArcSin[(e*x)/d]))/(15*d*e^4*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
86,1,58,93,0.0787129,"\int \frac{x^2 (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x^2*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(2 d^2-6 d e x+7 e^2 x^2\right)}{15 d e^3 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{d (d+e x)^3}{5 e^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8 (d+e x)^2}{15 e^3 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{7 (d+e x)}{15 d e^3 \sqrt{d^2-e^2 x^2}}",1,"((d + e*x)*(2*d^2 - 6*d*e*x + 7*e^2*x^2))/(15*d*e^3*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
87,1,55,86,0.1755425,"\int \frac{x (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(x*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","-\frac{(d+e x) \left(d^2-3 d e x+e^2 x^2\right)}{5 d^2 e^2 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^3}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 (d+e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{x}{5 d^2 e \sqrt{d^2-e^2 x^2}}",1,"-1/5*((d + e*x)*(d^2 - 3*d*e*x + e^2*x^2))/(d^2*e^2*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
88,1,58,103,0.0590661,"\int \frac{(d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(7 d^2-6 d e x+2 e^2 x^2\right)}{15 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}",1,"((d + e*x)*(7*d^2 - 6*d*e*x + 2*e^2*x^2))/(15*d^3*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
89,1,81,114,0.0586554,"\int \frac{(d+e x)^3}{x \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(x*(d^2 - e^2*x^2)^(7/2)),x]","\frac{9 d^5+45 d^4 e x-55 d^2 e^3 x^3+3 d^5 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+22 e^5 x^5}{15 d^4 \left(d^2-e^2 x^2\right)^{5/2}}","\frac{4 (d+e x)}{5 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{5 d+11 e x}{15 d^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{15 d+22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^4}",1,"(9*d^5 + 45*d^4*e*x - 55*d^2*e^3*x^3 + 22*e^5*x^5 + 3*d^5*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2])/(15*d^4*(d^2 - e^2*x^2)^(5/2))","C",1
90,1,96,145,0.0559625,"\int \frac{(d+e x)^3}{x^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(x^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{-5 d^6+d^5 e x+45 d^4 e^2 x^2-60 d^2 e^4 x^4+3 d^5 e x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+24 e^6 x^6}{5 d^5 x \left(d^2-e^2 x^2\right)^{5/2}}","\frac{4 e (d+e x)}{5 d \left(d^2-e^2 x^2\right)^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{e (15 d+19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{3 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^5}+\frac{e (5 d+7 e x)}{5 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(-5*d^6 + d^5*e*x + 45*d^4*e^2*x^2 - 60*d^2*e^4*x^4 + 24*e^6*x^6 + 3*d^5*e*x*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2])/(5*d^5*x*(d^2 - e^2*x^2)^(5/2))","C",1
91,1,119,182,0.0692958,"\int \frac{(d+e x)^3}{x^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{e \left(-45 d^6+285 d^4 e^2 x^2-380 d^2 e^4 x^4+9 d^5 e x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+3 d^5 e x \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right)+152 e^6 x^6\right)}{15 d^6 x \left(d^2-e^2 x^2\right)^{5/2}}","\frac{4 e^2 (d+e x)}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(e*(-45*d^6 + 285*d^4*e^2*x^2 - 380*d^2*e^4*x^4 + 152*e^6*x^6 + 9*d^5*e*x*Hypergeometric2F1[-5/2, 1, -3/2, 1 - (e^2*x^2)/d^2] + 3*d^5*e*x*Hypergeometric2F1[-5/2, 2, -3/2, 1 - (e^2*x^2)/d^2]))/(15*d^6*x*(d^2 - e^2*x^2)^(5/2))","C",1
92,1,91,147,0.1395192,"\int \frac{x^4 \sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[(x^4*Sqrt[d^2 - e^2*x^2])/(d + e*x),x]","\frac{45 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(64 d^4-45 d^3 e x+32 d^2 e^2 x^2-30 d e^3 x^3+24 e^4 x^4\right)}{120 e^5}","\frac{x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{4 e^2}+\frac{4 d^2 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}+\frac{3 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^5}+\frac{d^3 (64 d-45 e x) \sqrt{d^2-e^2 x^2}}{120 e^5}",1,"(Sqrt[d^2 - e^2*x^2]*(64*d^4 - 45*d^3*e*x + 32*d^2*e^2*x^2 - 30*d*e^3*x^3 + 24*e^4*x^4) + 45*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e^5)","A",1
93,1,80,118,0.1014461,"\int \frac{x^3 \sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[(x^3*Sqrt[d^2 - e^2*x^2])/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-16 d^3+9 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right)-9 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{24 e^4}","-\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{3 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^4}",1,"(Sqrt[d^2 - e^2*x^2]*(-16*d^3 + 9*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3) - 9*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e^4)","A",1
94,1,69,86,0.0727646,"\int \frac{x^2 \sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[(x^2*Sqrt[d^2 - e^2*x^2])/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(4 d^2-3 d e x+2 e^2 x^2\right)+3 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{6 e^3}","\frac{d (2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(4*d^2 - 3*d*e*x + 2*e^2*x^2) + 3*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^3)","A",1
95,1,57,62,0.0587462,"\int \frac{x \sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[(x*Sqrt[d^2 - e^2*x^2])/(d + e*x),x]","\frac{(e x-2 d) \sqrt{d^2-e^2 x^2}-d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^2}","-\frac{(2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^2}",1,"((-2*d + e*x)*Sqrt[d^2 - e^2*x^2] - d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^2)","A",1
96,1,43,46,0.0234332,"\int \frac{\sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2}+d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}","\frac{\sqrt{d^2-e^2 x^2}}{e}+\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e}",1,"(Sqrt[d^2 - e^2*x^2] + d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e","A",1
97,1,46,46,0.0381675,"\int \frac{\sqrt{d^2-e^2 x^2}}{x (d+e x)} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)),x]","-\log \left(\sqrt{d^2-e^2 x^2}+d\right)-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\log (x)","-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + Log[x] - Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
98,1,53,51,0.0641652,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^2 (d+e x)} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^2*(d + e*x)),x]","-\frac{\sqrt{d^2-e^2 x^2}-e x \log \left(\sqrt{d^2-e^2 x^2}+d\right)+e x \log (x)}{d x}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d}-\frac{\sqrt{d^2-e^2 x^2}}{d x}",1,"-((Sqrt[d^2 - e^2*x^2] + e*x*Log[x] - e*x*Log[d + Sqrt[d^2 - e^2*x^2]])/(d*x))","A",1
99,1,70,82,0.1040564,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^3 (d+e x)} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^3*(d + e*x)),x]","-\frac{(d-2 e x) \sqrt{d^2-e^2 x^2}+e^2 x^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)-e^2 x^2 \log (x)}{2 d^2 x^2}","\frac{e \sqrt{d^2-e^2 x^2}}{d^2 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d x^2}-\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^2}",1,"-1/2*((d - 2*e*x)*Sqrt[d^2 - e^2*x^2] - e^2*x^2*Log[x] + e^2*x^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(d^2*x^2)","A",1
100,1,84,114,0.1082716,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^4 (d+e x)} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)),x]","\frac{\left(-2 d^2+3 d e x-4 e^2 x^2\right) \sqrt{d^2-e^2 x^2}+3 e^3 x^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)-3 e^3 x^3 \log (x)}{6 d^3 x^3}","\frac{e \sqrt{d^2-e^2 x^2}}{2 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{2 e^2 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^3}",1,"((-2*d^2 + 3*d*e*x - 4*e^2*x^2)*Sqrt[d^2 - e^2*x^2] - 3*e^3*x^3*Log[x] + 3*e^3*x^3*Log[d + Sqrt[d^2 - e^2*x^2]])/(6*d^3*x^3)","A",1
101,1,95,143,0.1273255,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^5 (d+e x)} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^5*(d + e*x)),x]","\frac{-9 e^4 x^4 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right)+9 e^4 x^4 \log (x)}{24 d^4 x^4}","-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d^4}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-6*d^3 + 8*d^2*e*x - 9*d*e^2*x^2 + 16*e^3*x^3) + 9*e^4*x^4*Log[x] - 9*e^4*x^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(24*d^4*x^4)","A",1
102,1,112,113,0.1184933,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^(3/2))/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 d^4 \sin ^{-1}\left(\frac{e x}{d}\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(16 d^4-15 d^3 e x+8 d^2 e^2 x^2+30 d e^3 x^3-24 e^4 x^4\right)\right)}{120 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d (4 d-3 e x) \left(d^2-e^2 x^2\right)^{3/2}}{12 e^3}-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{5 e^3}+\frac{d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}+\frac{d^3 x \sqrt{d^2-e^2 x^2}}{8 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(16*d^4 - 15*d^3*e*x + 8*d^2*e^2*x^2 + 30*d*e^3*x^3 - 24*e^4*x^4) + 15*d^4*ArcSin[(e*x)/d]))/(120*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
103,1,135,201,0.1740021,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]","\frac{945 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(1024 d^8-945 d^7 e x+512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-5040 d e^7 x^7+4480 e^8 x^8\right)}{40320 e^5}","\frac{x^4 \left(d^2-e^2 x^2\right)^{5/2}}{9 e}-\frac{d x^3 \left(d^2-e^2 x^2\right)^{5/2}}{8 e^2}+\frac{4 d^2 x^2 \left(d^2-e^2 x^2\right)^{5/2}}{63 e^3}+\frac{3 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e^5}+\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left(d^2-e^2 x^2\right)^{3/2}}{64 e^4}+\frac{d^3 (128 d-315 e x) \left(d^2-e^2 x^2\right)^{5/2}}{5040 e^5}",1,"(Sqrt[d^2 - e^2*x^2]*(1024*d^8 - 945*d^7*e*x + 512*d^6*e^2*x^2 - 630*d^5*e^3*x^3 + 384*d^4*e^4*x^4 + 7560*d^3*e^5*x^5 - 6400*d^2*e^6*x^6 - 5040*d*e^7*x^7 + 4480*e^8*x^8) + 945*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40320*e^5)","A",1
104,1,124,172,0.1307902,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-256 d^7+105 d^6 e x-128 d^5 e^2 x^2+70 d^4 e^3 x^3+1024 d^3 e^4 x^4-840 d^2 e^5 x^5-640 d e^6 x^6+560 e^7 x^7\right)-105 d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{4480 e^4}","-\frac{d x^2 \left(d^2-e^2 x^2\right)^{5/2}}{7 e^2}+\frac{x^3 \left(d^2-e^2 x^2\right)^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left(d^2-e^2 x^2\right)^{5/2}}{560 e^4}-\frac{3 d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e^4}-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left(d^2-e^2 x^2\right)^{3/2}}{64 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(-256*d^7 + 105*d^6*e*x - 128*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 1024*d^3*e^4*x^4 - 840*d^2*e^5*x^5 - 640*d*e^6*x^6 + 560*e^7*x^7) - 105*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4480*e^4)","A",1
105,1,113,140,0.1016943,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]","\frac{105 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(96 d^6-105 d^5 e x+48 d^4 e^2 x^2+490 d^3 e^3 x^3-384 d^2 e^4 x^4-280 d e^5 x^5+240 e^6 x^6\right)}{1680 e^3}","\frac{d (6 d-5 e x) \left(d^2-e^2 x^2\right)^{5/2}}{30 e^3}-\frac{\left(d^2-e^2 x^2\right)^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^3}+\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left(d^2-e^2 x^2\right)^{3/2}}{24 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(96*d^6 - 105*d^5*e*x + 48*d^4*e^2*x^2 + 490*d^3*e^3*x^3 - 384*d^2*e^4*x^4 - 280*d*e^5*x^5 + 240*e^6*x^6) + 105*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1680*e^3)","A",1
106,1,102,116,0.0865444,"\int \frac{x \left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-48 d^5+15 d^4 e x+96 d^3 e^2 x^2-70 d^2 e^3 x^3-48 d e^4 x^4+40 e^5 x^5\right)-15 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{240 e^2}","-\frac{d^2 x \left(d^2-e^2 x^2\right)^{3/2}}{24 e}-\frac{(6 d-5 e x) \left(d^2-e^2 x^2\right)^{5/2}}{30 e^2}-\frac{d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^2}-\frac{d^4 x \sqrt{d^2-e^2 x^2}}{16 e}",1,"(Sqrt[d^2 - e^2*x^2]*(-48*d^5 + 15*d^4*e*x + 96*d^3*e^2*x^2 - 70*d^2*e^3*x^3 - 48*d*e^4*x^4 + 40*e^5*x^5) - 15*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^2)","A",1
107,1,91,100,0.0549766,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x),x]","\frac{15 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(8 d^4+25 d^3 e x-16 d^2 e^2 x^2-10 d e^3 x^3+8 e^4 x^4\right)}{40 e}","\frac{1}{4} d x \left(d^2-e^2 x^2\right)^{3/2}+\frac{\left(d^2-e^2 x^2\right)^{5/2}}{5 e}+\frac{3 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}+\frac{3}{8} d^3 x \sqrt{d^2-e^2 x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(8*d^4 + 25*d^3*e*x - 16*d^2*e^2*x^2 - 10*d*e^3*x^3 + 8*e^4*x^4) + 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(40*e)","A",1
108,1,108,113,0.0876053,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)),x]","d^4 \log (x)-d^4 \log \left(\sqrt{d^2-e^2 x^2}+d\right)-\frac{3}{8} d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{1}{24} \sqrt{d^2-e^2 x^2} \left(32 d^3-15 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right)","\frac{1}{8} d^2 (8 d-3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d-3 e x) \left(d^2-e^2 x^2\right)^{3/2}-\frac{3}{8} d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(Sqrt[d^2 - e^2*x^2]*(32*d^3 - 15*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3))/24 - (3*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + d^4*Log[x] - d^4*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
109,1,114,115,0.1253859,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^2 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)),x]","-d^3 e \log (x)+d^3 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)-\frac{3}{2} d^3 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(-\frac{d^3}{x}-\frac{4 d^2 e}{3}-\frac{1}{2} d e^2 x+\frac{e^3 x^2}{3}\right)","-\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+d^3 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"Sqrt[d^2 - e^2*x^2]*((-4*d^2*e)/3 - d^3/x - (d*e^2*x)/2 + (e^3*x^2)/3) - (3*d^3*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - d^3*e*Log[x] + d^3*e*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
110,1,119,121,0.1552128,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^3 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)),x]","\frac{1}{2} \left(3 d^2 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+3 d^2 e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-3 d^2 e^2 \log (x)+\frac{\sqrt{d^2-e^2 x^2} \left(-d^3+2 d^2 e x-2 d e^2 x^2+e^3 x^3\right)}{x^2}\right)","\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"((Sqrt[d^2 - e^2*x^2]*(-d^3 + 2*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3))/x^2 + 3*d^2*e^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 3*d^2*e^2*Log[x] + 3*d^2*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/2","A",1
111,1,116,120,0.1541627,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^4 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)),x]","-\frac{3}{2} d e^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+d e^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\left(-\frac{d^3}{3 x^3}+\frac{d^2 e}{2 x^2}+\frac{4 d e^2}{3 x}+e^3\right) \sqrt{d^2-e^2 x^2}+\frac{3}{2} d e^3 \log (x)","\frac{e^2 (2 d+3 e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(2 d-3 e x) \left(d^2-e^2 x^2\right)^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{3}{2} d e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(e^3 - d^3/(3*x^3) + (d^2*e)/(2*x^2) + (4*d*e^2)/(3*x))*Sqrt[d^2 - e^2*x^2] + d*e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (3*d*e^3*Log[x])/2 - (3*d*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/2","A",1
112,1,111,119,0.1850946,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^5 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)),x]","\frac{1}{24} \left(-9 e^4 \log \left(\sqrt{d^2-e^2 x^2}+d\right)-24 e^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(-6 d^3+8 d^2 e x+15 d e^2 x^2-32 e^3 x^3\right)}{x^4}+9 e^4 \log (x)\right)","\frac{e^2 (3 d-8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{(3 d-4 e x) \left(d^2-e^2 x^2\right)^{3/2}}{12 x^4}+e^4 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)-\frac{3}{8} e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"((Sqrt[d^2 - e^2*x^2]*(-6*d^3 + 8*d^2*e*x + 15*d*e^2*x^2 - 32*e^3*x^3))/x^4 - 24*e^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + 9*e^4*Log[x] - 9*e^4*Log[d + Sqrt[d^2 - e^2*x^2]])/24","A",1
113,1,106,108,0.1473871,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^6 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)),x]","\frac{15 e^5 x^5 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-8 d^4+10 d^3 e x+16 d^2 e^2 x^2-25 d e^3 x^3-8 e^4 x^4\right)-15 e^5 x^5 \log (x)}{40 d x^5}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{5 d x^5}+\frac{e \left(d^2-e^2 x^2\right)^{3/2}}{4 x^4}+\frac{3 e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{8 x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-8*d^4 + 10*d^3*e*x + 16*d^2*e^2*x^2 - 25*d*e^3*x^3 - 8*e^4*x^4) - 15*e^5*x^5*Log[x] + 15*e^5*x^5*Log[d + Sqrt[d^2 - e^2*x^2]])/(40*d*x^5)","A",1
114,1,117,143,0.1769538,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^7 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x]","\frac{-15 e^6 x^6 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-40 d^5+48 d^4 e x+70 d^3 e^2 x^2-96 d^2 e^3 x^3-15 d e^4 x^4+48 e^5 x^5\right)+15 e^6 x^6 \log (x)}{240 d^2 x^6}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{6 d x^6}+\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{5 d^2 x^5}-\frac{e^2 \left(d^2-e^2 x^2\right)^{3/2}}{24 d x^4}-\frac{e^6 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{16 d^2}+\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-40*d^5 + 48*d^4*e*x + 70*d^3*e^2*x^2 - 96*d^2*e^3*x^3 - 15*d*e^4*x^4 + 48*e^5*x^5) + 15*e^6*x^6*Log[x] - 15*e^6*x^6*Log[d + Sqrt[d^2 - e^2*x^2]])/(240*d^2*x^6)","A",1
115,1,128,172,0.1900264,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^8 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]","\frac{105 e^7 x^7 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-240 d^6+280 d^5 e x+384 d^4 e^2 x^2-490 d^3 e^3 x^3-48 d^2 e^4 x^4+105 d e^5 x^5-96 e^6 x^6\right)-105 e^7 x^7 \log (x)}{1680 d^3 x^7}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{7 d x^7}+\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{6 d^2 x^6}-\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left(d^2-e^2 x^2\right)^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left(d^2-e^2 x^2\right)^{5/2}}{35 d^3 x^5}+\frac{e^7 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{16 d^3}",1,"(Sqrt[d^2 - e^2*x^2]*(-240*d^6 + 280*d^5*e*x + 384*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 48*d^2*e^4*x^4 + 105*d*e^5*x^5 - 96*e^6*x^6) - 105*e^7*x^7*Log[x] + 105*e^7*x^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(1680*d^3*x^7)","A",1
116,1,139,201,0.2124517,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^9 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x]","\frac{-105 e^8 x^8 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-560 d^7+640 d^6 e x+840 d^5 e^2 x^2-1024 d^4 e^3 x^3-70 d^3 e^4 x^4+128 d^2 e^5 x^5-105 d e^6 x^6+256 e^7 x^7\right)+105 e^8 x^8 \log (x)}{4480 d^4 x^8}","-\frac{\left(d^2-e^2 x^2\right)^{5/2}}{8 d x^8}+\frac{e \left(d^2-e^2 x^2\right)^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{128 d^4}+\frac{2 e^3 \left(d^2-e^2 x^2\right)^{5/2}}{35 d^4 x^5}-\frac{e^2 \left(d^2-e^2 x^2\right)^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left(d^2-e^2 x^2\right)^{3/2}}{64 d^3 x^4}",1,"(Sqrt[d^2 - e^2*x^2]*(-560*d^7 + 640*d^6*e*x + 840*d^5*e^2*x^2 - 1024*d^4*e^3*x^3 - 70*d^3*e^4*x^4 + 128*d^2*e^5*x^5 - 105*d*e^6*x^6 + 256*e^7*x^7) + 105*e^8*x^8*Log[x] - 105*e^8*x^8*Log[d + Sqrt[d^2 - e^2*x^2]])/(4480*d^4*x^8)","A",1
117,1,26,27,0.037801,"\int \frac{x \sqrt{1-x^2}}{1+x} \, dx","Integrate[(x*Sqrt[1 - x^2])/(1 + x),x]","\left(\frac{x}{2}-1\right) \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1}(x)","-\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x)",1,"(-1 + x/2)*Sqrt[1 - x^2] - ArcSin[x]/2","A",1
118,1,49,51,0.0368236,"\int \frac{\left(1-a^2 x^2\right)^{3/2}}{x^2 (1-a x)} \, dx","Integrate[(1 - a^2*x^2)^(3/2)/(x^2*(1 - a*x)),x]","\frac{\sqrt{1-a^2 x^2} (a x-1)}{x}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-a \sin ^{-1}(a x)","-\frac{\sqrt{1-a^2 x^2} (1-a x)}{x}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-a \sin ^{-1}(a x)",1,"((-1 + a*x)*Sqrt[1 - a^2*x^2])/x - a*ArcSin[a*x] - a*ArcTanh[Sqrt[1 - a^2*x^2]]","A",1
119,1,91,118,0.0882752,"\int \frac{x^4}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^4/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-16 d^3-7 d^2 e x+d e^2 x^2-2 e^3 x^3\right)-9 d^3 (d+e x) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{6 e^5 (d+e x)}","\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{3 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^5}",1,"(Sqrt[d^2 - e^2*x^2]*(-16*d^3 - 7*d^2*e*x + d*e^2*x^2 - 2*e^3*x^3) - 9*d^3*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^5*(d + e*x))","A",1
120,1,80,91,0.0592258,"\int \frac{x^3}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^3/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{\sqrt{d^2-e^2 x^2} \left(4 d^2+d e x-e^2 x^2\right)+3 d^2 (d+e x) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^4 (d+e x)}","\frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{3 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^4}",1,"(Sqrt[d^2 - e^2*x^2]*(4*d^2 + d*e*x - e^2*x^2) + 3*d^2*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^4*(d + e*x))","A",1
121,1,59,77,0.0703378,"\int \frac{x^2}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^2/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\frac{\sqrt{d^2-e^2 x^2} (2 d+e x)}{d+e x}+d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}","-\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}",1,"-((((2*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d + e*x) + d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^3)","A",1
122,1,49,52,0.0292508,"\int \frac{x}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{\frac{\sqrt{d^2-e^2 x^2}}{d+e x}+\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^2}","\frac{\sqrt{d^2-e^2 x^2}}{e^2 (d+e x)}+\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^2}",1,"(Sqrt[d^2 - e^2*x^2]/(d + e*x) + ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2","A",1
123,1,32,31,0.0060408,"\int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2}}{d^2 e+d e^2 x}","-\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)}",1,"-(Sqrt[d^2 - e^2*x^2]/(d^2*e + d*e^2*x))","A",1
124,1,52,54,0.0374947,"\int \frac{1}{x (d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x*(d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{\frac{\sqrt{d^2-e^2 x^2}}{d+e x}-\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^2}","\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^2}",1,"(Sqrt[d^2 - e^2*x^2]/(d + e*x) - ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^2","A",1
125,1,62,81,0.0529066,"\int \frac{1}{x^2 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)-\frac{(d+2 e x) \sqrt{d^2-e^2 x^2}}{x (d+e x)}}{d^3}","\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^3}",1,"(-(((d + 2*e*x)*Sqrt[d^2 - e^2*x^2])/(x*(d + e*x))) + e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^3","A",1
126,1,127,113,0.3169969,"\int \frac{1}{x^3 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x^3*(d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","-\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^4}-\frac{d^3+d e^2 x^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left(\sqrt{1-\frac{e^2 x^2}{d^2}}\right)-2 d^2 e x-3 d e^2 x^2+4 e^3 x^3}{2 d^4 x^2 \sqrt{d^2-e^2 x^2}}","\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}",1,"-((e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^4) - (d^3 - 2*d^2*e*x - 3*d*e^2*x^2 + 4*e^3*x^3 + d*e^2*x^2*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(2*d^4*x^2*Sqrt[d^2 - e^2*x^2])","A",1
127,1,106,128,0.1704947,"\int \frac{x^5}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^5/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(16 d^4+d^3 e x-23 d^2 e^2 x^2-3 d e^3 x^3+3 e^4 x^4\right)}{(e x-d) (d+e x)^2}-15 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{6 e^6}","\frac{x^4 (d-e x)}{3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{(16 d-15 e x) \sqrt{d^2-e^2 x^2}}{6 e^6}-\frac{5 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^6}-\frac{x^2 (4 d-5 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}",1,"((Sqrt[d^2 - e^2*x^2]*(16*d^4 + d^3*e*x - 23*d^2*e^2*x^2 - 3*d*e^3*x^3 + 3*e^4*x^4))/((-d + e*x)*(d + e*x)^2) - 15*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^6)","A",1
128,1,93,113,0.135271,"\int \frac{x^4}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^4/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{3 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(8 d^3+5 d^2 e x-7 d e^2 x^2-3 e^3 x^3\right)}{(d-e x) (d+e x)^2}}{3 e^5}","\frac{x^3 (d-e x)}{3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{8 \sqrt{d^2-e^2 x^2}}{3 e^5}+\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}-\frac{x (3 d-4 e x)}{3 e^4 \sqrt{d^2-e^2 x^2}}",1,"((Sqrt[d^2 - e^2*x^2]*(8*d^3 + 5*d^2*e*x - 7*d*e^2*x^2 - 3*e^3*x^3))/((d - e*x)*(d + e*x)^2) + 3*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e^5)","A",1
129,1,80,89,0.1217665,"\int \frac{x^3}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^3/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(-2 d^2+d e x+4 e^2 x^2\right)}{(d-e x) (d+e x)^2}-3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{3 e^4}","\frac{x^2 (d-e x)}{3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 d-3 e x}{3 e^4 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^4}",1,"((Sqrt[d^2 - e^2*x^2]*(-2*d^2 + d*e*x + 4*e^2*x^2))/((d - e*x)*(d + e*x)^2) - 3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e^4)","A",1
130,1,60,60,0.0516268,"\int \frac{x^2}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^2/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(2 d^2+2 d e x-e^2 x^2\right)}{3 d e^3 (d-e x) (d+e x)^2}","\frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}}-\frac{x^2}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(2*d^2 + 2*d*e*x - e^2*x^2))/(3*d*e^3*(d - e*x)*(d + e*x)^2)","A",1
131,1,56,58,0.0449319,"\int \frac{x}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(d^2+d e x+e^2 x^2\right)}{3 d^2 e^2 (d-e x) (d+e x)^2}","\frac{x}{3 d^2 e \sqrt{d^2-e^2 x^2}}+\frac{1}{3 e^2 (d+e x) \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(d^2 + d*e*x + e^2*x^2))/(3*d^2*e^2*(d - e*x)*(d + e*x)^2)","A",1
132,1,58,58,0.0326079,"\int \frac{1}{(d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","-\frac{\left(d^2-2 d e x-2 e^2 x^2\right) \sqrt{d^2-e^2 x^2}}{3 d^3 e (d-e x) (d+e x)^2}","\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}}",1,"-1/3*((d^2 - 2*d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(d^3*e*(d - e*x)*(d + e*x)^2)","A",1
133,1,83,88,0.1025019,"\int \frac{1}{x (d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(4 d^2+d e x-2 e^2 x^2\right)}{(d-e x) (d+e x)^2}-3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+3 \log (x)}{3 d^4}","\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^4}",1,"(((4*d^2 + d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/((d - e*x)*(d + e*x)^2) + 3*Log[x] - 3*Log[d + Sqrt[d^2 - e^2*x^2]])/(3*d^4)","A",1
134,1,101,120,0.1188796,"\int \frac{1}{x^2 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{3 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(3 d^3+7 d^2 e x-5 d e^2 x^2-8 e^3 x^3\right)}{x (e x-d) (d+e x)^2}-3 e \log (x)}{3 d^5}","\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^5}+\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}",1,"((Sqrt[d^2 - e^2*x^2]*(3*d^3 + 7*d^2*e*x - 5*d*e^2*x^2 - 8*e^3*x^3))/(x*(-d + e*x)*(d + e*x)^2) - 3*e*Log[x] + 3*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(3*d^5)","A",1
135,1,115,152,0.1037205,"\int \frac{1}{x^3 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]","\frac{-15 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(3 d^4-3 d^3 e x-23 d^2 e^2 x^2+d e^3 x^3+16 e^4 x^4\right)}{x^2 (e x-d) (d+e x)^2}+15 e^2 \log (x)}{6 d^6}","\frac{1}{3 d^2 x^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{8 e \sqrt{d^2-e^2 x^2}}{3 d^6 x}-\frac{5 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^6}-\frac{5 \sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{5 d-4 e x}{3 d^4 x^2 \sqrt{d^2-e^2 x^2}}",1,"((Sqrt[d^2 - e^2*x^2]*(3*d^4 - 3*d^3*e*x - 23*d^2*e^2*x^2 + d*e^3*x^3 + 16*e^4*x^4))/(x^2*(-d + e*x)*(d + e*x)^2) + 15*e^2*Log[x] - 15*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(6*d^6)","A",1
136,1,128,162,0.2395674,"\int \frac{x^7}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{105 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(96 d^6-9 d^5 e x-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4+15 d e^5 x^5-15 e^6 x^6\right)}{(d-e x)^2 (d+e x)^3}}{30 e^8}","\frac{x^6 (d-e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^8}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(96*d^6 - 9*d^5*e*x - 249*d^4*e^2*x^2 - 4*d^3*e^3*x^3 + 176*d^2*e^4*x^4 + 15*d*e^5*x^5 - 15*e^6*x^6))/((d - e*x)^2*(d + e*x)^3) + 105*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8)","A",1
137,1,115,148,0.1924061,"\int \frac{x^6}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{15 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(48 d^5+33 d^4 e x-87 d^3 e^2 x^2-52 d^2 e^3 x^3+38 d e^4 x^4+15 e^5 x^5\right)}{(d-e x)^2 (d+e x)^3}}{15 e^7}","\frac{x^5 (d-e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^7}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*((Sqrt[d^2 - e^2*x^2]*(48*d^5 + 33*d^4*e*x - 87*d^3*e^2*x^2 - 52*d^2*e^3*x^3 + 38*d*e^4*x^4 + 15*e^5*x^5))/((d - e*x)^2*(d + e*x)^3) + 15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^7","A",1
138,1,103,122,0.1445126,"\int \frac{x^5}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^5/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{15 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(8 d^4-7 d^3 e x-27 d^2 e^2 x^2+8 d e^3 x^3+23 e^4 x^4\right)}{(d-e x)^2 (d+e x)^3}}{15 e^6}","\frac{x^4 (d-e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^6}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(8*d^4 - 7*d^3*e*x - 27*d^2*e^2*x^2 + 8*d*e^3*x^3 + 23*e^4*x^4))/((d - e*x)^2*(d + e*x)^3) + 15*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6)","A",1
139,1,82,85,0.0857256,"\int \frac{x^4}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^4/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(8 d^4+8 d^3 e x-12 d^2 e^2 x^2-12 d e^3 x^3+3 e^4 x^4\right)}{15 d e^5 (d-e x)^2 (d+e x)^3}","-\frac{x^4 (d-e x)}{5 d e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{4}{5 e^5 \sqrt{d^2-e^2 x^2}}+\frac{4 d^2}{15 e^5 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(8*d^4 + 8*d^3*e*x - 12*d^2*e^2*x^2 - 12*d*e^3*x^3 + 3*e^4*x^4))/(d*e^5*(d - e*x)^2*(d + e*x)^3)","A",1
140,1,82,91,0.0712362,"\int \frac{x^3}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^3/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-2 d^4-2 d^3 e x+3 d^2 e^2 x^2+3 d e^3 x^3+3 e^4 x^4\right)}{15 d^2 e^4 (d-e x)^2 (d+e x)^3}","\frac{x^2 (d-e x)}{5 e^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{2 d-3 e x}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-2*d^4 - 2*d^3*e*x + 3*d^2*e^2*x^2 + 3*d*e^3*x^3 + 3*e^4*x^4))/(15*d^2*e^4*(d - e*x)^2*(d + e*x)^3)","A",1
141,1,82,95,0.0604584,"\int \frac{x^2}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x^2/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(2 d^4+2 d^3 e x-3 d^2 e^2 x^2+2 d e^3 x^3+2 e^4 x^4\right)}{15 d^3 e^3 (d-e x)^2 (d+e x)^3}","-\frac{x^2}{5 d e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 (d+e x)}{15 d e^3 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(2*d^4 + 2*d^3*e*x - 3*d^2*e^2*x^2 + 2*d*e^3*x^3 + 2*e^4*x^4))/(15*d^3*e^3*(d - e*x)^2*(d + e*x)^3)","A",1
142,1,82,85,0.0549257,"\int \frac{x}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[x/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(3 d^4+3 d^3 e x+3 d^2 e^2 x^2-2 d e^3 x^3-2 e^4 x^4\right)}{15 d^4 e^2 (d-e x)^2 (d+e x)^3}","\frac{x}{15 d^2 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{1}{5 e^2 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(3*d^4 + 3*d^3*e*x + 3*d^2*e^2*x^2 - 2*d*e^3*x^3 - 2*e^4*x^4))/(15*d^4*e^2*(d - e*x)^2*(d + e*x)^3)","A",1
143,1,82,82,0.0391455,"\int \frac{1}{(d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(3 d^4-12 d^3 e x-12 d^2 e^2 x^2+8 d e^3 x^3+8 e^4 x^4\right)}{15 d^5 e (d-e x)^2 (d+e x)^3}","-\frac{1}{5 d e (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(3*d^4 - 12*d^3*e*x - 12*d^2*e^2*x^2 + 8*d*e^3*x^3 + 8*e^4*x^4))/(d^5*e*(d - e*x)^2*(d + e*x)^3)","A",1
144,1,106,119,0.0900205,"\int \frac{1}{x (d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{-15 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(23 d^4+8 d^3 e x-27 d^2 e^2 x^2-7 d e^3 x^3+8 e^4 x^4\right)}{(d-e x)^2 (d+e x)^3}+15 \log (x)}{15 d^6}","\frac{1}{5 d^2 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^6}+\frac{5 d-4 e x}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(23*d^4 + 8*d^3*e*x - 27*d^2*e^2*x^2 - 7*d*e^3*x^3 + 8*e^4*x^4))/((d - e*x)^2*(d + e*x)^3) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^6)","A",1
145,1,122,154,0.1240652,"\int \frac{1}{x^2 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{-15 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(15 d^5+38 d^4 e x-52 d^3 e^2 x^2-87 d^2 e^3 x^3+33 d e^4 x^4+48 e^5 x^5\right)}{x (d-e x)^2 (d+e x)^3}+15 e \log (x)}{15 d^7}","\frac{1}{5 d^2 x (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^7}+\frac{8 d-5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{6 d-5 e x}{15 d^4 x \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*((Sqrt[d^2 - e^2*x^2]*(15*d^5 + 38*d^4*e*x - 52*d^3*e^2*x^2 - 87*d^2*e^3*x^3 + 33*d*e^4*x^4 + 48*e^5*x^5))/(x*(d - e*x)^2*(d + e*x)^3) + 15*e*Log[x] - 15*e*Log[d + Sqrt[d^2 - e^2*x^2]])/d^7","A",1
146,1,137,186,0.1303542,"\int \frac{1}{x^3 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","\frac{-105 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(-15 d^6+15 d^5 e x+176 d^4 e^2 x^2-4 d^3 e^3 x^3-249 d^2 e^4 x^4-9 d e^5 x^5+96 e^6 x^6\right)}{x^2 (d-e x)^2 (d+e x)^3}+105 e^2 \log (x)}{30 d^8}","\frac{1}{5 d^2 x^2 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(-15*d^6 + 15*d^5*e*x + 176*d^4*e^2*x^2 - 4*d^3*e^3*x^3 - 249*d^2*e^4*x^4 - 9*d*e^5*x^5 + 96*e^6*x^6))/(x^2*(d - e*x)^2*(d + e*x)^3) + 105*e^2*Log[x] - 105*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^8)","A",1
147,1,148,215,0.1615855,"\int \frac{1}{x^4 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}} \, dx","Integrate[1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]","-\frac{-105 e^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(10 d^7-5 d^6 e x+75 d^5 e^2 x^2+236 d^4 e^3 x^3-244 d^3 e^4 x^4-489 d^2 e^5 x^5+151 d e^6 x^6+256 e^7 x^7\right)}{x^3 (d-e x)^2 (d+e x)^3}+105 e^3 \log (x)}{30 d^9}","\frac{1}{5 d^2 x^3 (d+e x) \left(d^2-e^2 x^2\right)^{3/2}}-\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}+\frac{7 e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^9}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}+\frac{8 d-7 e x}{15 d^4 x^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/30*((Sqrt[d^2 - e^2*x^2]*(10*d^7 - 5*d^6*e*x + 75*d^5*e^2*x^2 + 236*d^4*e^3*x^3 - 244*d^3*e^4*x^4 - 489*d^2*e^5*x^5 + 151*d*e^6*x^6 + 256*e^7*x^7))/(x^3*(d - e*x)^2*(d + e*x)^3) + 105*e^3*Log[x] - 105*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/d^9","A",1
148,1,104,118,0.1200809,"\int \frac{x^3}{(d+e x) \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[x^3/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(2 d^6+2 d^5 e x-5 d^4 e^2 x^2-5 d^3 e^3 x^3-5 d^2 e^4 x^4+2 d e^5 x^5+2 e^6 x^6\right)}{35 d^4 e^4 (d-e x)^3 (d+e x)^4}","\frac{x^2 (d-e x)}{7 e^2 \left(d^2-e^2 x^2\right)^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{x}{35 d^2 e^3 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}}",1,"-1/35*(Sqrt[d^2 - e^2*x^2]*(2*d^6 + 2*d^5*e*x - 5*d^4*e^2*x^2 - 5*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 2*d*e^5*x^5 + 2*e^6*x^6))/(d^4*e^4*(d - e*x)^3*(d + e*x)^4)","A",1
149,1,104,123,0.0802434,"\int \frac{x^2}{(d+e x) \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[x^2/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(6 d^6+6 d^5 e x-15 d^4 e^2 x^2+20 d^3 e^3 x^3+20 d^2 e^4 x^4-8 d e^5 x^5-8 e^6 x^6\right)}{105 d^5 e^3 (d-e x)^3 (d+e x)^4}","-\frac{x^2}{7 d e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 (d+2 e x)}{35 d e^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(6*d^6 + 6*d^5*e*x - 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 20*d^2*e^4*x^4 - 8*d*e^5*x^5 - 8*e^6*x^6))/(105*d^5*e^3*(d - e*x)^3*(d + e*x)^4)","A",1
150,1,54,66,0.0505583,"\int \frac{x^3}{(1+a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[x^3/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]","\frac{\sqrt{1-a^2 x^2} \left(-a^2 x^2+a x+4\right)+3 (a x+1) \sin ^{-1}(a x)}{2 a^4 (a x+1)}","\frac{3 \sin ^{-1}(a x)}{2 a^4}+\frac{x^2 (1-a x)}{a^2 \sqrt{1-a^2 x^2}}+\frac{(4-3 a x) \sqrt{1-a^2 x^2}}{2 a^4}",1,"(Sqrt[1 - a^2*x^2]*(4 + a*x - a^2*x^2) + 3*(1 + a*x)*ArcSin[a*x])/(2*a^4*(1 + a*x))","A",1
151,1,37,55,0.0530107,"\int \frac{x^2}{(1+a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[x^2/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]","-\frac{\frac{\sqrt{1-a^2 x^2} (a x+2)}{a x+1}+\sin ^{-1}(a x)}{a^3}","-\frac{\sin ^{-1}(a x)}{a^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 (a x+1)}-\frac{\sqrt{1-a^2 x^2}}{a^3}",1,"-((((2 + a*x)*Sqrt[1 - a^2*x^2])/(1 + a*x) + ArcSin[a*x])/a^3)","A",1
152,1,31,34,0.0235936,"\int \frac{x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[x/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]","\frac{\frac{\sqrt{1-a^2 x^2}}{a x+1}+\sin ^{-1}(a x)}{a^2}","\frac{\sqrt{1-a^2 x^2}}{a^2 (a x+1)}+\frac{\sin ^{-1}(a x)}{a^2}",1,"(Sqrt[1 - a^2*x^2]/(1 + a*x) + ArcSin[a*x])/a^2","A",1
153,1,25,26,0.0058695,"\int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[1/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]","-\frac{\sqrt{1-a^2 x^2}}{a^2 x+a}","-\frac{\sqrt{1-a^2 x^2}}{a (a x+1)}",1,"-(Sqrt[1 - a^2*x^2]/(a + a^2*x))","A",1
154,1,41,41,0.0278666,"\int \frac{1}{x (1-a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[1/(x*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]","\frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)","\frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"Sqrt[1 - a^2*x^2]/(1 - a*x) - ArcTanh[Sqrt[1 - a^2*x^2]]","A",1
155,1,50,64,0.0413068,"\int \frac{1}{x^2 (1-a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[1/(x^2*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]","\frac{(1-2 a x) \sqrt{1-a^2 x^2}}{x (a x-1)}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)","-\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"((1 - 2*a*x)*Sqrt[1 - a^2*x^2])/(x*(-1 + a*x)) - a*ArcTanh[Sqrt[1 - a^2*x^2]]","A",1
156,1,63,90,0.0547822,"\int \frac{1}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx","Integrate[1/(x^3*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]","\frac{1}{2} \left(\frac{\left(-4 a^2 x^2+a x+1\right) \sqrt{1-a^2 x^2}}{x^2 (a x-1)}-3 a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)\right)","-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)",1,"(((1 + a*x - 4*a^2*x^2)*Sqrt[1 - a^2*x^2])/(x^2*(-1 + a*x)) - 3*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2","A",1
157,1,135,229,0.213261,"\int \frac{x^5 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(x^5*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{\sqrt{d^2-e^2 x^2} \left(-512 d^8+315 d^7 e x-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6-1008 d e^7 x^7+448 e^8 x^8\right)-315 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{4032 e^6}","-\frac{1}{9} x^6 \left(d^2-e^2 x^2\right)^{3/2}+\frac{d x^5 \left(d^2-e^2 x^2\right)^{3/2}}{4 e}-\frac{5 d^2 x^4 \left(d^2-e^2 x^2\right)^{3/2}}{21 e^2}-\frac{5 d^9 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{64 e^6}-\frac{5 d^7 x \sqrt{d^2-e^2 x^2}}{64 e^5}-\frac{d^5 (256 d-315 e x) \left(d^2-e^2 x^2\right)^{3/2}}{2016 e^6}-\frac{4 d^4 x^2 \left(d^2-e^2 x^2\right)^{3/2}}{21 e^4}+\frac{5 d^3 x^3 \left(d^2-e^2 x^2\right)^{3/2}}{24 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(-512*d^8 + 315*d^7*e*x - 256*d^6*e^2*x^2 + 210*d^5*e^3*x^3 - 192*d^4*e^4*x^4 + 168*d^3*e^5*x^5 + 512*d^2*e^6*x^6 - 1008*d*e^7*x^7 + 448*e^8*x^8) - 315*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4032*e^6)","A",1
158,1,124,200,0.1546905,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{1365 d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(2048 d^7-1365 d^6 e x+1024 d^5 e^2 x^2-910 d^4 e^3 x^3+768 d^3 e^4 x^4+1960 d^2 e^5 x^5-3840 d e^6 x^6+1680 e^7 x^7\right)}{13440 e^5}","-\frac{1}{8} x^5 \left(d^2-e^2 x^2\right)^{3/2}+\frac{2 d x^4 \left(d^2-e^2 x^2\right)^{3/2}}{7 e}-\frac{13 d^2 x^3 \left(d^2-e^2 x^2\right)^{3/2}}{48 e^2}+\frac{13 d^8 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{128 e^5}+\frac{13 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^4 (1024 d-1365 e x) \left(d^2-e^2 x^2\right)^{3/2}}{6720 e^5}+\frac{8 d^3 x^2 \left(d^2-e^2 x^2\right)^{3/2}}{35 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(2048*d^7 - 1365*d^6*e*x + 1024*d^5*e^2*x^2 - 910*d^4*e^3*x^3 + 768*d^3*e^4*x^4 + 1960*d^2*e^5*x^5 - 3840*d*e^6*x^6 + 1680*e^7*x^7) + 1365*d^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(13440*e^5)","A",1
159,1,113,171,0.1274766,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{\sqrt{d^2-e^2 x^2} \left(-176 d^6+105 d^5 e x-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4-280 d e^5 x^5+120 e^6 x^6\right)-105 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{840 e^4}","-\frac{11 d^2 x^2 \left(d^2-e^2 x^2\right)^{3/2}}{35 e^2}-\frac{1}{7} x^4 \left(d^2-e^2 x^2\right)^{3/2}+\frac{d x^3 \left(d^2-e^2 x^2\right)^{3/2}}{3 e}-\frac{d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^4}-\frac{d^5 x \sqrt{d^2-e^2 x^2}}{8 e^3}-\frac{d^3 (88 d-105 e x) \left(d^2-e^2 x^2\right)^{3/2}}{420 e^4}",1,"(Sqrt[d^2 - e^2*x^2]*(-176*d^6 + 105*d^5*e*x - 88*d^4*e^2*x^2 + 70*d^3*e^3*x^3 + 144*d^2*e^4*x^4 - 280*d*e^5*x^5 + 120*e^6*x^6) - 105*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(840*e^4)","A",1
160,1,102,142,0.1138447,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{45 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(64 d^5-45 d^4 e x+32 d^3 e^2 x^2+50 d^2 e^3 x^3-96 d e^4 x^4+40 e^5 x^5\right)}{240 e^3}","\frac{2 d x^2 \left(d^2-e^2 x^2\right)^{3/2}}{5 e}-\frac{1}{6} x^3 \left(d^2-e^2 x^2\right)^{3/2}+\frac{d^2 (32 d-45 e x) \left(d^2-e^2 x^2\right)^{3/2}}{120 e^3}+\frac{3 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^3}+\frac{3 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(64*d^5 - 45*d^4*e*x + 32*d^3*e^2*x^2 + 50*d^2*e^3*x^3 - 96*d*e^4*x^4 + 40*e^5*x^5) + 45*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^3)","A",1
161,1,91,136,0.0755217,"\int \frac{x \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(x*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{\sqrt{d^2-e^2 x^2} \left(-28 d^4+15 d^3 e x+16 d^2 e^2 x^2-30 d e^3 x^3+12 e^4 x^4\right)-15 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{60 e^2}","-\frac{d x \left(d^2-e^2 x^2\right)^{3/2}}{6 e}-\frac{\left(d^2-e^2 x^2\right)^{7/2}}{3 e^2 (d+e x)^2}-\frac{2 \left(d^2-e^2 x^2\right)^{5/2}}{15 e^2}-\frac{d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{4 e^2}-\frac{d^3 x \sqrt{d^2-e^2 x^2}}{4 e}",1,"(Sqrt[d^2 - e^2*x^2]*(-28*d^4 + 15*d^3*e*x + 16*d^2*e^2*x^2 - 30*d*e^3*x^3 + 12*e^4*x^4) - 15*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(60*e^2)","A",1
162,1,80,108,0.0477892,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2,x]","\frac{15 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\sqrt{d^2-e^2 x^2} \left(16 d^3+9 d^2 e x-16 d e^2 x^2+6 e^3 x^3\right)}{24 e}","\frac{5}{8} d^2 x \sqrt{d^2-e^2 x^2}+\frac{5 d \left(d^2-e^2 x^2\right)^{3/2}}{12 e}+\frac{(d-e x) \left(d^2-e^2 x^2\right)^{3/2}}{4 e}+\frac{5 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e}",1,"(Sqrt[d^2 - e^2*x^2]*(16*d^3 + 9*d^2*e*x - 16*d*e^2*x^2 + 6*e^3*x^3) + 15*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e)","A",1
163,1,96,96,0.0960172,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)^2),x]","d^3 \log (x)+\sqrt{d^2-e^2 x^2} \left(\frac{2 d^2}{3}-d e x+\frac{e^2 x^2}{3}\right)-d^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+d^3 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)","d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left(d^2-e^2 x^2\right)^{3/2}+d^3 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)-d^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"Sqrt[d^2 - e^2*x^2]*((2*d^2)/3 - d*e*x + (e^2*x^2)/3) - d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + d^3*Log[x] - d^3*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
164,1,100,105,0.1269542,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^2 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)^2),x]","\left(-\frac{d^2}{x}-2 d e+\frac{e^2 x}{2}\right) \sqrt{d^2-e^2 x^2}+2 d^2 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)-\frac{1}{2} d^2 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-2 d^2 e \log (x)","-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{x}-\frac{1}{2} e (4 d+e x) \sqrt{d^2-e^2 x^2}-\frac{1}{2} d^2 e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+2 d^2 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(-2*d*e - d^2/x + (e^2*x)/2)*Sqrt[d^2 - e^2*x^2] - (d^2*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - 2*d^2*e*Log[x] + 2*d^2*e*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
165,1,102,110,0.1395973,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^3 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^2),x]","\left(-\frac{d^2}{2 x^2}+\frac{2 d e}{x}+e^2\right) \sqrt{d^2-e^2 x^2}-\frac{1}{2} d e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+2 d e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{1}{2} d e^2 \log (x)","\frac{e (4 d+e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{2 x^2}+2 d e^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{1}{2} d e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"(e^2 - d^2/(2*x^2) + (2*d*e)/x)*Sqrt[d^2 - e^2*x^2] + 2*d*e^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (d*e^2*Log[x])/2 - (d*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/2","A",1
166,1,96,102,0.1778852,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^4 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)^2),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(d^2-3 d e x+2 e^2 x^2\right)}{3 x^3}-e^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+e^3 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)+e^3 \log (x)","\frac{e (d-e x) \sqrt{d^2-e^2 x^2}}{x^2}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{3 x^3}+e^3 \left(-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)\right)-e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"-1/3*(Sqrt[d^2 - e^2*x^2]*(d^2 - 3*d*e*x + 2*e^2*x^2))/x^3 - e^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + e^3*Log[x] - e^3*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
167,1,95,108,0.1746451,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^5 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^2),x]","-\frac{-15 e^4 x^4 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(6 d^3-16 d^2 e x+9 d e^2 x^2+16 e^3 x^3\right)+15 e^4 x^4 \log (x)}{24 d x^4}","-\frac{5 e^2 \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{4 x^4}+\frac{2 e \left(d^2-e^2 x^2\right)^{3/2}}{3 d x^3}+\frac{5 e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d}",1,"-1/24*(Sqrt[d^2 - e^2*x^2]*(6*d^3 - 16*d^2*e*x + 9*d*e^2*x^2 + 16*e^3*x^3) + 15*e^4*x^4*Log[x] - 15*e^4*x^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(d*x^4)","A",1
168,1,106,140,0.167639,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^6 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^2),x]","\frac{-15 e^5 x^5 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-12 d^4+30 d^3 e x-16 d^2 e^2 x^2-15 d e^3 x^3+28 e^4 x^4\right)+15 e^5 x^5 \log (x)}{60 d^2 x^5}","-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{5 x^5}+\frac{e \left(d^2-e^2 x^2\right)^{3/2}}{2 d x^4}-\frac{7 e^2 \left(d^2-e^2 x^2\right)^{3/2}}{15 d^2 x^3}-\frac{e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{4 d^2}+\frac{e^3 \sqrt{d^2-e^2 x^2}}{4 d x^2}",1,"(Sqrt[d^2 - e^2*x^2]*(-12*d^4 + 30*d^3*e*x - 16*d^2*e^2*x^2 - 15*d*e^3*x^3 + 28*e^4*x^4) + 15*e^5*x^5*Log[x] - 15*e^5*x^5*Log[d + Sqrt[d^2 - e^2*x^2]])/(60*d^2*x^5)","A",1
169,1,117,169,0.2341121,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^7 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)^2),x]","-\frac{-45 e^6 x^6 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(40 d^5-96 d^4 e x+50 d^3 e^2 x^2+32 d^2 e^3 x^3-45 d e^4 x^4+64 e^5 x^5\right)+45 e^6 x^6 \log (x)}{240 d^3 x^6}","-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{6 x^6}+\frac{2 e \left(d^2-e^2 x^2\right)^{3/2}}{5 d x^5}-\frac{3 e^2 \left(d^2-e^2 x^2\right)^{3/2}}{8 d^2 x^4}-\frac{3 e^4 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{3 e^6 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{16 d^3}+\frac{4 e^3 \left(d^2-e^2 x^2\right)^{3/2}}{15 d^3 x^3}",1,"-1/240*(Sqrt[d^2 - e^2*x^2]*(40*d^5 - 96*d^4*e*x + 50*d^3*e^2*x^2 + 32*d^2*e^3*x^3 - 45*d*e^4*x^4 + 64*e^5*x^5) + 45*e^6*x^6*Log[x] - 45*e^6*x^6*Log[d + Sqrt[d^2 - e^2*x^2]])/(d^3*x^6)","A",1
170,1,128,198,0.234188,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^8 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]","\frac{-105 e^7 x^7 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\sqrt{d^2-e^2 x^2} \left(-120 d^6+280 d^5 e x-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4-105 d e^5 x^5+176 e^6 x^6\right)+105 e^7 x^7 \log (x)}{840 d^4 x^7}","-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{7 x^7}+\frac{e \left(d^2-e^2 x^2\right)^{3/2}}{3 d x^6}-\frac{11 e^2 \left(d^2-e^2 x^2\right)^{3/2}}{35 d^2 x^5}-\frac{e^7 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d^4}-\frac{22 e^4 \left(d^2-e^2 x^2\right)^{3/2}}{105 d^4 x^3}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{e^3 \left(d^2-e^2 x^2\right)^{3/2}}{4 d^3 x^4}",1,"(Sqrt[d^2 - e^2*x^2]*(-120*d^6 + 280*d^5*e*x - 144*d^4*e^2*x^2 - 70*d^3*e^3*x^3 + 88*d^2*e^4*x^4 - 105*d*e^5*x^5 + 176*e^6*x^6) + 105*e^7*x^7*Log[x] - 105*e^7*x^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(840*d^4*x^7)","A",1
171,1,106,123,0.1584047,"\int \frac{x^4}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^4/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\sqrt{d^2-e^2 x^2} \left(-\frac{d^2}{10 e^5 (d+e x)^3}+\frac{31 d}{60 e^5 (d+e x)^2}-\frac{1}{8 e^5 (e x-d)}-\frac{193}{120 e^5 (d+e x)}\right)-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}","\frac{17 d^2 (d-e x)}{15 e^5 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{2 (15 d-13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}-\frac{d^3 (d-e x)^2}{5 e^5 \left(d^2-e^2 x^2\right)^{5/2}}",1,"Sqrt[d^2 - e^2*x^2]*(-1/8*1/(e^5*(-d + e*x)) - d^2/(10*e^5*(d + e*x)^3) + (31*d)/(60*e^5*(d + e*x)^2) - 193/(120*e^5*(d + e*x))) - ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e^5","A",1
172,1,70,99,0.0772036,"\int \frac{x^3}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^3/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(2 d^3+4 d^2 e x+d e^2 x^2-2 e^3 x^3\right)}{5 d e^4 (d-e x) (d+e x)^3}","\frac{d^2 (d-e x)^2}{5 e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{4 d (d-e x)}{5 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{5 d-2 e x}{5 d e^4 \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(2*d^3 + 4*d^2*e*x + d*e^2*x^2 - 2*e^3*x^3))/(5*d*e^4*(d - e*x)*(d + e*x)^3)","A",1
173,1,70,89,0.0639247,"\int \frac{x^2}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x^2/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(4 d^3+8 d^2 e x+2 d e^2 x^2+e^3 x^3\right)}{15 d^2 e^3 (d-e x) (d+e x)^3}","\frac{x}{15 d^2 e^2 \sqrt{d^2-e^2 x^2}}-\frac{d}{5 e^3 (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{7}{15 e^3 (d+e x) \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(4*d^3 + 8*d^2*e*x + 2*d*e^2*x^2 + e^3*x^3))/(15*d^2*e^3*(d - e*x)*(d + e*x)^3)","A",1
174,1,69,91,0.0542742,"\int \frac{x}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[x/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(d^3+2 d^2 e x+8 d e^2 x^2+4 e^3 x^3\right)}{15 d^3 e^2 (d-e x) (d+e x)^3}","-\frac{2}{15 d e^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{1}{5 e^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(d^3 + 2*d^2*e*x + 8*d*e^2*x^2 + 4*e^3*x^3))/(15*d^3*e^2*(d - e*x)*(d + e*x)^3)","A",1
175,1,70,91,0.0367884,"\int \frac{1}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-2 d^3+d^2 e x+4 d e^2 x^2+2 e^3 x^3\right)}{5 d^4 e (d-e x) (d+e x)^3}","-\frac{1}{5 d^2 e (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{1}{5 d e (d+e x)^2 \sqrt{d^2-e^2 x^2}}+\frac{2 x}{5 d^4 \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-2*d^3 + d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3))/(5*d^4*e*(d - e*x)*(d + e*x)^3)","A",1
176,1,95,118,0.0874549,"\int \frac{1}{x (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{-15 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(26 d^3+22 d^2 e x-17 d e^2 x^2-16 e^3 x^3\right)}{(d-e x) (d+e x)^3}+15 \log (x)}{15 d^5}","\frac{2 (d-e x)}{5 d \left(d^2-e^2 x^2\right)^{5/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^5}+\frac{5 d-8 e x}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(26*d^3 + 22*d^2*e*x - 17*d*e^2*x^2 - 16*e^3*x^3))/((d - e*x)*(d + e*x)^3) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^5)","A",1
177,1,112,146,0.1080363,"\int \frac{1}{x^2 (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{30 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(15 d^4+76 d^3 e x+32 d^2 e^2 x^2-82 d e^3 x^3-56 e^4 x^4\right)}{x (e x-d) (d+e x)^3}-30 e \log (x)}{15 d^6}","-\frac{2 e (d-e x)}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{e (30 d-41 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^6}-\frac{e (10 d-13 e x)}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(15*d^4 + 76*d^3*e*x + 32*d^2*e^2*x^2 - 82*d*e^3*x^3 - 56*e^4*x^4))/(x*(-d + e*x)*(d + e*x)^3) - 30*e*Log[x] + 30*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^6)","A",1
178,1,127,183,0.1311561,"\int \frac{1}{x^3 (d+e x)^2 \left(d^2-e^2 x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]","\frac{-45 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(5 d^5-10 d^4 e x-94 d^3 e^2 x^2-58 d^2 e^3 x^3+83 d e^4 x^4+64 e^5 x^5\right)}{x^2 (e x-d) (d+e x)^3}+45 e^2 \log (x)}{10 d^7}","\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^7}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{2 e^2 (d-e x)}{5 d^3 \left(d^2-e^2 x^2\right)^{5/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(5*d^5 - 10*d^4*e*x - 94*d^3*e^2*x^2 - 58*d^2*e^3*x^3 + 83*d*e^4*x^4 + 64*e^5*x^5))/(x^2*(-d + e*x)*(d + e*x)^3) + 45*e^2*Log[x] - 45*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(10*d^7)","A",1
179,1,98,177,0.1926409,"\int \frac{x^5}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^5/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","\frac{195 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(304 d^4+717 d^3 e x+479 d^2 e^2 x^2+45 d e^3 x^3-15 e^4 x^4\right)}{(d+e x)^3}}{30 e^6}","\frac{127 d^2 (d-e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{13 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^6}-\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d-e x)^3}{5 e^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{23 d^3 (d-e x)^2}{15 e^6 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(304*d^4 + 717*d^3*e*x + 479*d^2*e^2*x^2 + 45*d*e^3*x^3 - 15*e^4*x^4))/(d + e*x)^3 + 195*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^6)","A",1
180,1,85,146,0.1502239,"\int \frac{x^4}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^4/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{15 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(24 d^3+57 d^2 e x+39 d e^2 x^2+5 e^3 x^3\right)}{(d+e x)^3}}{5 e^5}","\frac{6 d^2 (d-e x)^2}{5 e^5 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{24 d (d-e x)}{5 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{e^5}-\frac{3 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}-\frac{d^3 (d-e x)^3}{5 e^5 \left(d^2-e^2 x^2\right)^{5/2}}",1,"-1/5*((Sqrt[d^2 - e^2*x^2]*(24*d^3 + 57*d^2*e*x + 39*d*e^2*x^2 + 5*e^3*x^3))/(d + e*x)^3 + 15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^5","A",1
181,1,73,120,0.113253,"\int \frac{x^3}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^3/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(22 d^2+51 d e x+32 e^2 x^2\right)}{(d+e x)^3}+15 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{15 e^4}","\frac{d^2 (d-e x)^3}{5 e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{13 d (d-e x)^2}{15 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{32 (d-e x)}{15 e^4 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^4}",1,"((Sqrt[d^2 - e^2*x^2]*(22*d^2 + 51*d*e*x + 32*e^2*x^2))/(d + e*x)^3 + 15*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^4)","A",1
182,1,52,95,0.061421,"\int \frac{x^2}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x^2/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(2 d^2+6 d e x+7 e^2 x^2\right)}{15 d e^3 (d+e x)^3}","-\frac{d \sqrt{d^2-e^2 x^2}}{5 e^3 (d+e x)^3}+\frac{8 \sqrt{d^2-e^2 x^2}}{15 e^3 (d+e x)^2}-\frac{7 \sqrt{d^2-e^2 x^2}}{15 d e^3 (d+e x)}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(2*d^2 + 6*d*e*x + 7*e^2*x^2))/(d*e^3*(d + e*x)^3)","A",1
183,1,49,97,0.0486484,"\int \frac{x}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[x/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(d^2+3 d e x+e^2 x^2\right)}{5 d^2 e^2 (d+e x)^3}","-\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3}",1,"-1/5*(Sqrt[d^2 - e^2*x^2]*(d^2 + 3*d*e*x + e^2*x^2))/(d^2*e^2*(d + e*x)^3)","A",1
184,1,52,100,0.0290308,"\int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(7 d^2+6 d e x+2 e^2 x^2\right)}{15 d^3 e (d+e x)^3}","-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(7*d^2 + 6*d*e*x + 2*e^2*x^2))/(d^3*e*(d + e*x)^3)","A",1
185,1,76,115,0.120064,"\int \frac{1}{x (d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x*(d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(32 d^2+51 d e x+22 e^2 x^2\right)}{(d+e x)^3}-15 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+15 \log (x)}{15 d^4}","\frac{5 d-11 e x}{15 d^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{4 (d-e x)}{5 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{15 d-22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^4}",1,"((Sqrt[d^2 - e^2*x^2]*(32*d^2 + 51*d*e*x + 22*e^2*x^2))/(d + e*x)^3 + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^4)","A",1
186,1,92,146,0.1768748,"\int \frac{1}{x^2 (d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x^2*(d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{-15 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(5 d^3+39 d^2 e x+57 d e^2 x^2+24 e^3 x^3\right)}{x (d+e x)^3}+15 e \log (x)}{5 d^5}","-\frac{4 e (d-e x)}{5 d \left(d^2-e^2 x^2\right)^{5/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{3 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^5}-\frac{e (5 d-7 e x)}{5 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/5*((Sqrt[d^2 - e^2*x^2]*(5*d^3 + 39*d^2*e*x + 57*d*e^2*x^2 + 24*e^3*x^3))/(x*(d + e*x)^3) + 15*e*Log[x] - 15*e*Log[d + Sqrt[d^2 - e^2*x^2]])/d^5","A",1
187,1,107,183,0.1728728,"\int \frac{1}{x^3 (d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[1/(x^3*(d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","\frac{-195 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(-15 d^4+45 d^3 e x+479 d^2 e^2 x^2+717 d e^3 x^3+304 e^4 x^4\right)}{x^2 (d+e x)^3}+195 e^2 \log (x)}{30 d^6}","\frac{4 e^2 (d-e x)}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{e^2 (90 d-107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d-31 e x)}{15 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(-15*d^4 + 45*d^3*e*x + 479*d^2*e^2*x^2 + 717*d*e^3*x^3 + 304*e^4*x^4))/(x^2*(d + e*x)^3) + 195*e^2*Log[x] - 195*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^6)","A",1
188,1,109,204,0.1844219,"\int \frac{x^5 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","\frac{270 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right)}{(d+e x)^3}}{15 e^6}","\frac{10 d^2 (d-e x)^2}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{59 d^2 \sqrt{d^2-e^2 x^2}}{3 e^6}-\frac{2 d x \sqrt{d^2-e^2 x^2}}{e^5}+\frac{x^2 \sqrt{d^2-e^2 x^2}}{3 e^4}+\frac{d^4 (d-e x)^4}{5 e^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8 d^3 (d-e x)^3}{5 e^6 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{18 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^6}",1,"((Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3 - 15*d*e^4*x^4 + 5*e^5*x^5))/(d + e*x)^3 + 270*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6)","A",1
189,1,98,160,0.179257,"\int \frac{x^4 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[(x^4*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","-\frac{285 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(448 d^4+1059 d^3 e x+713 d^2 e^2 x^2+75 d e^3 x^3-15 e^4 x^4\right)}{(d+e x)^3}}{30 e^5}","\frac{19 d^2 (d-e x)^3}{15 e^5 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^5}-\frac{d^3 (d-e x)^4}{5 e^5 \left(d^2-e^2 x^2\right)^{5/2}}",1,"-1/30*((Sqrt[d^2 - e^2*x^2]*(448*d^4 + 1059*d^3*e*x + 713*d^2*e^2*x^2 + 75*d*e^3*x^3 - 15*e^4*x^4))/(d + e*x)^3 + 285*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^5","A",1
190,1,85,148,0.1303057,"\int \frac{x^3 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[(x^3*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","\frac{60 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(94 d^3+222 d^2 e x+149 d e^2 x^2+15 e^3 x^3\right)}{(d+e x)^3}}{15 e^4}","\frac{d^2 \left(d^2-e^2 x^2\right)^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left(d^2-e^2 x^2\right)^{3/2}}{15 e^4 (d+e x)^3}+\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^4}",1,"((Sqrt[d^2 - e^2*x^2]*(94*d^3 + 222*d^2*e*x + 149*d*e^2*x^2 + 15*e^3*x^3))/(d + e*x)^3 + 60*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^4)","A",1
191,1,73,115,0.1217708,"\int \frac{x^2 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[(x^2*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","-\frac{\frac{\sqrt{d^2-e^2 x^2} \left(8 d^2+19 d e x+13 e^2 x^2\right)}{(d+e x)^3}+5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{5 e^3}","\frac{3 \left(d^2-e^2 x^2\right)^{3/2}}{5 e^3 (d+e x)^3}-\frac{d \left(d^2-e^2 x^2\right)^{3/2}}{5 e^3 (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}",1,"-1/5*((Sqrt[d^2 - e^2*x^2]*(8*d^2 + 19*d*e*x + 13*e^2*x^2))/(d + e*x)^3 + 5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^3","A",1
192,1,50,64,0.0494008,"\int \frac{x \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[(x*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","-\frac{\left(d^2+3 d e x-4 e^2 x^2\right) \sqrt{d^2-e^2 x^2}}{15 d e^2 (d+e x)^3}","\frac{\left(d^2-e^2 x^2\right)^{3/2}}{5 e^2 (d+e x)^4}-\frac{4 \left(d^2-e^2 x^2\right)^{3/2}}{15 d e^2 (d+e x)^3}",1,"-1/15*((d^2 + 3*d*e*x - 4*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(d*e^2*(d + e*x)^3)","A",1
193,1,51,67,0.0281486,"\int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(d + e*x)^4,x]","\frac{\sqrt{d^2-e^2 x^2} \left(-4 d^2+3 d e x+e^2 x^2\right)}{15 d^2 e (d+e x)^3}","-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{15 d^2 e (d+e x)^3}-\frac{\left(d^2-e^2 x^2\right)^{3/2}}{5 d e (d+e x)^4}",1,"(Sqrt[d^2 - e^2*x^2]*(-4*d^2 + 3*d*e*x + e^2*x^2))/(15*d^2*e*(d + e*x)^3)","A",1
194,1,76,110,0.1367741,"\int \frac{\sqrt{d^2-e^2 x^2}}{x (d+e x)^4} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)^4),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(13 d^2+19 d e x+8 e^2 x^2\right)}{(d+e x)^3}-5 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+5 \log (x)}{5 d^3}","-\frac{4 e x}{5 d \left(d^2-e^2 x^2\right)^{3/2}}+\frac{8 d (d-e x)}{5 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{5 d-8 e x}{5 d^3 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^3}",1,"((Sqrt[d^2 - e^2*x^2]*(13*d^2 + 19*d*e*x + 8*e^2*x^2))/(d + e*x)^3 + 5*Log[x] - 5*Log[d + Sqrt[d^2 - e^2*x^2]])/(5*d^3)","A",1
195,1,92,143,0.2091234,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^2 (d+e x)^4} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^2*(d + e*x)^4),x]","-\frac{-60 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(15 d^3+149 d^2 e x+222 d e^2 x^2+94 e^3 x^3\right)}{x (d+e x)^3}+60 e \log (x)}{15 d^4}","-\frac{4 e (5 d-8 e x)}{15 d^2 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{8 e (d-e x)}{5 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{e (60 d-79 e x)}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{4 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^4}",1,"-1/15*((Sqrt[d^2 - e^2*x^2]*(15*d^3 + 149*d^2*e*x + 222*d*e^2*x^2 + 94*e^3*x^3))/(x*(d + e*x)^3) + 60*e*Log[x] - 60*e*Log[d + Sqrt[d^2 - e^2*x^2]])/d^4","A",1
196,1,107,183,0.2290847,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^3*(d + e*x)^4),x]","\frac{-285 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(-15 d^4+75 d^3 e x+713 d^2 e^2 x^2+1059 d e^3 x^3+448 e^4 x^4\right)}{x^2 (d+e x)^3}+285 e^2 \log (x)}{30 d^5}","\frac{8 e^2 (d-e x)}{5 d \left(d^2-e^2 x^2\right)^{5/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{19 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^5}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left(d^2-e^2 x^2\right)^{3/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(-15*d^4 + 75*d^3*e*x + 713*d^2*e^2*x^2 + 1059*d*e^3*x^3 + 448*e^4*x^4))/(x^2*(d + e*x)^3) + 285*e^2*Log[x] - 285*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^5)","A",1
197,1,118,210,0.2661879,"\int \frac{\sqrt{d^2-e^2 x^2}}{x^4 (d+e x)^4} \, dx","Integrate[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)^4),x]","-\frac{-270 e^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(5 d^5-15 d^4 e x+70 d^3 e^2 x^2+674 d^2 e^3 x^3+1002 d e^4 x^4+424 e^5 x^5\right)}{x^3 (d+e x)^3}+270 e^3 \log (x)}{15 d^6}","-\frac{8 e^3 (d-e x)}{5 d^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{29 e^2 \sqrt{d^2-e^2 x^2}}{3 d^6 x}-\frac{e^3 (80 d-93 e x)}{5 d^6 \sqrt{d^2-e^2 x^2}}+\frac{18 e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^6}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^5 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^4 x^3}-\frac{4 e^3 (5 d-6 e x)}{5 d^4 \left(d^2-e^2 x^2\right)^{3/2}}",1,"-1/15*((Sqrt[d^2 - e^2*x^2]*(5*d^5 - 15*d^4*e*x + 70*d^3*e^2*x^2 + 674*d^2*e^3*x^3 + 1002*d*e^4*x^4 + 424*e^5*x^5))/(x^3*(d + e*x)^3) + 270*e^3*Log[x] - 270*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/d^6","A",1
198,1,131,252,0.2320018,"\int \frac{x^5 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(x^5*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]","\frac{1365 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(2144 d^7+779 d^6 e x-293 d^5 e^2 x^2+162 d^4 e^3 x^3-106 d^3 e^4 x^4+76 d^2 e^5 x^5-44 d e^6 x^6+12 e^7 x^7\right)}{d+e x}}{84 e^6}","\frac{1}{7} x^6 \sqrt{d^2-e^2 x^2}-\frac{2 d x^5 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{11 d^2 x^4 \sqrt{d^2-e^2 x^2}}{7 e^2}+\frac{65 d^7 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{4 e^6}+\frac{515 d^6 \sqrt{d^2-e^2 x^2}}{21 e^6}-\frac{49 d^5 x \sqrt{d^2-e^2 x^2}}{4 e^5}+\frac{d^4 (d-e x)^4}{e^6 \sqrt{d^2-e^2 x^2}}+\frac{121 d^4 x^2 \sqrt{d^2-e^2 x^2}}{21 e^4}-\frac{17 d^3 x^3 \sqrt{d^2-e^2 x^2}}{6 e^3}",1,"((Sqrt[d^2 - e^2*x^2]*(2144*d^7 + 779*d^6*e*x - 293*d^5*e^2*x^2 + 162*d^4*e^3*x^3 - 106*d^3*e^4*x^4 + 76*d^2*e^5*x^5 - 44*d*e^6*x^6 + 12*e^7*x^7))/(d + e*x) + 1365*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(84*e^6)","A",1
199,1,125,224,0.1632142,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]","\frac{\sqrt{d^2-e^2 x^2} \left(-5632 d^6-2047 d^5 e x+769 d^4 e^2 x^2-426 d^3 e^3 x^3+278 d^2 e^4 x^4-152 d e^5 x^5+40 e^6 x^6\right)-3585 d^6 (d+e x) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{240 e^5 (d+e x)}","\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{4 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{47 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{239 d^6 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{16 e^5}-\frac{337 d^5 \sqrt{d^2-e^2 x^2}}{15 e^5}+\frac{175 d^4 x \sqrt{d^2-e^2 x^2}}{16 e^4}-\frac{d^3 (d-e x)^4}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{71 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(-5632*d^6 - 2047*d^5*e*x + 769*d^4*e^2*x^2 - 426*d^3*e^3*x^3 + 278*d^2*e^4*x^4 - 152*d*e^5*x^5 + 40*e^6*x^6) - 3585*d^6*(d + e*x)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^5*(d + e*x))","A",1
200,1,109,192,0.1355404,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]","\frac{135 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+\frac{\sqrt{d^2-e^2 x^2} \left(212 d^5+77 d^4 e x-29 d^3 e^2 x^2+16 d^2 e^3 x^3-8 d e^4 x^4+2 e^5 x^5\right)}{d+e x}}{10 e^4}","\frac{18 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}+\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{e}+\frac{d^2 (d-e x)^4}{e^4 \sqrt{d^2-e^2 x^2}}+\frac{27 d^5 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^4}+\frac{101 d^4 \sqrt{d^2-e^2 x^2}}{5 e^4}-\frac{19 d^3 x \sqrt{d^2-e^2 x^2}}{2 e^3}",1,"((Sqrt[d^2 - e^2*x^2]*(212*d^5 + 77*d^4*e*x - 29*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 8*d*e^4*x^4 + 2*e^5*x^5))/(d + e*x) + 135*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(10*e^4)","A",1
201,1,103,182,0.1186725,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]","\sqrt{d^2-e^2 x^2} \left(-\frac{8 d^4}{e^3 (d+e x)}-\frac{32 d^3}{3 e^3}+\frac{31 d^2 x}{8 e^2}-\frac{4 d x^2}{3 e}+\frac{x^3}{4}\right)-\frac{95 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}","-\frac{95 d^2 (d-e x) \sqrt{d^2-e^2 x^2}}{24 e^3}-\frac{19 d (d-e x)^2 \sqrt{d^2-e^2 x^2}}{12 e^3}-\frac{d (d-e x)^4}{e^3 \sqrt{d^2-e^2 x^2}}-\frac{(d-e x)^3 \sqrt{d^2-e^2 x^2}}{4 e^3}-\frac{95 d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}-\frac{95 d^3 \sqrt{d^2-e^2 x^2}}{8 e^3}",1,"Sqrt[d^2 - e^2*x^2]*((-32*d^3)/(3*e^3) + (31*d^2*x)/(8*e^2) - (4*d*x^2)/(3*e) + x^3/4 - (8*d^4)/(e^3*(d + e*x))) - (95*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)","A",1
202,1,83,130,0.1014493,"\int \frac{x \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(x*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^4,x]","\frac{1}{3} \sqrt{d^2-e^2 x^2} \left(\frac{24 d^3}{e^2 (d+e x)}+\frac{23 d^2}{e^2}-\frac{6 d x}{e}+x^2\right)+\frac{10 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^2}","\frac{\left(d^2-e^2 x^2\right)^{7/2}}{e^2 (d+e x)^4}+\frac{8 \left(d^2-e^2 x^2\right)^{5/2}}{e^2 (d+e x)^2}+\frac{20 \left(d^2-e^2 x^2\right)^{3/2}}{3 e^2}+\frac{10 d x \sqrt{d^2-e^2 x^2}}{e}+\frac{10 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^2}",1,"(Sqrt[d^2 - e^2*x^2]*((23*d^2)/e^2 - (6*d*x)/e + x^2 + (24*d^3)/(e^2*(d + e*x))))/3 + (10*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2","A",1
203,1,75,113,0.0647685,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4,x]","\sqrt{d^2-e^2 x^2} \left(-\frac{8 d^2}{e (d+e x)}-\frac{4 d}{e}+\frac{x}{2}\right)-\frac{15 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}","-\frac{2 \left(d^2-e^2 x^2\right)^{5/2}}{e (d+e x)^3}-\frac{5 \left(d^2-e^2 x^2\right)^{3/2}}{2 e (d+e x)}-\frac{15 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{15 d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e}",1,"Sqrt[d^2 - e^2*x^2]*((-4*d)/e + x/2 - (8*d^2)/(e*(d + e*x))) - (15*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)","A",1
204,1,79,89,0.1472895,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)^4),x]","\sqrt{d^2-e^2 x^2} \left(\frac{8 d}{d+e x}+1\right)-d \log \left(\sqrt{d^2-e^2 x^2}+d\right)+4 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+d \log (x)","\frac{8 d (d-e x)}{\sqrt{d^2-e^2 x^2}}+\sqrt{d^2-e^2 x^2}+4 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-d \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"Sqrt[d^2 - e^2*x^2]*(1 + (8*d)/(d + e*x)) + 4*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + d*Log[x] - d*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
205,1,84,94,0.1870945,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^2 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)^4),x]","\sqrt{d^2-e^2 x^2} \left(-\frac{8 e}{d+e x}-\frac{1}{x}\right)+4 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)-e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-4 e \log (x)","-\frac{8 e (d-e x)}{\sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{x}-e \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)+4 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)",1,"Sqrt[d^2 - e^2*x^2]*(-x^(-1) - (8*e)/(d + e*x)) - e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 4*e*Log[x] + 4*e*Log[d + Sqrt[d^2 - e^2*x^2]]","A",1
206,1,85,110,0.2212147,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^3 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^4),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(-d^2+7 d e x+24 e^2 x^2\right)}{x^2 (d+e x)}-15 e^2 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+15 e^2 \log (x)}{2 d}","\frac{8 e^2 (d-e x)}{d \sqrt{d^2-e^2 x^2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d x}-\frac{\sqrt{d^2-e^2 x^2}}{2 x^2}-\frac{15 e^2 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d}",1,"((Sqrt[d^2 - e^2*x^2]*(-d^2 + 7*d*e*x + 24*e^2*x^2))/(x^2*(d + e*x)) + 15*e^2*Log[x] - 15*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(2*d)","A",1
207,1,94,137,0.2469396,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^4 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)^4),x]","-\frac{-30 e^3 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(d^3-5 d^2 e x+17 d e^2 x^2+47 e^3 x^3\right)}{x^3 (d+e x)}+30 e^3 \log (x)}{3 d^2}","-\frac{23 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{8 e^3 (d-e x)}{d^2 \sqrt{d^2-e^2 x^2}}+\frac{10 e^3 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^2}",1,"-1/3*((Sqrt[d^2 - e^2*x^2]*(d^3 - 5*d^2*e*x + 17*d*e^2*x^2 + 47*e^3*x^3))/(x^3*(d + e*x)) + 30*e^3*Log[x] - 30*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/d^2","A",1
208,1,107,170,0.2679779,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^5 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x]","\frac{-285 e^4 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(-6 d^4+26 d^3 e x-61 d^2 e^2 x^2+163 d e^3 x^3+448 e^4 x^4\right)}{x^4 (d+e x)}+285 e^4 \log (x)}{24 d^3}","-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}+\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{95 e^4 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{8 d^3}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}",1,"((Sqrt[d^2 - e^2*x^2]*(-6*d^4 + 26*d^3*e*x - 61*d^2*e^2*x^2 + 163*d*e^3*x^3 + 448*e^4*x^4))/(x^4*(d + e*x)) + 285*e^4*Log[x] - 285*e^4*Log[d + Sqrt[d^2 - e^2*x^2]])/(24*d^3)","A",1
209,1,118,196,0.3310324,"\int \frac{\left(d^2-e^2 x^2\right)^{5/2}}{x^6 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]","-\frac{-135 e^5 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right)}{x^5 (d+e x)}+135 e^5 \log (x)}{10 d^4}","-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}+\frac{27 e^5 \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{2 d^4}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}",1,"-1/10*((Sqrt[d^2 - e^2*x^2]*(2*d^5 - 8*d^4*e*x + 16*d^3*e^2*x^2 - 29*d^2*e^3*x^3 + 77*d*e^4*x^4 + 212*e^5*x^5))/(x^5*(d + e*x)) + 135*e^5*Log[x] - 135*e^5*Log[d + Sqrt[d^2 - e^2*x^2]])/d^4","A",1
210,1,50,95,0.1137668,"\int \frac{x^2 \sqrt{1-a^2 x^2}}{(1-a x)^4} \, dx","Integrate[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^4,x]","\frac{\frac{\left(-13 a^2 x^2+19 a x-8\right) \sqrt{1-a^2 x^2}}{(a x-1)^3}-5 \sin ^{-1}(a x)}{5 a^3}","-\frac{\sin ^{-1}(a x)}{a^3}-\frac{3 \left(1-a^2 x^2\right)^{3/2}}{5 a^3 (1-a x)^3}+\frac{\left(1-a^2 x^2\right)^{3/2}}{5 a^3 (1-a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)}",1,"(((-8 + 19*a*x - 13*a^2*x^2)*Sqrt[1 - a^2*x^2])/(-1 + a*x)^3 - 5*ArcSin[a*x])/(5*a^3)","A",1
211,1,50,88,0.0715097,"\int \frac{x^2 \sqrt{1-a^2 x^2}}{(1-a x)^5} \, dx","Integrate[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^5,x]","\frac{\sqrt{1-a^2 x^2} \left(23 a^3 x^3+13 a^2 x^2-8 a x+2\right)}{105 a^3 (a x-1)^4}","\frac{23 \left(1-a^2 x^2\right)^{3/2}}{105 a^3 (1-a x)^3}-\frac{12 \left(1-a^2 x^2\right)^{3/2}}{35 a^3 (1-a x)^4}+\frac{\left(1-a^2 x^2\right)^{3/2}}{7 a^3 (1-a x)^5}",1,"(Sqrt[1 - a^2*x^2]*(2 - 8*a*x + 13*a^2*x^2 + 23*a^3*x^3))/(105*a^3*(-1 + a*x)^4)","A",1
212,1,137,209,0.1762532,"\int \frac{x^3}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(90 d^9+360 d^8 e x+315 d^7 e^2 x^2-540 d^6 e^3 x^3+160 d^5 e^4 x^4+776 d^4 e^5 x^5+384 d^3 e^6 x^6-224 d^2 e^7 x^7-256 d e^8 x^8-64 e^9 x^9\right)}{5005 d^7 e^4 (d-e x)^3 (d+e x)^7}","\frac{d^2}{13 e^4 (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}}-\frac{32 x}{5005 d^5 e^3 \left(d^2-e^2 x^2\right)^{3/2}}-\frac{24 x}{5005 d^3 e^3 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(90*d^9 + 360*d^8*e*x + 315*d^7*e^2*x^2 - 540*d^6*e^3*x^3 + 160*d^5*e^4*x^4 + 776*d^4*e^5*x^5 + 384*d^3*e^6*x^6 - 224*d^2*e^7*x^7 - 256*d*e^8*x^8 - 64*e^9*x^9))/(5005*d^7*e^4*(d - e*x)^3*(d + e*x)^7)","A",1
213,1,137,209,0.1000317,"\int \frac{x^2}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[x^2/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(200 d^9+800 d^8 e x+700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+448 d e^8 x^8+112 e^9 x^9\right)}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7}","-\frac{d}{13 e^3 (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}}+\frac{56 x}{6435 d^6 e^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{14 x}{2145 d^4 e^2 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(200*d^9 + 800*d^8*e*x + 700*d^7*e^2*x^2 + 945*d^6*e^3*x^3 - 280*d^5*e^4*x^4 - 1358*d^4*e^5*x^5 - 672*d^3*e^6*x^6 + 392*d^2*e^7*x^7 + 448*d*e^8*x^8 + 112*e^9*x^9))/(6435*d^8*e^3*(d - e*x)^3*(d + e*x)^7)","A",1
214,1,137,211,0.0883516,"\int \frac{x}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[x/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-5 d^9-20 d^8 e x+3200 d^7 e^2 x^2+4320 d^6 e^3 x^3-1280 d^5 e^4 x^4-6208 d^4 e^5 x^5-3072 d^3 e^6 x^6+1792 d^2 e^7 x^7+2048 d e^8 x^8+512 e^9 x^9\right)}{6435 d^9 e^2 (d-e x)^3 (d+e x)^7}","-\frac{32}{1287 d^2 e^2 (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{4}{143 d e^2 (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{1}{13 e^2 (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{512 x}{6435 d^9 e \sqrt{d^2-e^2 x^2}}+\frac{256 x}{6435 d^7 e \left(d^2-e^2 x^2\right)^{3/2}}+\frac{64 x}{2145 d^5 e \left(d^2-e^2 x^2\right)^{5/2}}-\frac{32}{1287 d^3 e^2 (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-5*d^9 - 20*d^8*e*x + 3200*d^7*e^2*x^2 + 4320*d^6*e^3*x^3 - 1280*d^5*e^4*x^4 - 6208*d^4*e^5*x^5 - 3072*d^3*e^6*x^6 + 1792*d^2*e^7*x^7 + 2048*d*e^8*x^8 + 512*e^9*x^9))/(6435*d^9*e^2*(d - e*x)^3*(d + e*x)^7)","A",1
215,1,137,205,0.065566,"\int \frac{1}{(d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(-180 d^9-5 d^8 e x+800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7+512 d e^8 x^8+128 e^9 x^9\right)}{715 d^{10} e (d-e x)^3 (d+e x)^7}","-\frac{9}{143 d^2 e (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{48 x}{715 d^6 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left(d^2-e^2 x^2\right)^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(-180*d^9 - 5*d^8*e*x + 800*d^7*e^2*x^2 + 1080*d^6*e^3*x^3 - 320*d^5*e^4*x^4 - 1552*d^4*e^5*x^5 - 768*d^3*e^6*x^6 + 448*d^2*e^7*x^7 + 512*d*e^8*x^8 + 128*e^9*x^9))/(715*d^10*e*(d - e*x)^3*(d + e*x)^7)","A",1
216,1,161,234,0.1722652,"\int \frac{1}{x (d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/(x*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","\frac{-4095 \log \left(\sqrt{d^2-e^2 x^2}+d\right)+\frac{\sqrt{d^2-e^2 x^2} \left(9839 d^9+22976 d^8 e x-4466 d^7 e^2 x^2-56304 d^6 e^3 x^3-34156 d^5 e^4 x^4+40240 d^4 e^5 x^5+45735 d^3 e^6 x^6-1540 d^2 e^7 x^7-16385 d e^8 x^8-5120 e^9 x^9\right)}{(d-e x)^3 (d+e x)^7}+4095 \log (x)}{4095 d^{11}}","-\frac{4 e x}{13 d \left(d^2-e^2 x^2\right)^{11/2}}+\frac{8 d (d-e x)}{13 \left(d^2-e^2 x^2\right)^{13/2}}+\frac{819 d-1024 e x}{819 d^{11} \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^{11}}+\frac{273 d-512 e x}{819 d^9 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{273 d-640 e x}{1365 d^7 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{117 d-320 e x}{819 d^5 \left(d^2-e^2 x^2\right)^{7/2}}+\frac{13 d-40 e x}{117 d^3 \left(d^2-e^2 x^2\right)^{9/2}}",1,"((Sqrt[d^2 - e^2*x^2]*(9839*d^9 + 22976*d^8*e*x - 4466*d^7*e^2*x^2 - 56304*d^6*e^3*x^3 - 34156*d^5*e^4*x^4 + 40240*d^4*e^5*x^5 + 45735*d^3*e^6*x^6 - 1540*d^2*e^7*x^7 - 16385*d*e^8*x^8 - 5120*e^9*x^9))/((d - e*x)^3*(d + e*x)^7) + 4095*Log[x] - 4095*Log[d + Sqrt[d^2 - e^2*x^2]])/(4095*d^11)","A",1
217,1,183,271,0.2104754,"\int \frac{1}{x^2 (d+e x)^4 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[1/(x^2*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]","-\frac{4 e \log (x)}{d^{12}}+\frac{4 e \log \left(\sqrt{d^2-e^2 x^2}+d\right)}{d^{12}}+\frac{\sqrt{d^2-e^2 x^2} \left(45045 d^{10}+546316 d^9 e x+1014094 d^8 e^2 x^2-700504 d^7 e^3 x^3-3157776 d^6 e^4 x^4-1301264 d^5 e^5 x^5+2748320 d^4 e^6 x^6+2496180 d^3 e^7 x^7-350000 d^2 e^8 x^8-1043500 d e^9 x^9-305920 e^{10} x^{10}\right)}{45045 d^{12} x (e x-d)^3 (d+e x)^7}","-\frac{4 e (13 d-24 e x)}{143 d^2 \left(d^2-e^2 x^2\right)^{11/2}}-\frac{8 e (d-e x)}{13 \left(d^2-e^2 x^2\right)^{13/2}}-\frac{e (36036 d-52175 e x)}{9009 d^{12} \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^{12} x}+\frac{4 e \tanh ^{-1}\left(\frac{\sqrt{d^2-e^2 x^2}}{d}\right)}{d^{12}}-\frac{e (12012 d-21583 e x)}{9009 d^{10} \left(d^2-e^2 x^2\right)^{3/2}}-\frac{e (12012 d-23225 e x)}{15015 d^8 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{e (5148 d-10111 e x)}{9009 d^6 \left(d^2-e^2 x^2\right)^{7/2}}-\frac{e (572 d-1103 e x)}{1287 d^4 \left(d^2-e^2 x^2\right)^{9/2}}",1,"(Sqrt[d^2 - e^2*x^2]*(45045*d^10 + 546316*d^9*e*x + 1014094*d^8*e^2*x^2 - 700504*d^7*e^3*x^3 - 3157776*d^6*e^4*x^4 - 1301264*d^5*e^5*x^5 + 2748320*d^4*e^6*x^6 + 2496180*d^3*e^7*x^7 - 350000*d^2*e^8*x^8 - 1043500*d*e^9*x^9 - 305920*e^10*x^10))/(45045*d^12*x*(-d + e*x)^3*(d + e*x)^7) - (4*e*Log[x])/d^12 + (4*e*Log[d + Sqrt[d^2 - e^2*x^2]])/d^12","A",1
218,1,93,102,0.0950118,"\int \frac{\sqrt{c-a c x} \sqrt{1-a^2 x^2}}{x^2} \, dx","Integrate[(Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/x^2,x]","\frac{\sqrt{1-a^2 x^2} \left(a \sqrt{c} x \tanh ^{-1}\left(\sqrt{c} \sqrt{\frac{a x+1}{c}}\right)-c (2 a x+1) \sqrt{\frac{a x+1}{c}}\right)}{x \sqrt{\frac{a x+1}{c}} \sqrt{c-a c x}}","-\frac{c^2 \left(1-a^2 x^2\right)^{3/2}}{x (c-a c x)^{3/2}}-\frac{a c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}+a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right)",1,"(Sqrt[1 - a^2*x^2]*(-(c*Sqrt[(1 + a*x)/c]*(1 + 2*a*x)) + a*Sqrt[c]*x*ArcTanh[Sqrt[c]*Sqrt[(1 + a*x)/c]]))/(x*Sqrt[(1 + a*x)/c]*Sqrt[c - a*c*x])","A",1
219,1,67,39,0.0346458,"\int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[Sqrt[c - a*c*x]/(x*Sqrt[1 - a^2*x^2]),x]","-\frac{2 \sqrt{c} \sqrt{\frac{a x}{c}+\frac{1}{c}} \sqrt{c-a c x} \tanh ^{-1}\left(\sqrt{c} \sqrt{\frac{a x}{c}+\frac{1}{c}}\right)}{\sqrt{1-a^2 x^2}}","-2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right)",1,"(-2*Sqrt[c]*Sqrt[c^(-1) + (a*x)/c]*Sqrt[c - a*c*x]*ArcTanh[Sqrt[c]*Sqrt[c^(-1) + (a*x)/c]])/Sqrt[1 - a^2*x^2]","A",1
220,1,35,35,0.0151795,"\int \frac{\sqrt{1-a x}}{\sqrt{x}} \, dx","Integrate[Sqrt[1 - a*x]/Sqrt[x],x]","\sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}","\sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}",1,"Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]","A",1
221,1,35,35,0.0091835,"\int \frac{\sqrt{1-a^2 x^2}}{\sqrt{x} \sqrt{1+a x}} \, dx","Integrate[Sqrt[1 - a^2*x^2]/(Sqrt[x]*Sqrt[1 + a*x]),x]","\sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}","\sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}",1,"Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]","A",1
222,1,34,34,0.0120403,"\int \frac{\sqrt{1+a x}}{\sqrt{x}} \, dx","Integrate[Sqrt[1 + a*x]/Sqrt[x],x]","\sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}","\sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}",1,"Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]","A",1
223,1,34,34,0.0116205,"\int \frac{\sqrt{1-a^2 x^2}}{\sqrt{x} \sqrt{1-a x}} \, dx","Integrate[Sqrt[1 - a^2*x^2]/(Sqrt[x]*Sqrt[1 - a*x]),x]","\sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}","\sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{\sqrt{a}}",1,"Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]","A",1
224,1,49,63,0.019711,"\int \sqrt{x} \sqrt{1-a x} \, dx","Integrate[Sqrt[x]*Sqrt[1 - a*x],x]","\frac{\sqrt{a} \sqrt{x} \sqrt{1-a x} (2 a x-1)+\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{4 a^{3/2}}","\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a}",1,"(Sqrt[a]*Sqrt[x]*Sqrt[1 - a*x]*(-1 + 2*a*x) + ArcSin[Sqrt[a]*Sqrt[x]])/(4*a^(3/2))","A",1
225,1,49,63,0.0073704,"\int \frac{\sqrt{x} \sqrt{1-a^2 x^2}}{\sqrt{1+a x}} \, dx","Integrate[(Sqrt[x]*Sqrt[1 - a^2*x^2])/Sqrt[1 + a*x],x]","\frac{\sqrt{a} \sqrt{x} \sqrt{1-a x} (2 a x-1)+\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{4 a^{3/2}}","\frac{\sin ^{-1}\left(\sqrt{a} \sqrt{x}\right)}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a}",1,"(Sqrt[a]*Sqrt[x]*Sqrt[1 - a*x]*(-1 + 2*a*x) + ArcSin[Sqrt[a]*Sqrt[x]])/(4*a^(3/2))","A",1
226,1,199,250,0.1969217,"\int (g x)^m (d+e x)^3 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]","\frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \left(e x \left(\frac{3 d^2 \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{m+2}+e x \left(\frac{3 d \, _2F_1\left(-\frac{5}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)}{m+3}+\frac{e x \, _2F_1\left(-\frac{5}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)}{m+4}\right)\right)+\frac{d^3 \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{m+1}\right)}{\sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{e \left(d^2-e^2 x^2\right)^{7/2} (g x)^{m+2}}{g^2 (m+9)}-\frac{3 d \left(d^2-e^2 x^2\right)^{7/2} (g x)^{m+1}}{g (m+8)}+\frac{d^7 (4 m+11) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^6 e (4 m+29) \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) (m+9) \sqrt{1-\frac{e^2 x^2}{d^2}}}",1,"(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*((d^3*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((3*d^2*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F1[-5/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) + (e*x*Hypergeometric2F1[-5/2, (4 + m)/2, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/Sqrt[1 - (e^2*x^2)/d^2]","A",1
227,1,174,206,0.1085893,"\int (g x)^m (d+e x)^2 \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(g*x)^m*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2),x]","\frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \left(2 d (m+3) \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)+e (m+2) x \, _2F_1\left(-\frac{5}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{(m+1) (m+2) (m+3) \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{\left(d^2-e^2 x^2\right)^{7/2} (g x)^{m+1}}{g (m+8)}+\frac{d^6 (2 m+9) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{2 d^5 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}",1,"(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] + e*(2 + m)*x*Hypergeometric2F1[-5/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/((1 + m)*(2 + m)*(3 + m)*Sqrt[1 - (e^2*x^2)/d^2])","A",1
228,1,121,162,0.0502414,"\int (g x)^m (d+e x) \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(g*x)^m*(d + e*x)*(d^2 - e^2*x^2)^(5/2),x]","\frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \left(d (m+2) \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)\right)}{(m+1) (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^5 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^4 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}",1,"(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(d*(2 + m)*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2]))/((1 + m)*(2 + m)*Sqrt[1 - (e^2*x^2)/d^2])","A",1
229,1,78,80,0.0184605,"\int (g x)^m \left(d^2-e^2 x^2\right)^{5/2} \, dx","Integrate[(g*x)^m*(d^2 - e^2*x^2)^(5/2),x]","\frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right)}{(m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^4 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}",1,"(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, 1 + (1 + m)/2, (e^2*x^2)/d^2])/((1 + m)*Sqrt[1 - (e^2*x^2)/d^2])","A",1
230,1,122,163,0.0564954,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^{5/2}}{d+e x} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]","\frac{d^2 x \sqrt{d^2-e^2 x^2} (g x)^m \left(d (m+2) \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \, _2F_1\left(-\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right)\right)}{(m+1) (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^3 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{d^2 e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left(-\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}",1,"(d^2*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(-(e*(1 + m)*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m/2, (e^2*x^2)/d^2]) + d*(2 + m)*Hypergeometric2F1[-3/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2]))/((1 + m)*(2 + m)*Sqrt[1 - (e^2*x^2)/d^2])","A",1
231,1,173,204,0.110712,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^2} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^2,x]","\frac{x \sqrt{d^2-e^2 x^2} (g x)^m \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \left(2 d (m+3) \, _2F_1\left(-\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)-e (m+2) x \, _2F_1\left(-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{(m+1) (m+2) (m+3) \sqrt{1-\frac{e^2 x^2}{d^2}}}","-\frac{2 d e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left(-\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^2 (2 m+5) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+4) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{\left(d^2-e^2 x^2\right)^{3/2} (g x)^{m+1}}{g (m+4)}",1,"(x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[-1/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hypergeometric2F1[-1/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/((1 + m)*(2 + m)*(3 + m)*Sqrt[1 - (e^2*x^2)/d^2])","A",1
232,1,245,250,0.1758452,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^{5/2}}{(d+e x)^3} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^(5/2))/(d + e*x)^3,x]","\frac{x \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(d^3 \left(m^3+9 m^2+26 m+24\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \left(3 d^2 \left(m^2+7 m+12\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)+e (m+2) x \left(e (m+3) x \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)-3 d (m+4) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)\right)}{(m+1) (m+2) (m+3) (m+4) (d-e x) (d+e x)}","-\frac{d^2 e (4 m+11) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) (m+3) \sqrt{d^2-e^2 x^2}}+\frac{e \sqrt{d^2-e^2 x^2} (g x)^{m+2}}{g^2 (m+3)}-\frac{3 d \sqrt{d^2-e^2 x^2} (g x)^{m+1}}{g (m+2)}+\frac{d^3 (4 m+5) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+2) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2]*(d^3*(24 + 26*m + 9*m^2 + m^3)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(3*d^2*(12 + 7*m + m^2)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] + e*(2 + m)*x*(-3*d*(4 + m)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2] + e*(3 + m)*x*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, (e^2*x^2)/d^2]))))/((1 + m)*(2 + m)*(3 + m)*(4 + m)*(d - e*x)*(d + e*x))","A",1
233,1,199,213,0.1857834,"\int \frac{(g x)^m (d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((g*x)^m*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(e x \left(\frac{3 d^2 \, _2F_1\left(\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{m+2}+e x \left(\frac{3 d \, _2F_1\left(\frac{7}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)}{m+3}+\frac{e x \, _2F_1\left(\frac{7}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)}{m+4}\right)\right)+\frac{d^3 \, _2F_1\left(\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{m+1}\right)}{d^6 \sqrt{d^2-e^2 x^2}}","\frac{4 (d+e x) (g x)^{m+1}}{5 g \left(d^2-e^2 x^2\right)^{5/2}}+\frac{e (7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^4 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(1-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^3 g (m+1) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*((d^3*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((3*d^2*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F1[7/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) + (e*x*Hypergeometric2F1[7/2, (4 + m)/2, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/(d^6*Sqrt[d^2 - e^2*x^2])","A",1
234,1,174,216,0.1085232,"\int \frac{(g x)^m (d+e x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((g*x)^m*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \left(2 d (m+3) \, _2F_1\left(\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)+e (m+2) x \, _2F_1\left(\frac{7}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{d^6 (m+1) (m+2) (m+3) \sqrt{d^2-e^2 x^2}}","\frac{2 (d+e x) (g x)^{m+1}}{5 d g \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 e (3-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^5 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(3-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^4 g (m+1) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] + e*(2 + m)*x*Hypergeometric2F1[7/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/(d^6*(1 + m)*(2 + m)*(3 + m)*Sqrt[d^2 - e^2*x^2])","A",1
235,1,121,124,0.0531468,"\int \frac{(g x)^m (d+e x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((g*x)^m*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(d (m+2) \, _2F_1\left(\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \, _2F_1\left(\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)\right)}{d^6 (m+1) (m+2) \sqrt{d^2-e^2 x^2}}","\frac{e (g x)^{m+2} \, _2F_1\left(1,\frac{m-3}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{d^2 g^2 (m+2) \left(d^2-e^2 x^2\right)^{5/2}}+\frac{(g x)^{m+1} \, _2F_1\left(1,\frac{m-4}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d g (m+1) \left(d^2-e^2 x^2\right)^{5/2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d*(2 + m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2]))/(d^6*(1 + m)*(2 + m)*Sqrt[d^2 - e^2*x^2])","A",1
236,1,78,80,0.0176725,"\int \frac{(g x)^m}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(g*x)^m/(d^2 - e^2*x^2)^(7/2),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \, _2F_1\left(\frac{7}{2},\frac{m+1}{2};\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right)}{d^6 (m+1) \sqrt{d^2-e^2 x^2}}","\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d^6 g (m+1) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (1 + m)/2, 1 + (1 + m)/2, (e^2*x^2)/d^2])/(d^6*(1 + m)*Sqrt[d^2 - e^2*x^2])","A",1
237,1,122,163,0.0606686,"\int \frac{(g x)^m}{(d+e x) \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(g*x)^m/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(d (m+2) \, _2F_1\left(\frac{9}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \, _2F_1\left(\frac{9}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right)\right)}{d^8 (m+1) (m+2) \sqrt{d^2-e^2 x^2}}","\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{9}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d^7 g (m+1) \sqrt{d^2-e^2 x^2}}-\frac{e \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{9}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{d^8 g^2 (m+2) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(-(e*(1 + m)*x*Hypergeometric2F1[9/2, 1 + m/2, 2 + m/2, (e^2*x^2)/d^2]) + d*(2 + m)*Hypergeometric2F1[9/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2]))/(d^8*(1 + m)*(2 + m)*Sqrt[d^2 - e^2*x^2])","A",1
238,1,176,217,0.1228945,"\int \frac{(g x)^m}{(d+e x)^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(g*x)^m/((d + e*x)^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \left(2 d (m+3) \, _2F_1\left(\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)-e (m+2) x \, _2F_1\left(\frac{11}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{d^{10} (m+1) (m+2) (m+3) \sqrt{d^2-e^2 x^2}}","\frac{2 (d-e x) (g x)^{m+1}}{9 d g \left(d^2-e^2 x^2\right)^{9/2}}-\frac{2 e (7-m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{9}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{9 d^9 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-2 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{9}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{9 d^8 g (m+1) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[11/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hypergeometric2F1[11/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/(d^10*(1 + m)*(2 + m)*(3 + m)*Sqrt[d^2 - e^2*x^2])","A",1
239,1,200,214,0.2066408,"\int \frac{(g x)^m}{(d+e x)^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(g*x)^m/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left(e x \left(e x \left(\frac{3 d \, _2F_1\left(\frac{13}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)}{m+3}-\frac{e x \, _2F_1\left(\frac{13}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)}{m+4}\right)-\frac{3 d^2 \, _2F_1\left(\frac{13}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{m+2}\right)+\frac{d^3 \, _2F_1\left(\frac{13}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{m+1}\right)}{d^{12} \sqrt{d^2-e^2 x^2}}","\frac{4 (d-e x) (g x)^{m+1}}{11 g \left(d^2-e^2 x^2\right)^{11/2}}-\frac{e (25-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left(\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{11 d^{10} g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left(\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{11 d^9 g (m+1) \sqrt{d^2-e^2 x^2}}",1,"(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*((d^3*Hypergeometric2F1[13/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((-3*d^2*Hypergeometric2F1[13/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F1[13/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) - (e*x*Hypergeometric2F1[13/2, (4 + m)/2, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/(d^12*Sqrt[d^2 - e^2*x^2])","A",1
240,1,132,148,0.0907891,"\int x^5 (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^5*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(2 e^7 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)-\frac{7 d \left(d^2-e^2 x^2\right) \left(2 d^4+2 d^2 e^2 (p+1) x^2+e^4 \left(p^2+3 p+2\right) x^4\right)}{(p+1) (p+2) (p+3)}\right)}{14 e^6}","\frac{1}{7} e x^7 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d \left(d^2-e^2 x^2\right)^{p+3}}{2 e^6 (p+3)}-\frac{d^5 \left(d^2-e^2 x^2\right)^{p+1}}{2 e^6 (p+1)}+\frac{d^3 \left(d^2-e^2 x^2\right)^{p+2}}{e^6 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-7*d*(d^2 - e^2*x^2)*(2*d^4 + 2*d^2*e^2*(1 + p)*x^2 + e^4*(2 + 3*p + p^2)*x^4))/((1 + p)*(2 + p)*(3 + p)) + (2*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(14*e^6)","A",1
241,1,129,147,0.0816446,"\int x^4 (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^4*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{1}{10} \left(d^2-e^2 x^2\right)^p \left(2 d x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{5 \left(d^2-e^2 x^2\right) \left(2 d^4+2 d^2 e^2 (p+1) x^2+e^4 \left(p^2+3 p+2\right) x^4\right)}{e^5 (p+1) (p+2) (p+3)}\right)","\frac{1}{5} d x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)+\frac{d^2 \left(d^2-e^2 x^2\right)^{p+2}}{e^5 (p+2)}-\frac{\left(d^2-e^2 x^2\right)^{p+3}}{2 e^5 (p+3)}-\frac{d^4 \left(d^2-e^2 x^2\right)^{p+1}}{2 e^5 (p+1)}",1,"((d^2 - e^2*x^2)^p*((-5*(d^2 - e^2*x^2)*(2*d^4 + 2*d^2*e^2*(1 + p)*x^2 + e^4*(2 + 3*p + p^2)*x^4))/(e^5*(1 + p)*(2 + p)*(3 + p)) + (2*d*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/10","A",1
242,1,106,120,0.0757282,"\int x^3 (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^3*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(2 e^5 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{5 d \left(d^2-e^2 x^2\right) \left(d^2+e^2 (p+1) x^2\right)}{(p+1) (p+2)}\right)}{10 e^4}","\frac{1}{5} e x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)+\frac{d \left(d^2-e^2 x^2\right)^{p+2}}{2 e^4 (p+2)}-\frac{d^3 \left(d^2-e^2 x^2\right)^{p+1}}{2 e^4 (p+1)}",1,"((d^2 - e^2*x^2)^p*((-5*d*(d^2 - e^2*x^2)*(d^2 + e^2*(1 + p)*x^2))/((1 + p)*(2 + p)) + (2*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(10*e^4)","A",1
243,1,103,119,0.0739525,"\int x^2 (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^2*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{1}{6} \left(d^2-e^2 x^2\right)^p \left(2 d x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{3 \left(d^2-e^2 x^2\right) \left(d^2+e^2 (p+1) x^2\right)}{e^3 (p+1) (p+2)}\right)","\frac{1}{3} d x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d^2 \left(d^2-e^2 x^2\right)^{p+1}}{2 e^3 (p+1)}+\frac{\left(d^2-e^2 x^2\right)^{p+2}}{2 e^3 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-3*(d^2 - e^2*x^2)*(d^2 + e^2*(1 + p)*x^2))/(e^3*(1 + p)*(2 + p)) + (2*d*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/6","A",1
244,1,89,89,0.0412275,"\int x (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{1}{3} e x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 e^2 (p+1)}","\frac{1}{3} e x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 e^2 (p+1)}",1,"-1/2*(d*(d^2 - e^2*x^2)^(1 + p))/(e^2*(1 + p)) + (e*x^3*(d^2 - e^2*x^2)^p*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(1 - (e^2*x^2)/d^2)^p)","A",1
245,1,83,83,0.0459089,"\int (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(d + e*x)*(d^2 - e^2*x^2)^p,x]","d x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e (p+1)}","d x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e (p+1)}",1,"-1/2*(d^2 - e^2*x^2)^(1 + p)/(e*(1 + p)) + (d*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p","A",1
246,1,104,104,0.0355667,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^p}{x} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^p)/x,x]","e x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d (p+1)}","e x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d (p+1)}",1,"(e*x*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*d*(1 + p))","A",1
247,1,108,108,0.0485491,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^p)/x^2,x]","-\frac{d \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{e \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (p+1)}","-\frac{d \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{e \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (p+1)}",1,"-((d*(d^2 - e^2*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p)) - (e*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*d^2*(1 + p))","A",1
248,1,106,110,0.0496692,"\int \frac{(d+e x) \left(d^2-e^2 x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)*(d^2 - e^2*x^2)^p)/x^3,x]","\frac{1}{2} e \left(d^2-e^2 x^2\right)^p \left(\frac{e \left(e^2 x^2-d^2\right) \, _2F_1\left(2,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{d^3 (p+1)}-\frac{2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}\right)","-\frac{e \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{e^2 \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d^3 (p+1)}",1,"(e*(d^2 - e^2*x^2)^p*((-2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (e*(-d^2 + e^2*x^2)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(d^3*(1 + p))))/2","A",1
249,1,159,178,0.1275077,"\int x^5 (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^5*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{7 \left(d^2-e^2 x^2\right)^4}{p+4}-\frac{28 d^2 \left(d^2-e^2 x^2\right)^3}{p+3}+4 d e^7 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)-\frac{14 d^6 \left(d^2-e^2 x^2\right)}{p+1}+\frac{35 d^4 \left(d^2-e^2 x^2\right)^2}{p+2}\right)}{14 e^6}","\frac{2}{7} d e x^7 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)-\frac{2 d^2 \left(d^2-e^2 x^2\right)^{p+3}}{e^6 (p+3)}+\frac{\left(d^2-e^2 x^2\right)^{p+4}}{2 e^6 (p+4)}-\frac{d^6 \left(d^2-e^2 x^2\right)^{p+1}}{e^6 (p+1)}+\frac{5 d^4 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^6 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-14*d^6*(d^2 - e^2*x^2))/(1 + p) + (35*d^4*(d^2 - e^2*x^2)^2)/(2 + p) - (28*d^2*(d^2 - e^2*x^2)^3)/(3 + p) + (7*(d^2 - e^2*x^2)^4)/(4 + p) + (4*d*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(14*e^6)","A",1
250,1,186,185,0.1240243,"\int x^4 (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^4*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{1}{35} \left(d^2-e^2 x^2\right)^p \left(5 e^2 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)+7 d^2 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{35 d \left(d^2-e^2 x^2\right)^3}{e^5 (p+3)}-\frac{35 d^5 \left(d^2-e^2 x^2\right)}{e^5 (p+1)}+\frac{70 d^3 \left(d^2-e^2 x^2\right)^2}{e^5 (p+2)}\right)","\frac{2 d^2 (p+6) x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 (2 p+7)}-\frac{x^5 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+7}-\frac{d \left(d^2-e^2 x^2\right)^{p+3}}{e^5 (p+3)}-\frac{d^5 \left(d^2-e^2 x^2\right)^{p+1}}{e^5 (p+1)}+\frac{2 d^3 \left(d^2-e^2 x^2\right)^{p+2}}{e^5 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-35*d^5*(d^2 - e^2*x^2))/(e^5*(1 + p)) + (70*d^3*(d^2 - e^2*x^2)^2)/(e^5*(2 + p)) - (35*d*(d^2 - e^2*x^2)^3)/(e^5*(3 + p)) + (7*d^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (5*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/35","A",1
251,1,138,149,0.1188678,"\int x^3 (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^3*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(4 d e^5 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{5 \left(d^2-e^2 x^2\right) \left(d^4 (p+5)+d^2 e^2 \left(p^2+6 p+5\right) x^2+e^4 \left(p^2+3 p+2\right) x^4\right)}{(p+1) (p+2) (p+3)}\right)}{10 e^4}","\frac{2}{5} d e x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)+\frac{3 d^2 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^4 (p+2)}-\frac{\left(d^2-e^2 x^2\right)^{p+3}}{2 e^4 (p+3)}-\frac{d^4 \left(d^2-e^2 x^2\right)^{p+1}}{e^4 (p+1)}",1,"((d^2 - e^2*x^2)^p*((-5*(d^2 - e^2*x^2)*(d^4*(5 + p) + d^2*e^2*(5 + 6*p + p^2)*x^2 + e^4*(2 + 3*p + p^2)*x^4))/((1 + p)*(2 + p)*(3 + p)) + (4*d*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(10*e^4)","A",1
252,1,168,155,0.1069133,"\int x^2 (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^2*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(-15 d \left(d^2-e^2 x^2\right) \left(d^2+e^2 (p+1) x^2\right) \left(1-\frac{e^2 x^2}{d^2}\right)^p+3 e^5 \left(p^2+3 p+2\right) x^5 \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)+5 d^2 e^3 \left(p^2+3 p+2\right) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)}{15 e^3 (p+1) (p+2)}","\frac{2 d^2 (p+4) x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)}{3 (2 p+5)}-\frac{x^3 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+5}+\frac{d \left(d^2-e^2 x^2\right)^{p+2}}{e^3 (p+2)}-\frac{d^3 \left(d^2-e^2 x^2\right)^{p+1}}{e^3 (p+1)}",1,"((d^2 - e^2*x^2)^p*(-15*d*(d^2 - e^2*x^2)*(1 - (e^2*x^2)/d^2)^p*(d^2 + e^2*(1 + p)*x^2) + 5*d^2*e^3*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2] + 3*e^5*(2 + 3*p + p^2)*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2]))/(15*e^3*(1 + p)*(2 + p)*(1 - (e^2*x^2)/d^2)^p)","A",1
253,1,110,118,0.073908,"\int x (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(4 d e^3 x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{3 \left(d^2-e^2 x^2\right) \left(d^2 (p+3)+e^2 (p+1) x^2\right)}{(p+1) (p+2)}\right)}{6 e^2}","\frac{2}{3} d e x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)-\frac{d^2 \left(d^2-e^2 x^2\right)^{p+1}}{e^2 (p+1)}+\frac{\left(d^2-e^2 x^2\right)^{p+2}}{2 e^2 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-3*(d^2 - e^2*x^2)*(d^2*(3 + p) + e^2*(1 + p)*x^2))/((1 + p)*(2 + p)) + (4*d*e^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(6*e^2)","A",1
254,1,134,71,0.0573565,"\int (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(3 d^2 e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-3 d \left(d^2-e^2 x^2\right) \left(1-\frac{e^2 x^2}{d^2}\right)^p+e^3 (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)}{3 e (p+1)}","-\frac{d 2^{p+2} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(-p-2,p+1;p+2;\frac{d-e x}{2 d}\right)}{e (p+1)}",1,"((d^2 - e^2*x^2)^p*(-3*d*(d^2 - e^2*x^2)*(1 - (e^2*x^2)/d^2)^p + 3*d^2*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + e^3*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2]))/(3*e*(1 + p)*(1 - (e^2*x^2)/d^2)^p)","A",1
255,1,103,128,0.0642028,"\int \frac{(d+e x)^2 \left(d^2-e^2 x^2\right)^p}{x} \, dx","Integrate[((d + e*x)^2*(d^2 - e^2*x^2)^p)/x,x]","\frac{1}{2} \left(d^2-e^2 x^2\right)^p \left(4 d e x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right) \left(\, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)+1\right)}{p+1}\right)","2 d e x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{\left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 (p+1)}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 (p+1)}",1,"((d^2 - e^2*x^2)^p*((4*d*e*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)*(1 + Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2]))/(1 + p)))/2","A",1
256,1,153,128,0.0719176,"\int \frac{(d+e x)^2 \left(d^2-e^2 x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)^2*(d^2 - e^2*x^2)^p)/x^2,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(e x \left(d e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\left(d^2-e^2 x^2\right) \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)\right)-d^3 (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)\right)}{d (p+1) x}","-2 e^2 p x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{e \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{d (p+1)}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{x}",1,"((d^2 - e^2*x^2)^p*(-(d^3*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2]) + e*x*(d*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] - (d^2 - e^2*x^2)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])))/(d*(1 + p)*x*(1 - (e^2*x^2)/d^2)^p)","A",1
257,1,131,139,0.0823836,"\int \frac{(d+e x)^2 \left(d^2-e^2 x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)^2*(d^2 - e^2*x^2)^p)/x^3,x]","\frac{e \left(d^2-e^2 x^2\right)^p \left(\frac{e \left(e^2 x^2-d^2\right) \left(\, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)+\, _2F_1\left(2,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)\right)}{p+1}-\frac{4 d^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}\right)}{2 d^2}","-\frac{2 d e \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{e^2 (1-p) \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (p+1)}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 x^2}",1,"(e*(d^2 - e^2*x^2)^p*((-4*d^3*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (e*(-d^2 + e^2*x^2)*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2] + Hypergeometric2F1[2, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2]))/(1 + p)))/(2*d^2)","A",1
258,1,205,222,0.2368417,"\int x^5 (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^5*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{630 \left(d^3-d e^2 x^2\right)^3}{p+3}+\frac{189 d \left(d^2-e^2 x^2\right)^4}{p+4}+14 e^9 x^9 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{9}{2},-p;\frac{11}{2};\frac{e^2 x^2}{d^2}\right)+54 d^2 e^7 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)-\frac{252 d^7 \left(d^2-e^2 x^2\right)}{p+1}+\frac{693 d^5 \left(d^2-e^2 x^2\right)^2}{p+2}\right)}{126 e^6}","\frac{2 d^2 e (3 p+17) x^7 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)}{7 (2 p+9)}-\frac{e x^7 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+9}+\frac{3 d \left(d^2-e^2 x^2\right)^{p+4}}{2 e^6 (p+4)}-\frac{2 d^7 \left(d^2-e^2 x^2\right)^{p+1}}{e^6 (p+1)}+\frac{11 d^5 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^6 (p+2)}-\frac{5 d^3 \left(d^2-e^2 x^2\right)^{p+3}}{e^6 (p+3)}",1,"((d^2 - e^2*x^2)^p*((-252*d^7*(d^2 - e^2*x^2))/(1 + p) + (693*d^5*(d^2 - e^2*x^2)^2)/(2 + p) + (189*d*(d^2 - e^2*x^2)^4)/(4 + p) - (630*(d^3 - d*e^2*x^2)^3)/(3 + p) + (54*d^2*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (14*e^9*x^9*Hypergeometric2F1[9/2, -p, 11/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(126*e^6)","A",1
259,1,219,218,0.2426775,"\int x^4 (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{1}{70} \left(d^2-e^2 x^2\right)^p \left(30 d e^2 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)+14 d^3 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{35 \left(d^2-e^2 x^2\right) \left(6 d^6 (p+5)+6 d^4 e^2 \left(p^2+6 p+5\right) x^2+3 d^2 e^4 \left(p^3+8 p^2+17 p+10\right) x^4+e^6 \left(p^3+6 p^2+11 p+6\right) x^6\right)}{e^5 (p+1) (p+2) (p+3) (p+4)}\right)","-\frac{3 d x^5 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+7}-\frac{3 d^2 \left(d^2-e^2 x^2\right)^{p+3}}{e^5 (p+3)}+\frac{\left(d^2-e^2 x^2\right)^{p+4}}{2 e^5 (p+4)}-\frac{2 d^6 \left(d^2-e^2 x^2\right)^{p+1}}{e^5 (p+1)}+\frac{9 d^4 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^5 (p+2)}+\frac{2 d^3 (p+11) x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 (2 p+7)}",1,"((d^2 - e^2*x^2)^p*((-35*(d^2 - e^2*x^2)*(6*d^6*(5 + p) + 6*d^4*e^2*(5 + 6*p + p^2)*x^2 + 3*d^2*e^4*(10 + 17*p + 8*p^2 + p^3)*x^4 + e^6*(6 + 11*p + 6*p^2 + p^3)*x^6))/(e^5*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (14*d^3*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (30*d*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/70","A",1
260,1,187,193,0.1806469,"\int x^3 (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^3*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(10 e^7 x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)+42 d^2 e^5 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{35 d \left(d^2-e^2 x^2\right) \left(d^4 (p+9)+d^2 e^2 \left(p^2+10 p+9\right) x^2+3 e^4 \left(p^2+3 p+2\right) x^4\right)}{(p+1) (p+2) (p+3)}\right)}{70 e^4}","\frac{2 d^2 e (3 p+13) x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 (2 p+7)}-\frac{e x^5 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+7}-\frac{3 d \left(d^2-e^2 x^2\right)^{p+3}}{2 e^4 (p+3)}-\frac{2 d^5 \left(d^2-e^2 x^2\right)^{p+1}}{e^4 (p+1)}+\frac{7 d^3 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^4 (p+2)}",1,"((d^2 - e^2*x^2)^p*((-35*d*(d^2 - e^2*x^2)*(d^4*(9 + p) + d^2*e^2*(9 + 10*p + p^2)*x^2 + 3*e^4*(2 + 3*p + p^2)*x^4))/((1 + p)*(2 + p)*(3 + p)) + (42*d^2*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (10*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(70*e^4)","A",1
261,1,187,189,0.1897461,"\int x^2 (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x^2*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{1}{30} \left(d^2-e^2 x^2\right)^p \left(18 d e^2 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)-\frac{15 \left(d^2-e^2 x^2\right) \left(d^4 (3 p+11)+d^2 e^2 \left(3 p^2+14 p+11\right) x^2+e^4 \left(p^2+3 p+2\right) x^4\right)}{e^3 (p+1) (p+2) (p+3)}+10 d^3 x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)","-\frac{3 d x^3 \left(d^2-e^2 x^2\right)^{p+1}}{2 p+5}+\frac{5 d^2 \left(d^2-e^2 x^2\right)^{p+2}}{2 e^3 (p+2)}-\frac{\left(d^2-e^2 x^2\right)^{p+3}}{2 e^3 (p+3)}-\frac{2 d^4 \left(d^2-e^2 x^2\right)^{p+1}}{e^3 (p+1)}+\frac{2 d^3 (p+7) x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)}{3 (2 p+5)}",1,"((d^2 - e^2*x^2)^p*((-15*(d^2 - e^2*x^2)*(d^4*(11 + 3*p) + d^2*e^2*(11 + 14*p + 3*p^2)*x^2 + e^4*(2 + 3*p + p^2)*x^4))/(e^3*(1 + p)*(2 + p)*(3 + p)) + (10*d^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (18*d*e^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/30","A",1
262,1,159,116,0.2616465,"\int x (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[x*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{5 d \left(d^2-e^2 x^2\right) \left(d^2 (p+5)+3 e^2 (p+1) x^2\right)}{(p+1) (p+2)}+2 e^5 x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)+10 d^2 e^3 x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)}{10 e^2}","-\frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^{p+1}}{e^2 (2 p+5)}-\frac{3 d^3 2^{p+3} \left(d^2-e^2 x^2\right)^{p+1} \left(\frac{e x}{d}+1\right)^{-p-1} \, _2F_1\left(-p-3,p+1;p+2;\frac{d-e x}{2 d}\right)}{e^2 (p+1) (2 p+5)}",1,"((d^2 - e^2*x^2)^p*((-5*d*(d^2 - e^2*x^2)*(d^2*(5 + p) + 3*e^2*(1 + p)*x^2))/((1 + p)*(2 + p)) + (10*d^2*e^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (2*e^5*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(10*e^2)","A",1
263,1,155,73,0.1632936,"\int (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","\frac{1}{2} \left(d^2-e^2 x^2\right)^p \left(2 d e^2 x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)+\frac{\left(e^2 x^2-d^2\right) \left(d^2 (3 p+7)+e^2 (p+1) x^2\right)}{e (p+1) (p+2)}+2 d^3 x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)\right)","-\frac{d^2 2^{p+3} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(-p-3,p+1;p+2;\frac{d-e x}{2 d}\right)}{e (p+1)}",1,"((d^2 - e^2*x^2)^p*(((-d^2 + e^2*x^2)*(d^2*(7 + 3*p) + e^2*(1 + p)*x^2))/(e*(1 + p)*(2 + p)) + (2*d^3*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (2*d*e^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/2","B",1
264,1,169,171,0.139595,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^p}{x} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^p)/x,x]","\frac{1}{6} \left(d^2-e^2 x^2\right)^p \left(18 d^2 e x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{3 d \left(d^2-e^2 x^2\right) \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{p+1}-\frac{9 d \left(d^2-e^2 x^2\right)}{p+1}+2 e^3 x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)","\frac{2 d^2 e (3 p+5) x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{2 p+3}-\frac{d \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 (p+1)}-\frac{e x \left(d^2-e^2 x^2\right)^{p+1}}{2 p+3}-\frac{3 d \left(d^2-e^2 x^2\right)^{p+1}}{2 (p+1)}",1,"((d^2 - e^2*x^2)^p*((-9*d*(d^2 - e^2*x^2))/(1 + p) + (18*d^2*e*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - (3*d*(d^2 - e^2*x^2)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(1 + p) + (2*e^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/6","A",1
265,1,158,159,0.0868856,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^p)/x^2,x]","\frac{\left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(e x \left(6 d e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\left(d^2-e^2 x^2\right) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(3 \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)+1\right)\right)-2 d^3 (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)\right)}{2 (p+1) x}","2 d e^2 (1-p) x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\frac{3 e \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 (p+1)}-\frac{e \left(d^2-e^2 x^2\right)^{p+1}}{2 (p+1)}-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{x}",1,"((d^2 - e^2*x^2)^p*(-2*d^3*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2] + e*x*(6*d*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] - (d^2 - e^2*x^2)*(1 - (e^2*x^2)/d^2)^p*(1 + 3*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2]))))/(2*(1 + p)*x*(1 - (e^2*x^2)/d^2)^p)","A",1
266,1,182,166,0.1022112,"\int \frac{(d+e x)^3 \left(d^2-e^2 x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^p)/x^3,x]","\frac{e \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(e x \left(2 d e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-\left(d^2-e^2 x^2\right) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(3 \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)+\, _2F_1\left(2,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)\right)\right)-6 d^3 (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)\right)}{2 d (p+1) x}","-\frac{e^2 (3-p) \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right)}{2 d (p+1)}-\frac{3 e \left(d^2-e^2 x^2\right)^{p+1}}{x}-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 x^2}-2 e^3 (3 p+1) x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)",1,"(e*(d^2 - e^2*x^2)^p*(-6*d^3*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2] + e*x*(2*d*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] - (d^2 - e^2*x^2)*(1 - (e^2*x^2)/d^2)^p*(3*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2] + Hypergeometric2F1[2, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2]))))/(2*d*(1 + p)*x*(1 - (e^2*x^2)/d^2)^p)","A",1
267,1,66,148,0.1073313,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x),x]","\frac{x^5 (d-e x)^p (d+e x)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} F_1\left(5;-p,1-p;6;\frac{e x}{d},-\frac{e x}{d}\right)}{5 d}","\frac{x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d}-\frac{d^2 \left(d^2-e^2 x^2\right)^{p+1}}{e^5 (p+1)}+\frac{\left(d^2-e^2 x^2\right)^{p+2}}{2 e^5 (p+2)}+\frac{d^4 \left(d^2-e^2 x^2\right)^p}{2 e^5 p}",1,"(x^5*(d - e*x)^p*(d + e*x)^p*AppellF1[5, -p, 1 - p, 6, (e*x)/d, -((e*x)/d)])/(5*d*(1 - (e^2*x^2)/d^2)^p)","C",0
268,1,245,121,0.289557,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x),x]","\frac{\left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(6 d^2 e (p+1) x \left(\frac{e x}{d}+1\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+3 d \left(d (d-e x) \left(2-\frac{2 e^2 x^2}{d^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\left(d^2 \left(\left(1-\frac{e^2 x^2}{d^2}\right)^p-1\right)-e^2 x^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p\right) \left(\frac{e x}{d}+1\right)^p\right)+2 e^3 (p+1) x^3 \left(\frac{e x}{d}+1\right)^p \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)\right)}{6 e^4 (p+1)}","-\frac{e x^5 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{5}{2},1-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^2}+\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 e^4 (p+1)}-\frac{d^3 \left(d^2-e^2 x^2\right)^p}{2 e^4 p}",1,"((d^2 - e^2*x^2)^p*(6*d^2*e*(1 + p)*x*(1 + (e*x)/d)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + 2*e^3*(1 + p)*x^3*(1 + (e*x)/d)^p*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2] + 3*d*((1 + (e*x)/d)^p*(-(e^2*x^2*(1 - (e^2*x^2)/d^2)^p) + d^2*(-1 + (1 - (e^2*x^2)/d^2)^p)) + d*(d - e*x)*(2 - (2*e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(6*e^4*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","B",1
269,1,198,119,0.2500819,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x),x]","-\frac{\left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(2 d e (p+1) x \left(\frac{e x}{d}+1\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+d (d-e x) \left(2-\frac{2 e^2 x^2}{d^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\left(d^2 \left(\left(1-\frac{e^2 x^2}{d^2}\right)^p-1\right)-e^2 x^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p\right) \left(\frac{e x}{d}+1\right)^p\right)}{2 e^3 (p+1)}","\frac{x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)}{3 d}+\frac{d^2 \left(d^2-e^2 x^2\right)^p}{2 e^3 p}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^3 (p+1)}",1,"-1/2*((d^2 - e^2*x^2)^p*((1 + (e*x)/d)^p*(-(e^2*x^2*(1 - (e^2*x^2)/d^2)^p) + d^2*(-1 + (1 - (e^2*x^2)/d^2)^p)) + 2*d*e*(1 + p)*x*(1 + (e*x)/d)^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + d*(d - e*x)*(2 - (2*e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^3*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","A",1
270,1,147,90,0.1027018,"\int \frac{x \left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(x*(d^2 - e^2*x^2)^p)/(d + e*x),x]","\frac{2^{p-1} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(2 e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+(d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{e^2 (p+1)}","-\frac{e x^3 \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^2}-\frac{d \left(d^2-e^2 x^2\right)^p}{2 e^2 p}",1,"(2^(-1 + p)*(d^2 - e^2*x^2)^p*(2*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^2*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","A",1
271,1,75,73,0.034006,"\int \frac{\left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[(d^2 - e^2*x^2)^p/(d + e*x),x]","-\frac{2^{p-1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d e (p+1)}","-\frac{2^{p-1} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e (p+1)}",1,"-((2^(-1 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
272,1,151,104,0.1102369,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x*(d + e*x)),x]","\frac{2^{p-1} \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+d (p+1) \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)\right)}{d^2 p (p+1)}","-\frac{e x \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{1}{2},1-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^2}-\frac{\left(d^2-e^2 x^2\right)^p \, _2F_1\left(1,p;p+1;1-\frac{e^2 x^2}{d^2}\right)}{2 d p}",1,"(2^(-1 + p)*(d^2 - e^2*x^2)^p*(p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*(1 + p)*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^2*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)","A",1
273,1,167,106,0.1777343,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^2 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{d e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{2 d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{e 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{2 d^3}","\frac{e \left(d^2-e^2 x^2\right)^p \, _2F_1\left(1,p;p+1;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 p}-\frac{\left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{d x}",1,"((d^2 - e^2*x^2)^p*((-2*d^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (2^p*e*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (d*e*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(2*d^3)","A",1
274,1,219,108,0.5652307,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^3 (d+e x)} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{2 d^2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \left(e^2 \left(\frac{(d-e x) \left(2-\frac{2 d^2}{e^2 x^2}\right)^p \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{d \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}\right)+\frac{d^3 \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}\right)\right)}{2 d^4}","\frac{e \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},1-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{d^2 x}-\frac{e^2 \left(d^2-e^2 x^2\right)^p \, _2F_1\left(2,p;p+1;1-\frac{e^2 x^2}{d^2}\right)}{2 d^3 p}",1,"((d^2 - e^2*x^2)^p*((2*d^2*e*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + ((d^3*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*x^2) + e^2*(((2 - (2*d^2)/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (d*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/p))/(1 - d^2/(e^2*x^2))^p))/(2*d^4)","B",1
275,1,66,179,0.147475,"\int \frac{x^5 \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^5*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{x^6 (d-e x)^p (d+e x)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} F_1\left(6;-p,2-p;7;\frac{e x}{d},-\frac{e x}{d}\right)}{6 d^2}","-\frac{2 d^2 \left(d^2-e^2 x^2\right)^{p+1}}{e^6 (p+1)}+\frac{\left(d^2-e^2 x^2\right)^{p+2}}{2 e^6 (p+2)}+\frac{d^6 \left(d^2-e^2 x^2\right)^{p-1}}{e^6 (1-p)}+\frac{5 d^4 \left(d^2-e^2 x^2\right)^p}{2 e^6 p}-\frac{2 e x^7 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{7}{2},2-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right)}{7 d^3}",1,"(x^6*(d - e*x)^p*(d + e*x)^p*AppellF1[6, -p, 2 - p, 7, (e*x)/d, -((e*x)/d)])/(6*d^2*(1 - (e^2*x^2)/d^2)^p)","C",0
276,1,66,184,0.1158418,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{x^5 (d-e x)^p (d+e x)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} F_1\left(5;-p,2-p;6;\frac{e x}{d},-\frac{e x}{d}\right)}{5 d^2}","\frac{2 (p+4) x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^2 (2 p+3)}-\frac{x^5 \left(d^2-e^2 x^2\right)^{p-1}}{2 p+3}+\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{e^5 (p+1)}-\frac{d^5 \left(d^2-e^2 x^2\right)^{p-1}}{e^5 (1-p)}-\frac{2 d^3 \left(d^2-e^2 x^2\right)^p}{e^5 p}",1,"(x^5*(d - e*x)^p*(d + e*x)^p*AppellF1[5, -p, 2 - p, 6, (e*x)/d, -((e*x)/d)])/(5*d^2*(1 - (e^2*x^2)/d^2)^p)","C",0
277,1,332,150,0.2898315,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{2^{p-2} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(-8 d e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)-6 d (d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)-d e x \left(1-\frac{e^2 x^2}{d^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)-2 d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p+2 e^2 x^2 \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p+2 d^2 \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p\right)}{e^4 (p+1)}","\frac{3 d^2 \left(d^2-e^2 x^2\right)^p}{2 e^4 p}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^4 (p+1)}+\frac{d^4 \left(d^2-e^2 x^2\right)^{p-1}}{e^4 (1-p)}-\frac{2 e x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^3}",1,"(2^(-2 + p)*(d^2 - e^2*x^2)^p*(2*d^2*(1/2 + (e*x)/(2*d))^p - 2*d^2*(1/2 + (e*x)/(2*d))^p*(1 - (e^2*x^2)/d^2)^p + 2*e^2*x^2*(1/2 + (e*x)/(2*d))^p*(1 - (e^2*x^2)/d^2)^p - 8*d*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] - 6*d*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d^2*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - d*e*x*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^4*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","B",1
278,1,177,156,0.1617041,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{2^{p-2} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(4 e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+(d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(4 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)-\, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)\right)}{e^3 (p+1)}","\frac{2 (p+2) x^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{3}{2},2-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^2 (2 p+1)}-\frac{x^3 \left(d^2-e^2 x^2\right)^{p-1}}{2 p+1}-\frac{d \left(d^2-e^2 x^2\right)^p}{e^3 p}-\frac{d^3 \left(d^2-e^2 x^2\right)^{p-1}}{e^3 (1-p)}",1,"(2^(-2 + p)*(d^2 - e^2*x^2)^p*(4*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*(4*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(e^3*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","A",1
279,1,102,115,0.0862994,"\int \frac{x \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{2^{p-2} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(\, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)-2 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d e^2 (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^2 (1-p) (d+e x)^2}-\frac{2^{p-1} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e^2 \left(1-p^2\right)}",1,"(2^(-2 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(-2*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d*e^2*(1 + p)*(1 + (e*x)/d)^p)","A",1
280,1,75,73,0.0404822,"\int \frac{\left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(d + e*x)^2,x]","-\frac{2^{p-2} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e (p+1)}","-\frac{2^{p-2} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^3 e (p+1)}",1,"-((2^(-2 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^2*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
281,1,201,128,0.1559928,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x*(d + e*x)^2),x]","\frac{2^{p-2} \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(2 p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+2 d (p+1) \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)\right)}{d^3 p (p+1)}","-\frac{\left(d^2-e^2 x^2\right)^p \, _2F_1\left(1,p;p+1;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 p}+\frac{\left(d^2-e^2 x^2\right)^{p-1}}{1-p}-\frac{2 e x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^3}",1,"(2^(-2 + p)*(d^2 - e^2*x^2)^p*(2*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2*d*(1 + p)*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^3*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)","A",1
282,1,223,137,0.3343007,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^2 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^2),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{4 d e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{4 d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{e 2^{p+2} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{4 d^4}","-\frac{e \left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(1,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{d (1-p)}-\frac{\left(d^2-e^2 x^2\right)^{p-1}}{x}+\frac{2 e^2 (2-p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},2-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^4}",1,"((d^2 - e^2*x^2)^p*((-4*d^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (2^(2 + p)*e*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (4*d*e*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(4*d^4)","A",1
283,1,283,143,0.4997192,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^3 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^2),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{6 d e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{8 d^2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{2 d^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{3 e^2 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e^2 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{4 d^5}","\frac{e^2 (3-p) \left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(1,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (1-p)}-\frac{\left(d^2-e^2 x^2\right)^{p-1}}{2 x^2}+\frac{2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{d^3 x}",1,"((d^2 - e^2*x^2)^p*((8*d^2*e*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (2*d^3*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (3*2^(1 + p)*e^2*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e^2*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (6*d*e^2*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(4*d^5)","A",1
284,1,334,145,0.4108674,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^4 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^2),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{36 d^2 e^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{24 d e^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{4 d^4 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}-\frac{12 d^3 e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{3 e^3 2^{p+3} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^3 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{12 d^6}","-\frac{\left(d^2-e^2 x^2\right)^{p-1}}{3 x^3}-\frac{2 e^2 (4-p) \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{1}{2},2-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^4 x}-\frac{e^3 \left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(2,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{d^3 (1-p)}",1,"((d^2 - e^2*x^2)^p*((-4*d^4*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) - (36*d^2*e^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) - (12*d^3*e*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (3*2^(3 + p)*e^3*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^p*e^3*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (24*d*e^3*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(12*d^6)","B",1
285,1,389,145,0.5009513,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^5 (d+e x)^2} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^2),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{30 d e^4 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{48 d^2 e^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{6 d^5 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right)}{(p-2) x^4}+\frac{8 d^4 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}+\frac{18 d^3 e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{15 e^4 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^4 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{12 d^7}","-\frac{\left(d^2-e^2 x^2\right)^{p-1}}{4 x^4}+\frac{e^4 (5-p) \left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(2,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{4 d^4 (1-p)}+\frac{2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{3}{2},2-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^3 x^3}",1,"((d^2 - e^2*x^2)^p*((8*d^4*e*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) + (48*d^2*e^3*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (18*d^3*e^2*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (15*2^(1 + p)*e^4*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (6*d^5*Hypergeometric2F1[2 - p, -p, 3 - p, d^2/(e^2*x^2)])/((-2 + p)*(1 - d^2/(e^2*x^2))^p*x^4) + (3*2^p*e^4*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (30*d*e^4*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(12*d^7)","B",1
286,1,245,220,0.306659,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]","-\frac{2^{p-3} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(24 d e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+(d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(24 d \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)-8 d \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+d \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+4 d \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p+4 e x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p\right)\right)}{e^5 (p+1)}","-\frac{3 d x^5 \left(d^2-e^2 x^2\right)^{p-2}}{2 p+1}+\frac{3 d^2 \left(d^2-e^2 x^2\right)^p}{e^5 p}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^5 (p+1)}-\frac{2 d^6 \left(d^2-e^2 x^2\right)^{p-2}}{e^5 (2-p)}+\frac{9 d^4 \left(d^2-e^2 x^2\right)^{p-1}}{2 e^5 (1-p)}+\frac{2 (p+8) x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^3 (2 p+1)}",1,"-((2^(-3 + p)*(d^2 - e^2*x^2)^p*(24*d*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*(4*d*(1/2 + (e*x)/(2*d))^p + 4*e*x*(1/2 + (e*x)/(2*d))^p + 24*d*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 8*d*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(e^5*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p))","A",1
287,1,202,194,0.2420348,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]","\frac{2^{p-3} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(8 e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+(d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(12 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)-6 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)\right)}{e^4 (p+1)}","\frac{e x^5 \left(d^2-e^2 x^2\right)^{p-2}}{2 p+1}-\frac{3 d \left(d^2-e^2 x^2\right)^p}{2 e^4 p}+\frac{2 d^5 \left(d^2-e^2 x^2\right)^{p-2}}{e^4 (2-p)}-\frac{2 e (3 p+4) x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^4 (2 p+1)}-\frac{7 d^3 \left(d^2-e^2 x^2\right)^{p-1}}{2 e^4 (1-p)}",1,"(2^(-3 + p)*(d^2 - e^2*x^2)^p*(8*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*(12*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 6*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(e^4*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","A",1
288,1,130,157,0.1154446,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]","-\frac{2^{p-3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(4 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)-4 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d e^3 (p+1)}","\frac{2^{p-3} (p+4) \left(d^2-e^2 x^2\right)^{p+1} \left(\frac{e x}{d}+1\right)^{-p-1} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e^3 (2-p) p (p+1)}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^3 p (d+e x)^2}-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 e^3 (2-p) (d+e x)^3}",1,"-((2^(-3 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(4*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 4*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d*e^3*(1 + p)*(1 + (e*x)/d)^p))","A",1
289,1,102,118,0.0851005,"\int \frac{x \left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]","\frac{2^{p-3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(\, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)-2 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d^2 e^2 (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^2 (2-p) (d+e x)^3}-\frac{3\ 2^{p-3} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^3 e^2 (2-p) (p+1)}",1,"(2^(-3 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(-2*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d^2*e^2*(1 + p)*(1 + (e*x)/d)^p)","A",1
290,1,75,73,0.0462166,"\int \frac{\left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(d + e*x)^3,x]","-\frac{2^{p-3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^3 e (p+1)}","-\frac{2^{p-3} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^4 e (p+1)}",1,"-((2^(-3 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^3*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
291,1,328,175,0.229083,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x (d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x*(d + e*x)^3),x]","\frac{2^{p-3} \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(4 p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+2 p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+d p \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)-e p x \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+4 d \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)+4 d p \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)\right)}{d^4 p (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(1,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{2 d (1-p)}-\frac{e x \left(d^2-e^2 x^2\right)^{p-2}}{3-2 p}+\frac{2 d \left(d^2-e^2 x^2\right)^{p-2}}{2-p}-\frac{2 e (4-3 p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^4 (3-2 p)}",1,"(2^(-3 + p)*(d^2 - e^2*x^2)^p*(4*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*p*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - e*p*(1 - d^2/(e^2*x^2))^p*x*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 4*d*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)] + 4*d*p*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^4*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)","A",1
292,1,280,166,0.4269419,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^2 (d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^3),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{12 d e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{8 d^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{3 e 2^{p+2} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e 2^{p+2} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{8 d^5}","-\frac{3 e \left(d^2-e^2 x^2\right)^{p-1} \, _2F_1\left(1,p-1;p;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (1-p)}-\frac{2 e \left(d^2-e^2 x^2\right)^{p-2}}{2-p}-\frac{d \left(d^2-e^2 x^2\right)^{p-2}}{x}+\frac{2 e^2 (4-p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^5}",1,"((d^2 - e^2*x^2)^p*((-8*d^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (3*2^(2 + p)*e*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^(2 + p)*e*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e*(-d + e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (12*d*e*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(8*d^5)","A",1
293,1,341,173,0.6494558,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^3 (d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^3),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{24 d e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{24 d^2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{4 d^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{3 e^2 2^{p+3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^2 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e^2 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{8 d^6}","\frac{e^2 (6-p) \left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{2 d (2-p)}+\frac{3 e \left(d^2-e^2 x^2\right)^{p-2}}{x}-\frac{d \left(d^2-e^2 x^2\right)^{p-2}}{2 x^2}-\frac{2 e^3 (8-3 p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^6}",1,"((d^2 - e^2*x^2)^p*((24*d^2*e*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (4*d^3*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (3*2^(3 + p)*e^2*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^(1 + p)*e^2*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e^2*(d - e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (24*d*e^2*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(8*d^6)","A",1
294,1,393,179,0.4990052,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^4 (d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^3),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{144 d^2 e^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{120 d e^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{8 d^4 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}-\frac{36 d^3 e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{15 e^3 2^{p+3} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^3 2^{p+3} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^3 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{24 d^7}","\frac{3 e \left(d^2-e^2 x^2\right)^{p-2}}{2 x^2}-\frac{d \left(d^2-e^2 x^2\right)^{p-2}}{3 x^3}-\frac{e^3 (10-3 p) \left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (2-p)}-\frac{2 e^2 (8-p) \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^5 x}",1,"((d^2 - e^2*x^2)^p*((-8*d^4*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) - (144*d^2*e^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) - (36*d^3*e*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (15*2^(3 + p)*e^3*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^(3 + p)*e^3*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^p*e^3*(-d + e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (120*d*e^3*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(24*d^7)","B",1
295,1,446,174,0.5769971,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^5 (d+e x)^3} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{60 d e^4 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{80 d^2 e^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{4 d^5 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right)}{(p-2) x^4}+\frac{8 d^4 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}+\frac{24 d^3 e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{15 e^4 2^{p+2} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{5 e^4 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e^4 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{8 d^8}","-\frac{d \left(d^2-e^2 x^2\right)^{p-2}}{4 x^4}+\frac{e \left(d^2-e^2 x^2\right)^{p-2}}{x^3}+\frac{2 e^3 (4-p) \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{1}{2},3-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{d^6 x}+\frac{e^4 (10-p) \left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(2,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{4 d^3 (2-p)}",1,"((d^2 - e^2*x^2)^p*((8*d^4*e*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) + (80*d^2*e^3*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (24*d^3*e^2*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (15*2^(2 + p)*e^4*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (4*d^5*Hypergeometric2F1[2 - p, -p, 3 - p, d^2/(e^2*x^2)])/((-2 + p)*(1 - d^2/(e^2*x^2))^p*x^4) + (5*2^(1 + p)*e^4*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e^4*(d - e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (60*d*e^4*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(8*d^8)","B",1
296,1,231,265,0.2887057,"\int \frac{x^4 \left(d^2-e^2 x^2\right)^p}{(d+e x)^4} \, dx","Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]","\frac{2^{p-4} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(16 e (p+1) x \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)+(d-e x) \left(1-\frac{e^2 x^2}{d^2}\right)^p \left(32 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)-24 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+8 \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)-\, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)\right)}{e^5 (p+1)}","\frac{d^2 (12 p+13) x^5 \left(d^2-e^2 x^2\right)^{p-3}}{1-4 p^2}-\frac{e^2 x^7 \left(d^2-e^2 x^2\right)^{p-3}}{2 p+1}-\frac{2 d \left(d^2-e^2 x^2\right)^p}{e^5 p}-\frac{4 d^7 \left(d^2-e^2 x^2\right)^{p-3}}{e^5 (3-p)}+\frac{10 d^5 \left(d^2-e^2 x^2\right)^{p-2}}{e^5 (2-p)}-\frac{4 \left(p^2+15 p+16\right) x^5 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{5}{2},4-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right)}{5 d^4 \left(1-4 p^2\right)}-\frac{8 d^3 \left(d^2-e^2 x^2\right)^{p-1}}{e^5 (1-p)}",1,"(2^(-4 + p)*(d^2 - e^2*x^2)^p*(16*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*(32*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 24*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 8*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])))/(e^5*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)","A",1
297,1,156,211,0.1991218,"\int \frac{x^3 \left(d^2-e^2 x^2\right)^p}{(d+e x)^4} \, dx","Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]","\frac{2^{p-4} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(-8 \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+12 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)-6 \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d e^4 (p+1)}","\frac{3\ 2^{p-2} (p+2) \left(d^2-e^2 x^2\right)^{p+1} \left(\frac{e x}{d}+1\right)^{-p-1} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^2 e^4 (1-2 p) (3-p) p (p+1)}-\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^4 p (d+e x)^2}-\frac{d (2 p+1) \left(d^2-e^2 x^2\right)^{p+1}}{e^4 (1-2 p) p (d+e x)^3}+\frac{d^2 \left(d^2-e^2 x^2\right)^{p+1}}{2 e^4 (3-p) (d+e x)^4}",1,"(2^(-4 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(-8*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 12*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 6*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d*e^4*(1 + p)*(1 + (e*x)/d)^p)","A",1
298,1,130,163,0.1139122,"\int \frac{x^2 \left(d^2-e^2 x^2\right)^p}{(d+e x)^4} \, dx","Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]","-\frac{2^{p-4} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(4 \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)-4 \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+\, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d^2 e^3 (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p+1}}{e^3 (1-2 p) (d+e x)^3}-\frac{d \left(d^2-e^2 x^2\right)^{p+1}}{2 e^3 (3-p) (d+e x)^4}-\frac{2^{p-3} (p+7) \left(d^2-e^2 x^2\right)^{p+1} \left(\frac{e x}{d}+1\right)^{-p-1} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^3 e^3 (1-2 p) (3-p) (p+1)}",1,"-((2^(-4 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(4*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 4*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d^2*e^3*(1 + p)*(1 + (e*x)/d)^p))","A",1
299,1,102,118,0.0859451,"\int \frac{x \left(d^2-e^2 x^2\right)^p}{(d+e x)^4} \, dx","Integrate[(x*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]","\frac{2^{p-4} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(\, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)-2 \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{d^3 e^2 (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p+1}}{2 e^2 (3-p) (d+e x)^4}-\frac{2^{p-2} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^4 e^2 (3-p) (p+1)}",1,"(2^(-4 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*(-2*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(d^3*e^2*(1 + p)*(1 + (e*x)/d)^p)","A",1
300,1,75,73,0.0399937,"\int \frac{\left(d^2-e^2 x^2\right)^p}{(d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(d + e*x)^4,x]","-\frac{2^{p-4} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^4 e (p+1)}","-\frac{2^{p-4} \left(\frac{e x}{d}+1\right)^{-p-1} \left(d^2-e^2 x^2\right)^{p+1} \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{d^5 e (p+1)}",1,"-((2^(-4 + p)*(d - e*x)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^4*e*(1 + p)*(1 + (e*x)/d)^p))","A",1
301,1,417,204,0.2793653,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x*(d + e*x)^4),x]","\frac{2^{p-4} \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \left(\frac{e x}{d}+1\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(8 p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+4 p (d-e x) \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+2 d p \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)-2 e p x \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+d p \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)-e p x \left(1-\frac{d^2}{e^2 x^2}\right)^p \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)+8 d \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)+8 d p \left(\frac{e x}{2 d}+\frac{1}{2}\right)^p \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)\right)}{d^5 p (p+1)}","\frac{\left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{2 (2-p)}-\frac{4 d e x \left(d^2-e^2 x^2\right)^{p-3}}{5-2 p}+\frac{4 d^2 \left(d^2-e^2 x^2\right)^{p-3}}{3-p}-\frac{\left(d^2-e^2 x^2\right)^{p-2}}{2 (2-p)}-\frac{8 e (2-p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^5 (5-2 p)}",1,"(2^(-4 + p)*(d^2 - e^2*x^2)^p*(8*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 4*p*(1 - d^2/(e^2*x^2))^p*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 2*d*p*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 2*e*p*(1 - d^2/(e^2*x^2))^p*x*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*p*(1 - d^2/(e^2*x^2))^p*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - e*p*(1 - d^2/(e^2*x^2))^p*x*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + 8*d*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)] + 8*d*p*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)]))/(d^5*p*(1 + p)*(1 - d^2/(e^2*x^2))^p*(1 + (e*x)/d)^p)","B",1
302,1,337,207,0.437127,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^2 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^4),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-32 d e (p+1) x \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)-16 d^2 p (p+1) \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)+e 2^{p+5} p x (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)+3 e 2^{p+2} p x (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)+e 2^{p+2} p x (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)+e 2^p p x (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)\right)}{16 d^6 p (p+1) x}","-\frac{2 e \left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{d (2-p)}+\frac{e^2 x \left(d^2-e^2 x^2\right)^{p-3}}{5-2 p}-\frac{4 d e \left(d^2-e^2 x^2\right)^{p-3}}{3-p}-\frac{d^2 \left(d^2-e^2 x^2\right)^{p-3}}{x}+\frac{4 e^2 \left(p^2-9 p+16\right) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^6 (5-2 p)}",1,"((d^2 - e^2*x^2)^p*((-16*d^2*p*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (2^(5 + p)*e*p*x*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(1 + (e*x)/d)^p + (3*2^(2 + p)*e*p*x*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(1 + (e*x)/d)^p + (2^(2 + p)*e*p*x*(-d + e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(1 + (e*x)/d)^p + (2^p*e*p*x*(-d + e*x)*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(1 + (e*x)/d)^p - (32*d*e*(1 + p)*x*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(1 - d^2/(e^2*x^2))^p))/(16*d^6*p*(1 + p)*x)","A",1
303,1,399,211,0.773081,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^3 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^4),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{80 d e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{64 d^2 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{8 d^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{5 e^2 2^{p+4} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^2 2^{p+3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^2 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{e^2 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{16 d^7}","\frac{e^2 (10-p) \left(d^2-e^2 x^2\right)^{p-2} \, _2F_1\left(1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right)}{2 d^2 (2-p)}+\frac{e^2 (11-p) \left(d^2-e^2 x^2\right)^{p-3}}{2 (3-p)}+\frac{4 d e \left(d^2-e^2 x^2\right)^{p-3}}{x}-\frac{d^2 \left(d^2-e^2 x^2\right)^{p-3}}{2 x^2}-\frac{8 e^3 (4-p) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{d^7}",1,"((d^2 - e^2*x^2)^p*((64*d^2*e*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (8*d^3*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (5*2^(4 + p)*e^2*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^(3 + p)*e^2*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^(1 + p)*e^2*(d - e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (2^p*e^2*(d - e*x)*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (80*d*e^2*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(16*d^7)","A",1
304,1,452,210,0.6155877,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^4 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^4),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(-\frac{480 d^2 e^2 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}-\frac{480 d e^3 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}-\frac{16 d^4 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}-\frac{96 d^3 e \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{15 e^3 2^{p+5} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{15 e^3 2^{p+3} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^3 2^{p+3} (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^3 2^p (e x-d) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{48 d^8}","-\frac{e^2 (27-2 p) \left(d^2-e^2 x^2\right)^{p-3}}{3 x}+\frac{2 d e \left(d^2-e^2 x^2\right)^{p-3}}{x^2}-\frac{d^2 \left(d^2-e^2 x^2\right)^{p-3}}{3 x^3}-\frac{2 e^3 (5-p) \left(d^2-e^2 x^2\right)^{p-3} \, _2F_1\left(1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right)}{d (3-p)}+\frac{4 e^4 \left(p^2-17 p+48\right) x \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^8}",1,"((d^2 - e^2*x^2)^p*((-16*d^4*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) - (480*d^2*e^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) - (96*d^3*e*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (15*2^(5 + p)*e^3*(-d + e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (15*2^(3 + p)*e^3*(-d + e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^(3 + p)*e^3*(-d + e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^p*e^3*(-d + e*x)*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (480*d*e^3*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(48*d^8)","B",1
305,1,505,216,0.7384605,"\int \frac{\left(d^2-e^2 x^2\right)^p}{x^5 (d+e x)^4} \, dx","Integrate[(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^4),x]","\frac{\left(d^2-e^2 x^2\right)^p \left(\frac{840 d e^4 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(-p,-p;1-p;\frac{d^2}{e^2 x^2}\right)}{p}+\frac{960 d^2 e^3 \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x}+\frac{24 d^5 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(2-p,-p;3-p;\frac{d^2}{e^2 x^2}\right)}{(p-2) x^4}+\frac{64 d^4 e \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{x^3}+\frac{240 d^3 e^2 \left(1-\frac{d^2}{e^2 x^2}\right)^{-p} \, _2F_1\left(1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right)}{(p-1) x^2}+\frac{105 e^4 2^{p+3} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(1-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{45 e^4 2^{p+2} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(2-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{15 e^4 2^{p+1} (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(3-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}+\frac{3 e^4 2^p (d-e x) \left(\frac{e x}{d}+1\right)^{-p} \, _2F_1\left(4-p,p+1;p+2;\frac{d-e x}{2 d}\right)}{p+1}\right)}{48 d^9}","-\frac{e^2 (17-p) \left(d^2-e^2 x^2\right)^{p-3}}{4 x^2}-\frac{d^2 \left(d^2-e^2 x^2\right)^{p-3}}{4 x^4}+\frac{4 d e \left(d^2-e^2 x^2\right)^{p-3}}{3 x^3}+\frac{e^4 \left(p^2-21 p+70\right) \left(d^2-e^2 x^2\right)^{p-3} \, _2F_1\left(1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right)}{4 d^2 (3-p)}+\frac{8 e^3 (6-p) \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2-e^2 x^2\right)^p \, _2F_1\left(-\frac{1}{2},4-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right)}{3 d^7 x}",1,"((d^2 - e^2*x^2)^p*((64*d^4*e*Hypergeometric2F1[-3/2, -p, -1/2, (e^2*x^2)/d^2])/(x^3*(1 - (e^2*x^2)/d^2)^p) + (960*d^2*e^3*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2])/(x*(1 - (e^2*x^2)/d^2)^p) + (240*d^3*e^2*Hypergeometric2F1[1 - p, -p, 2 - p, d^2/(e^2*x^2)])/((-1 + p)*(1 - d^2/(e^2*x^2))^p*x^2) + (105*2^(3 + p)*e^4*(d - e*x)*Hypergeometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (24*d^5*Hypergeometric2F1[2 - p, -p, 3 - p, d^2/(e^2*x^2)])/((-2 + p)*(1 - d^2/(e^2*x^2))^p*x^4) + (45*2^(2 + p)*e^4*(d - e*x)*Hypergeometric2F1[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (15*2^(1 + p)*e^4*(d - e*x)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (3*2^p*e^4*(d - e*x)*Hypergeometric2F1[4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) + (840*d*e^4*Hypergeometric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(48*d^9)","B",1
306,1,194,264,0.1666709,"\int (g x)^m (d+e x)^3 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]","x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(e x \left(\frac{3 d^2 \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{m+2}+e x \left(\frac{3 d \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)}{m+3}+\frac{e x \, _2F_1\left(\frac{m+4}{2},-p;\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)}{m+4}\right)\right)+\frac{d^3 \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{m+1}\right)","\frac{2 d^2 e (2 m+3 p+7) (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2) (m+2 p+4)}-\frac{e (g x)^{m+2} \left(d^2-e^2 x^2\right)^{p+1}}{g^2 (m+2 p+4)}-\frac{3 d (g x)^{m+1} \left(d^2-e^2 x^2\right)^{p+1}}{g (m+2 p+3)}+\frac{2 d^3 (2 m+p+3) (g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+2 p+3)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*((d^3*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((3*d^2*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) + (e*x*Hypergeometric2F1[(4 + m)/2, -p, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/(1 - (e^2*x^2)/d^2)^p","A",1
307,1,169,206,0.0852169,"\int (g x)^m (d+e x)^2 \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \left(2 d (m+3) \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)+e (m+2) x \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{(m+1) (m+2) (m+3)}","\frac{2 d e (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2)}+\frac{2 d^2 (m+p+2) (g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1) (m+2 p+3)}-\frac{(g x)^{m+1} \left(d^2-e^2 x^2\right)^{p+1}}{g (m+2 p+3)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2] + e*(2 + m)*x*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, (e^2*x^2)/d^2])))/((1 + m)*(2 + m)*(3 + m)*(1 - (e^2*x^2)/d^2)^p)","A",1
308,1,116,153,0.0332138,"\int (g x)^m (d+e x) \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)*(d^2 - e^2*x^2)^p,x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d (m+2) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)+e (m+1) x \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)\right)}{(m+1) (m+2)}","\frac{e (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{g^2 (m+2)}+\frac{d (g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*(d*(2 + m)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2]))/((1 + m)*(2 + m)*(1 - (e^2*x^2)/d^2)^p)","A",1
309,1,73,75,0.0088246,"\int (g x)^m \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(g*x)^m*(d^2 - e^2*x^2)^p,x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right)}{m+1}","\frac{(g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{g (m+1)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, (e^2*x^2)/d^2])/((1 + m)*(1 - (e^2*x^2)/d^2)^p)","A",1
310,1,124,163,0.0514543,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^p}{d+e x} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^p)/(d + e*x),x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d (m+2) \, _2F_1\left(\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \, _2F_1\left(\frac{m}{2}+1,1-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right)\right)}{d^2 (m+1) (m+2)}","\frac{(g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d g (m+1)}-\frac{e (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},1-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{d^2 g^2 (m+2)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*(-(e*(1 + m)*x*Hypergeometric2F1[1 + m/2, 1 - p, 2 + m/2, (e^2*x^2)/d^2]) + d*(2 + m)*Hypergeometric2F1[(1 + m)/2, 1 - p, (3 + m)/2, (e^2*x^2)/d^2]))/(d^2*(1 + m)*(2 + m)*(1 - (e^2*x^2)/d^2)^p)","A",1
311,1,180,214,0.1053481,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^p}{(d+e x)^2} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)-e (m+1) x \left(2 d (m+3) \, _2F_1\left(\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)-e (m+2) x \, _2F_1\left(\frac{m+3}{2},2-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)\right)\right)}{d^4 (m+1) (m+2) (m+3)}","-\frac{2 (m+p) (g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d^2 g (m+1) (-m-2 p+1)}+\frac{(g x)^{m+1} \left(d^2-e^2 x^2\right)^{p-1}}{g (-m-2 p+1)}-\frac{2 e (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{d^3 g^2 (m+2)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[(2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hypergeometric2F1[(3 + m)/2, 2 - p, (5 + m)/2, (e^2*x^2)/d^2])))/(d^4*(1 + m)*(2 + m)*(3 + m)*(1 - (e^2*x^2)/d^2)^p)","A",1
312,1,206,275,0.1932675,"\int \frac{(g x)^m \left(d^2-e^2 x^2\right)^p}{(d+e x)^3} \, dx","Integrate[((g*x)^m*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]","\frac{x (g x)^m \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \left(e x \left(e x \left(\frac{3 d \, _2F_1\left(\frac{m+3}{2},3-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right)}{m+3}-\frac{e x \, _2F_1\left(\frac{m+4}{2},3-p;\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right)}{m+4}\right)-\frac{3 d^2 \, _2F_1\left(\frac{m+2}{2},3-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{m+2}\right)+\frac{d^3 \, _2F_1\left(\frac{m+1}{2},3-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{m+1}\right)}{d^6}","-\frac{e (g x)^{m+2} \left(d^2-e^2 x^2\right)^{p-2}}{g^2 (-m-2 p+2)}+\frac{3 d (g x)^{m+1} \left(d^2-e^2 x^2\right)^{p-2}}{g (-m-2 p+3)}-\frac{2 e (-2 m-3 p+2) (g x)^{m+2} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+2}{2},3-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right)}{d^4 g^2 (m+2) (-m-2 p+2)}-\frac{2 (2 m+p) (g x)^{m+1} \left(d^2-e^2 x^2\right)^p \left(1-\frac{e^2 x^2}{d^2}\right)^{-p} \, _2F_1\left(\frac{m+1}{2},3-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right)}{d^3 g (m+1) (-m-2 p+3)}",1,"(x*(g*x)^m*(d^2 - e^2*x^2)^p*((d^3*Hypergeometric2F1[(1 + m)/2, 3 - p, (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((-3*d^2*Hypergeometric2F1[(2 + m)/2, 3 - p, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2F1[(3 + m)/2, 3 - p, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) - (e*x*Hypergeometric2F1[(4 + m)/2, 3 - p, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/(d^6*(1 - (e^2*x^2)/d^2)^p)","A",1
313,1,77,89,0.0413118,"\int \frac{(g x)^m \left(1-a^2 x^2\right)^p}{1+a x} \, dx","Integrate[((g*x)^m*(1 - a^2*x^2)^p)/(1 + a*x),x]","x (g x)^m \left(\frac{\, _2F_1\left(\frac{m+1}{2},1-p;\frac{m+3}{2};a^2 x^2\right)}{m+1}-\frac{a x \, _2F_1\left(\frac{m}{2}+1,1-p;\frac{m}{2}+2;a^2 x^2\right)}{m+2}\right)","\frac{(g x)^{m+1} \, _2F_1\left(\frac{m+1}{2},1-p;\frac{m+3}{2};a^2 x^2\right)}{g (m+1)}-\frac{a (g x)^{m+2} \, _2F_1\left(\frac{m+2}{2},1-p;\frac{m+4}{2};a^2 x^2\right)}{g^2 (m+2)}",1,"x*(g*x)^m*(-((a*x*Hypergeometric2F1[1 + m/2, 1 - p, 2 + m/2, a^2*x^2])/(2 + m)) + Hypergeometric2F1[(1 + m)/2, 1 - p, (3 + m)/2, a^2*x^2]/(1 + m))","A",1
314,1,90,96,0.1124447,"\int (g x)^m (d+e x)^n \left(d^2-e^2 x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)^n*(d^2 - e^2*x^2)^p,x]","\frac{x (g x)^m (d-e x)^p \left(\frac{d-e x}{d}\right)^{-p} (d+e x)^{n+p} \left(\frac{d+e x}{d}\right)^{-n-p} F_1\left(m+1;-p,-n-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right)}{m+1}","\frac{(g x)^{m+1} (d+e x)^n \left(1-\frac{e x}{d}\right)^{-p} \left(d^2-e^2 x^2\right)^p \left(\frac{e x}{d}+1\right)^{-n-p} F_1\left(m+1;-p,-n-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right)}{g (m+1)}",1,"(x*(g*x)^m*(d - e*x)^p*(d + e*x)^(n + p)*((d + e*x)/d)^(-n - p)*AppellF1[1 + m, -p, -n - p, 2 + m, (e*x)/d, -((e*x)/d)])/((1 + m)*((d - e*x)/d)^p)","A",0
315,1,60,214,0.0429053,"\int \frac{x \sqrt{1+x}}{1+x^2} \, dx","Integrate[(x*Sqrt[1 + x])/(1 + x^2),x]","2 \sqrt{x+1}-\sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1-i}}\right)-\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1+i}}\right)","2 \sqrt{x+1}+\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \log \left(x-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)-\frac{1}{2} \sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \log \left(x+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{x+1}+\sqrt{2}+1\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{x+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{\sqrt{2 \left(1+\sqrt{2}\right)}}-\frac{\tan ^{-1}\left(\frac{2 \sqrt{x+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{\sqrt{2 \left(1+\sqrt{2}\right)}}",1,"2*Sqrt[1 + x] - Sqrt[1 - I]*ArcTanh[Sqrt[1 + x]/Sqrt[1 - I]] - Sqrt[1 + I]*ArcTanh[Sqrt[1 + x]/Sqrt[1 + I]]","C",1
316,1,259,255,0.6128171,"\int \frac{x^4 \sqrt{a+c x^2}}{d+e x} \, dx","Integrate[(x^4*Sqrt[a + c*x^2])/(d + e*x),x]","\frac{e \sqrt{a+c x^2} \left(-16 a^2 e^4+a c e^2 \left(40 d^2-15 d e x+8 e^2 x^2\right)+2 c^2 \left(60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right)\right)-120 c^{5/2} d^5 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-120 c^2 d^4 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+\frac{15 \sqrt{a} \sqrt{c} d e^2 \sqrt{a+c x^2} \left(a e^2-4 c d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}}{120 c^2 e^6}","-\frac{d \left(-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{3/2} e^6}+\frac{\left(a+c x^2\right)^{3/2} \left(47 c d^2-8 a e^2\right)}{60 c^2 e^3}-\frac{d^4 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^6}+\frac{d \sqrt{a+c x^2} \left(8 c d^3-e x \left(4 c d^2-a e^2\right)\right)}{8 c e^5}-\frac{13 d \left(a+c x^2\right)^{3/2} (d+e x)}{20 c e^3}+\frac{\left(a+c x^2\right)^{3/2} (d+e x)^2}{5 c e^3}",1,"(e*Sqrt[a + c*x^2]*(-16*a^2*e^4 + a*c*e^2*(40*d^2 - 15*d*e*x + 8*e^2*x^2) + 2*c^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + (15*Sqrt[a]*Sqrt[c]*d*e^2*(-4*c*d^2 + a*e^2)*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a] - 120*c^(5/2)*d^5*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - 120*c^2*d^4*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(120*c^2*e^6)","A",1
317,1,225,211,0.4004324,"\int \frac{x^3 \sqrt{a+c x^2}}{d+e x} \, dx","Integrate[(x^3*Sqrt[a + c*x^2])/(d + e*x),x]","\frac{24 c^{3/2} d^4 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+24 c d^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+e \sqrt{a+c x^2} \left(a e^2 (3 e x-8 d)+c \left(-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right)\right)}{24 c e^5}-\frac{\sqrt{a} \sqrt{a+c x^2} \left(a e^2-4 c d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 c^{3/2} e^3 \sqrt{\frac{c x^2}{a}+1}}","\frac{\left(-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{3/2} e^5}+\frac{d^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^5}-\frac{\sqrt{a+c x^2} \left(8 c d^3-e x \left(4 c d^2-a e^2\right)\right)}{8 c e^4}-\frac{7 d \left(a+c x^2\right)^{3/2}}{12 c e^2}+\frac{\left(a+c x^2\right)^{3/2} (d+e x)}{4 c e^2}",1,"-1/8*(Sqrt[a]*(-4*c*d^2 + a*e^2)*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(c^(3/2)*e^3*Sqrt[1 + (c*x^2)/a]) + (e*Sqrt[a + c*x^2]*(a*e^2*(-8*d + 3*e*x) + c*(-24*d^3 + 12*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3)) + 24*c^(3/2)*d^4*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + 24*c*d^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(24*c*e^5)","A",1
318,1,193,153,0.318829,"\int \frac{x^2 \sqrt{a+c x^2}}{d+e x} \, dx","Integrate[(x^2*Sqrt[a + c*x^2])/(d + e*x),x]","\frac{-6 c^{3/2} d^3 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+e \sqrt{a+c x^2} \left(2 a e^2+c \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-6 c d^2 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)-\frac{3 \sqrt{a} \sqrt{c} d e^2 \sqrt{a+c x^2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}}{6 c e^4}","-\frac{d^2 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^4}-\frac{d \left(a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 \sqrt{c} e^4}+\frac{d \sqrt{a+c x^2} (2 d-e x)}{2 e^3}+\frac{\left(a+c x^2\right)^{3/2}}{3 c e}",1,"(e*Sqrt[a + c*x^2]*(2*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - (3*Sqrt[a]*Sqrt[c]*d*e^2*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a] - 6*c^(3/2)*d^3*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - 6*c*d^2*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(6*c*e^4)","A",1
319,1,175,127,0.2754662,"\int \frac{x \sqrt{a+c x^2}}{d+e x} \, dx","Integrate[(x*Sqrt[a + c*x^2])/(d + e*x),x]","\frac{\frac{a^{3/2} e^2 \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{c} \sqrt{a+c x^2}}+2 d \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+2 \sqrt{c} d^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-2 d e \sqrt{a+c x^2}+e^2 x \sqrt{a+c x^2}}{2 e^3}","\frac{\left(a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 \sqrt{c} e^3}+\frac{d \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^3}-\frac{\sqrt{a+c x^2} (2 d-e x)}{2 e^2}",1,"(-2*d*e*Sqrt[a + c*x^2] + e^2*x*Sqrt[a + c*x^2] + (a^(3/2)*e^2*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[a + c*x^2]) + 2*Sqrt[c]*d^2*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + 2*d*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*e^3)","A",1
320,1,99,103,0.0241817,"\int \frac{\sqrt{a+c x^2}}{d+e x} \, dx","Integrate[Sqrt[a + c*x^2]/(d + e*x),x]","\frac{-\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)-\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+e \sqrt{a+c x^2}}{e^2}","-\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2}-\frac{\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e^2}+\frac{\sqrt{a+c x^2}}{e}",1,"(e*Sqrt[a + c*x^2] - Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^2","A",1
321,1,113,116,0.0484405,"\int \frac{\sqrt{a+c x^2}}{x (d+e x)} \, dx","Integrate[Sqrt[a + c*x^2]/(x*(d + e*x)),x]","\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)-\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d e}","\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d e}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{e}",1,"(Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])] - Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(d*e)","A",1
322,1,178,105,0.2379428,"\int \frac{\sqrt{a+c x^2}}{x^2 (d+e x)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^2*(d + e*x)),x]","\frac{-\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)-\frac{d \sqrt{a+c x^2}}{x}+\frac{\sqrt{a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a+c x^2}}-\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}","-\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^2}+\frac{\sqrt{a} e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^2}-\frac{\sqrt{a+c x^2}}{d x}",1,"(-((d*Sqrt[a + c*x^2])/x) + (Sqrt[a]*Sqrt[c]*d*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a + c*x^2] - Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])] + Sqrt[a]*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^2","A",1
323,1,283,160,0.3965115,"\int \frac{\sqrt{a+c x^2}}{x^3 (d+e x)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^3*(d + e*x)),x]","-\frac{-2 e x^2 \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+c d^2 x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)+2 \sqrt{a} \sqrt{c} d e x^2 \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)-2 \sqrt{c} d e x^2 \sqrt{a+c x^2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+2 \sqrt{a} e^2 x^2 \sqrt{a+c x^2} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)+a d^2-2 a d e x+c d^2 x^2-2 c d e x^3}{2 d^3 x^2 \sqrt{a+c x^2}}","-\frac{\sqrt{a} e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{e \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^3}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 \sqrt{a} d}",1,"-1/2*(a*d^2 - 2*a*d*e*x + c*d^2*x^2 - 2*c*d*e*x^3 + 2*Sqrt[a]*Sqrt[c]*d*e*x^2*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] - 2*Sqrt[c]*d*e*x^2*Sqrt[a + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - 2*e*Sqrt[c*d^2 + a*e^2]*x^2*Sqrt[a + c*x^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])] + 2*Sqrt[a]*e^2*x^2*Sqrt[a + c*x^2]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]] + c*d^2*x^2*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]])/(d^3*x^2*Sqrt[a + c*x^2])","A",1
324,1,301,191,1.0202759,"\int \frac{\sqrt{a+c x^2}}{x^4 (d+e x)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^4*(d + e*x)),x]","-\frac{\frac{2 d^3 \left(a+c x^2\right)^{3/2}}{a x^3}+6 e^2 \left(\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)-\frac{3 d^2 e \left(c x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)+a+c x^2\right)}{x^2 \sqrt{a+c x^2}}+\frac{6 d e^2 \left(-\sqrt{a} \sqrt{c} x \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+a+c x^2\right)}{x \sqrt{a+c x^2}}-6 e^3 \sqrt{a+c x^2}+6 e^3 \left(\sqrt{a+c x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)\right)}{6 d^4}","\frac{\sqrt{a} e^3 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^4}-\frac{e^2 \sqrt{a+c x^2}}{d^3 x}+\frac{e \sqrt{a+c x^2}}{2 d^2 x^2}+\frac{c e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 \sqrt{a} d^2}-\frac{e^2 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^4}-\frac{\left(a+c x^2\right)^{3/2}}{3 a d x^3}",1,"-1/6*(-6*e^3*Sqrt[a + c*x^2] + (2*d^3*(a + c*x^2)^(3/2))/(a*x^3) + (6*d*e^2*(a + c*x^2 - Sqrt[a]*Sqrt[c]*x*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(x*Sqrt[a + c*x^2]) + 6*e^2*(Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]) + 6*e^3*(Sqrt[a + c*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]) - (3*d^2*e*(a + c*x^2 + c*x^2*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]]))/(x^2*Sqrt[a + c*x^2]))/d^4","A",1
325,1,344,274,1.0974315,"\int \frac{\sqrt{a+c x^2}}{x^5 (d+e x)} \, dx","Integrate[Sqrt[a + c*x^2]/(x^5*(d + e*x)),x]","\frac{-\frac{2 c^2 d^4 \left(a+c x^2\right)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{a}+1\right)}{a^3}+\frac{2 d^3 e \left(a+c x^2\right)^{3/2}}{a x^3}-\frac{3 d^2 e^2 \left(c x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)+a+c x^2\right)}{x^2 \sqrt{a+c x^2}}+6 e^3 \left(\sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)+\sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)+\frac{6 d e^3 \left(-\sqrt{a} \sqrt{c} x \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+a+c x^2\right)}{x \sqrt{a+c x^2}}-6 e^4 \sqrt{a+c x^2}+6 e^4 \left(\sqrt{a+c x^2}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)\right)}{6 d^5}","\frac{c^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{8 a^{3/2} d}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{d^5}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}-\frac{c e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 \sqrt{a} d^3}+\frac{e \left(a+c x^2\right)^{3/2}}{3 a d^2 x^3}+\frac{e^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^5}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{\sqrt{a+c x^2}}{4 d x^4}",1,"(-6*e^4*Sqrt[a + c*x^2] + (2*d^3*e*(a + c*x^2)^(3/2))/(a*x^3) + (6*d*e^3*(a + c*x^2 - Sqrt[a]*Sqrt[c]*x*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(x*Sqrt[a + c*x^2]) + 6*e^3*(Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]) + 6*e^4*(Sqrt[a + c*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]) - (3*d^2*e^2*(a + c*x^2 + c*x^2*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]]))/(x^2*Sqrt[a + c*x^2]) - (2*c^2*d^4*(a + c*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x^2)/a])/a^3)/(6*d^5)","C",1
326,1,149,195,0.2272246,"\int \frac{x^4}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[x^4/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{-\frac{3 d \left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2}}+\frac{e \sqrt{a+c x^2} \left(-4 a e^2+6 c d^2-3 c d e x+2 c e^2 x^2\right)}{c^2}-\frac{6 d^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}}{6 e^4}","-\frac{d \left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2} e^4}+\frac{\sqrt{a+c x^2} \left(11 c d^2-4 a e^2\right)}{6 c^2 e^3}-\frac{d^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^4 \sqrt{a e^2+c d^2}}-\frac{7 d \sqrt{a+c x^2} (d+e x)}{6 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^3}",1,"((e*Sqrt[a + c*x^2]*(6*c*d^2 - 4*a*e^2 - 3*c*d*e*x + 2*c*e^2*x^2))/c^2 - (3*d*(2*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/c^(3/2) - (6*d^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/Sqrt[c*d^2 + a*e^2])/(6*e^4)","A",1
327,1,131,152,0.2218754,"\int \frac{x^3}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[x^3/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{\left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+\sqrt{c} \left(\frac{2 c d^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}+e \sqrt{a+c x^2} (e x-2 d)\right)}{2 c^{3/2} e^3}","\frac{\left(2 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2} e^3}+\frac{d^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^3 \sqrt{a e^2+c d^2}}-\frac{3 d \sqrt{a+c x^2}}{2 c e^2}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^2}",1,"((2*c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[c]*(e*(-2*d + e*x)*Sqrt[a + c*x^2] + (2*c*d^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/Sqrt[c*d^2 + a*e^2]))/(2*c^(3/2)*e^3)","A",1
328,1,105,109,0.0735539,"\int \frac{x^2}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[x^2/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{-\frac{d^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}}{e^2}","-\frac{d^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2 \sqrt{a e^2+c d^2}}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} e^2}+\frac{\sqrt{a+c x^2}}{c e}",1,"((e*Sqrt[a + c*x^2])/c - (d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c] - (d^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/Sqrt[c*d^2 + a*e^2])/e^2","A",1
329,1,86,86,0.0259749,"\int \frac{x}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[x/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e \sqrt{a e^2+c d^2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} e}","\frac{d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e \sqrt{a e^2+c d^2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} e}",1,"ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]/(Sqrt[c]*e) + (d*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*Sqrt[c*d^2 + a*e^2])","A",1
330,1,54,54,0.0068988,"\int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + c*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}","-\frac{\tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}",1,"-(ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]/Sqrt[c*d^2 + a*e^2])","A",1
331,1,86,86,0.0452312,"\int \frac{1}{x (d+e x) \sqrt{a+c x^2}} \, dx","Integrate[1/(x*(d + e*x)*Sqrt[a + c*x^2]),x]","\frac{e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d \sqrt{a e^2+c d^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d \sqrt{a e^2+c d^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(e*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d*Sqrt[c*d^2 + a*e^2]) - ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]/(Sqrt[a]*d)","A",1
332,1,107,111,0.0825843,"\int \frac{1}{x^2 (d+e x) \sqrt{a+c x^2}} \, dx","Integrate[1/(x^2*(d + e*x)*Sqrt[a + c*x^2]),x]","\frac{-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}-\frac{d \sqrt{a+c x^2}}{a x}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a}}}{d^2}","-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^2 \sqrt{a e^2+c d^2}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^2}-\frac{\sqrt{a+c x^2}}{a d x}",1,"(-((d*Sqrt[a + c*x^2])/(a*x)) - (e^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/Sqrt[c*d^2 + a*e^2] + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a])/d^2","A",1
333,1,163,168,0.6422796,"\int \frac{1}{x^3 (d+e x) \sqrt{a+c x^2}} \, dx","Integrate[1/(x^3*(d + e*x)*Sqrt[a + c*x^2]),x]","\frac{\frac{2 e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\sqrt{a e^2+c d^2}}+\frac{d \left(c d x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x^2}{a}+1}\right)-\left(a+c x^2\right) (d-2 e x)\right)}{a x^2 \sqrt{a+c x^2}}-\frac{2 e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a}}}{2 d^3}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{3/2} d}-\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^3}+\frac{e \sqrt{a+c x^2}}{a d^2 x}+\frac{e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^3 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{2 a d x^2}",1,"((2*e^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/Sqrt[c*d^2 + a*e^2] - (2*e^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a] + (d*(-((d - 2*e*x)*(a + c*x^2)) + c*d*x^2*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]]))/(a*x^2*Sqrt[a + c*x^2]))/(2*d^3)","A",1
334,1,179,146,0.4438339,"\int \frac{x^4}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[x^4/((d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{\frac{e \left(2 a^2 e^2+a c \left(d^2+d e x+e^2 x^2\right)+c^2 d^2 x^2\right)}{c^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{\sqrt{a} d \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2} \sqrt{a+c x^2}}-\frac{d^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}}{e^2}","-\frac{d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2} e^2}+\frac{a (a e+c d x)}{c^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{\sqrt{a+c x^2}}{c^2 e}-\frac{d^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2 \left(a e^2+c d^2\right)^{3/2}}",1,"((e*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + d*e*x + e^2*x^2)))/(c^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (Sqrt[a]*d*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(c^(3/2)*Sqrt[a + c*x^2]) - (d^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2))/e^2","A",1
335,1,153,123,0.2798134,"\int \frac{x^3}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[x^3/((d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{\frac{\sqrt{c} \left(a e (d-e x) \sqrt{a e^2+c d^2}+c d^3 \sqrt{a+c x^2} \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)\right)}{\left(a e^2+c d^2\right)^{3/2}}+\sqrt{a} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2} e \sqrt{a+c x^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2} e}+\frac{a (d-e x)}{c \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{d^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e \left(a e^2+c d^2\right)^{3/2}}",1,"(Sqrt[a]*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + (Sqrt[c]*(a*e*Sqrt[c*d^2 + a*e^2]*(d - e*x) + c*d^3*Sqrt[a + c*x^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]))/(c*d^2 + a*e^2)^(3/2))/(c^(3/2)*e*Sqrt[a + c*x^2])","A",1
336,1,95,95,0.0835338,"\int \frac{x^2}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[x^2/((d + e*x)*(a + c*x^2)^(3/2)),x]","-\frac{a e+c d x}{c \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{d^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}","-\frac{a e+c d x}{c \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{d^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"-((a*e + c*d*x)/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) - (d^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2)","A",1
337,1,88,88,0.053063,"\int \frac{x}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[x/((d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{e x-d}{\sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{d e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}","\frac{d e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{d-e x}{\sqrt{a+c x^2} \left(a e^2+c d^2\right)}",1,"(-d + e*x)/((c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + (d*e*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2)","A",1
338,1,94,94,0.0454105,"\int \frac{1}{(d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{a e+c d x}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}","\frac{a e+c d x}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (e^2*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2)","A",1
339,1,132,147,0.1684171,"\int \frac{1}{x (d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/(x*(d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{-\frac{e (a e+c d x)}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a \sqrt{a+c x^2}}}{d}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{e (a e+c d x)}{a d \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d \left(a e^2+c d^2\right)^{3/2}}+\frac{1}{a d \sqrt{a+c x^2}}",1,"(-((e*(a*e + c*d*x))/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) + (e^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2) + Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/a]/(a*Sqrt[a + c*x^2]))/d","C",1
340,1,163,194,0.4046011,"\int \frac{1}{x^2 (d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(d + e*x)*(a + c*x^2)^(3/2)),x]","-\frac{\frac{d \left(a+2 c x^2\right)}{a^2 x \sqrt{a+c x^2}}-\frac{e^2 (a e+c d x)}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{e \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a \sqrt{a+c x^2}}}{d^2}","\frac{e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{e^2 (a e+c d x)}{a d^2 \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{e^4 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^2 \left(a e^2+c d^2\right)^{3/2}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}}",1,"-((-((e^2*(a*e + c*d*x))/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) + (d*(a + 2*c*x^2))/(a^2*x*Sqrt[a + c*x^2]) + (e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2) + (e*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/a])/(a*Sqrt[a + c*x^2]))/d^2)","C",1
341,1,203,276,0.3422125,"\int \frac{1}{x^3 (d+e x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(d + e*x)*(a + c*x^2)^(3/2)),x]","\frac{-\frac{c d^2 \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a^2 \sqrt{a+c x^2}}+\frac{d e \left(a+2 c x^2\right)}{a^2 x \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{e^3 (a e+c d x)}{a \sqrt{a+c x^2} \left(a e^2+c d^2\right)}+\frac{e^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right)}{a \sqrt{a+c x^2}}}{d^3}","-\frac{e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{a^{3/2} d^3}+\frac{3 c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{5/2} d}+\frac{2 c e x}{a^2 d^2 \sqrt{a+c x^2}}-\frac{3 c}{2 a^2 d \sqrt{a+c x^2}}+\frac{e^2}{a d^3 \sqrt{a+c x^2}}+\frac{e}{a d^2 x \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^3 \left(a e^2+c d^2\right)^{3/2}}-\frac{e^3 (a e+c d x)}{a d^3 \sqrt{a+c x^2} \left(a e^2+c d^2\right)}-\frac{1}{2 a d x^2 \sqrt{a+c x^2}}",1,"(-((e^3*(a*e + c*d*x))/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) + (d*e*(a + 2*c*x^2))/(a^2*x*Sqrt[a + c*x^2]) + (e^5*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2) + (e^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/a])/(a*Sqrt[a + c*x^2]) - (c*d^2*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x^2)/a])/(a^2*Sqrt[a + c*x^2]))/d^3","C",1
342,1,230,244,0.5129754,"\int \frac{x^5}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[x^5/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{-\frac{3 d \left(4 c d^2-a e^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{3/2}}+e \sqrt{a+c x^2} \left(-\frac{2 a e^2}{c^2}+\frac{3 d^5}{(d+e x) \left(a e^2+c d^2\right)}+\frac{9 d^2-3 d e x+e^2 x^2}{c}\right)-\frac{3 d^4 \left(5 a e^2+4 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{3 d^4 \left(5 a e^2+4 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}}{3 e^5}","-\frac{d \left(4 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2} e^5}+\frac{\sqrt{a+c x^2} \left(13 c d^2-2 a e^2\right)}{3 c^2 e^4}+\frac{d^5 \sqrt{a+c x^2}}{e^4 (d+e x) \left(a e^2+c d^2\right)}-\frac{d^4 \left(5 a e^2+4 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^5 \left(a e^2+c d^2\right)^{3/2}}-\frac{5 d \sqrt{a+c x^2} (d+e x)}{3 c e^4}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^4}",1,"(e*Sqrt[a + c*x^2]*((-2*a*e^2)/c^2 + (3*d^5)/((c*d^2 + a*e^2)*(d + e*x)) + (9*d^2 - 3*d*e*x + e^2*x^2)/c) + (3*d^4*(4*c*d^2 + 5*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - (3*d*(4*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(3/2) - (3*d^4*(4*c*d^2 + 5*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(3*e^5)","A",1
343,1,208,204,0.3661234,"\int \frac{x^4}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[x^4/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{\frac{\left(6 c d^2-a e^2\right) \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{c^{3/2}}+e \sqrt{a+c x^2} \left(\frac{e x-4 d}{c}-\frac{2 d^4}{(d+e x) \left(a e^2+c d^2\right)}\right)+\frac{2 d^3 \left(4 a e^2+3 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{2 d^3 \left(4 a e^2+3 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}}{2 e^4}","\frac{\left(6 c d^2-a e^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2} e^4}-\frac{d^4 \sqrt{a+c x^2}}{e^3 (d+e x) \left(a e^2+c d^2\right)}+\frac{d^3 \left(4 a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^4 \left(a e^2+c d^2\right)^{3/2}}-\frac{5 d \sqrt{a+c x^2}}{2 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^3}",1,"(e*Sqrt[a + c*x^2]*((-4*d + e*x)/c - (2*d^4)/((c*d^2 + a*e^2)*(d + e*x))) - (2*d^3*(3*c*d^2 + 4*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + ((6*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/c^(3/2) + (2*d^3*(3*c*d^2 + 4*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(2*e^4)","A",1
344,1,184,160,0.2744084,"\int \frac{x^3}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[x^3/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{-\frac{d^2 \left(3 a e^2+2 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{d^2 \left(3 a e^2+2 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}+e \sqrt{a+c x^2} \left(\frac{d^3}{(d+e x) \left(a e^2+c d^2\right)}+\frac{1}{c}\right)-\frac{2 d \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{\sqrt{c}}}{e^3}","-\frac{d^2 \left(3 a e^2+2 c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^3 \left(a e^2+c d^2\right)^{3/2}}+\frac{d^3 \sqrt{a+c x^2}}{e^2 (d+e x) \left(a e^2+c d^2\right)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} e^3}+\frac{\sqrt{a+c x^2}}{c e^2}",1,"(e*Sqrt[a + c*x^2]*(c^(-1) + d^3/((c*d^2 + a*e^2)*(d + e*x))) + (d^2*(2*c*d^2 + 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - (2*d*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c] - (d^2*(2*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/e^3","A",1
345,1,172,137,0.3225945,"\int \frac{x^2}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[x^2/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{d \left(\frac{\left(2 a e^2+c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{d e \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}\right)-\frac{\left(2 a d e^2+c d^3\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{\log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{\sqrt{c}}}{e^2}","-\frac{d^2 \sqrt{a+c x^2}}{e (d+e x) \left(a e^2+c d^2\right)}+\frac{d \left(2 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{e^2 \left(a e^2+c d^2\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} e^2}",1,"(-(((c*d^3 + 2*a*d*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2)) + Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]]/Sqrt[c] + d*(-((d*e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x))) + ((c*d^2 + 2*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2)))/e^2","A",1
346,1,90,90,0.0461244,"\int \frac{x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[x/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{d \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{a e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}","\frac{d \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{a e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"(d*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)) - (a*e*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2)","A",1
347,1,115,91,0.0737085,"\int \frac{1}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + c*x^2]),x]","-\frac{e \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{c d \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{c d \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}","-\frac{e \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}-\frac{c d \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}",1,"-((e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x))) + (c*d*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - (c*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2)","A",1
348,1,178,179,0.221648,"\int \frac{1}{x (d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[1/(x*(d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{\frac{d e^2 \sqrt{a+c x^2}}{(d+e x) \left(a e^2+c d^2\right)}+\frac{e \left(a e^2+2 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{e \left(a e^2+2 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{\log \left(\sqrt{a} \sqrt{a+c x^2}+a\right)}{\sqrt{a}}+\frac{\log (x)}{\sqrt{a}}}{d^2}","\frac{e^2 \sqrt{a+c x^2}}{d (d+e x) \left(a e^2+c d^2\right)}+\frac{e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^2 \sqrt{a e^2+c d^2}}+\frac{c e \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^2}",1,"((d*e^2*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)) + Log[x]/Sqrt[a] - (e*(2*c*d^2 + a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - Log[a + Sqrt[a]*Sqrt[a + c*x^2]]/Sqrt[a] + (e*(2*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/d^2","A",1
349,1,197,212,0.3278872,"\int \frac{1}{x^2 (d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[1/(x^2*(d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{-\frac{e^2 \left(2 a e^2+3 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}+\frac{e^2 \left(2 a e^2+3 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}-d \sqrt{a+c x^2} \left(\frac{e^3}{(d+e x) \left(a e^2+c d^2\right)}+\frac{1}{a x}\right)+\frac{2 e \log \left(\sqrt{a} \sqrt{a+c x^2}+a\right)}{\sqrt{a}}-\frac{2 e \log (x)}{\sqrt{a}}}{d^3}","\frac{2 e \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^3}-\frac{c e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d \left(a e^2+c d^2\right)^{3/2}}-\frac{e^3 \sqrt{a+c x^2}}{d^2 (d+e x) \left(a e^2+c d^2\right)}-\frac{\sqrt{a+c x^2}}{a d^2 x}-\frac{2 e^2 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^3 \sqrt{a e^2+c d^2}}",1,"(-(d*Sqrt[a + c*x^2]*(1/(a*x) + e^3/((c*d^2 + a*e^2)*(d + e*x)))) - (2*e*Log[x])/Sqrt[a] + (e^2*(3*c*d^2 + 2*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + (2*e*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/Sqrt[a] - (e^2*(3*c*d^2 + 2*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/d^3","A",1
350,1,229,268,0.4483118,"\int \frac{1}{x^3 (d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[1/(x^3*(d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{\frac{\left(c d^2-6 a e^2\right) \log \left(\sqrt{a} \sqrt{a+c x^2}+a\right)}{a^{3/2}}+\frac{\log (x) \left(6 a e^2-c d^2\right)}{a^{3/2}}+d \sqrt{a+c x^2} \left(\frac{2 e^4}{(d+e x) \left(a e^2+c d^2\right)}-\frac{d-4 e x}{a x^2}\right)+\frac{2 e^3 \left(3 a e^2+4 c d^2\right) \log \left(\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right)}{\left(a e^2+c d^2\right)^{3/2}}-\frac{2 e^3 \left(3 a e^2+4 c d^2\right) \log (d+e x)}{\left(a e^2+c d^2\right)^{3/2}}}{2 d^4}","\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{2 a^{3/2} d^2}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right)}{\sqrt{a} d^4}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{c e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^2 \left(a e^2+c d^2\right)^{3/2}}-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{3 e^3 \tanh ^{-1}\left(\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right)}{d^4 \sqrt{a e^2+c d^2}}+\frac{e^4 \sqrt{a+c x^2}}{d^3 (d+e x) \left(a e^2+c d^2\right)}",1,"(d*Sqrt[a + c*x^2]*(-((d - 4*e*x)/(a*x^2)) + (2*e^4)/((c*d^2 + a*e^2)*(d + e*x))) + ((-(c*d^2) + 6*a*e^2)*Log[x])/a^(3/2) - (2*e^3*(4*c*d^2 + 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + ((c*d^2 - 6*a*e^2)*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/a^(3/2) + (2*e^3*(4*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(2*d^4)","A",1
351,1,114,135,0.0924899,"\int x^2 (a+b x)^n \left(c+d x^2\right) \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^2),x]","\frac{(a+b x)^{n+1} \left(\frac{(a+b x)^2 \left(6 a^2 d+b^2 c\right)}{n+3}-\frac{2 a (a+b x) \left(2 a^2 d+b^2 c\right)}{n+2}+\frac{a^4 d+a^2 b^2 c}{n+1}+\frac{d (a+b x)^4}{n+5}-\frac{4 a d (a+b x)^3}{n+4}\right)}{b^5}","\frac{a^2 \left(a^2 d+b^2 c\right) (a+b x)^{n+1}}{b^5 (n+1)}-\frac{2 a \left(2 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{\left(6 a^2 d+b^2 c\right) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)}",1,"((a + b*x)^(1 + n)*((a^2*b^2*c + a^4*d)/(1 + n) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x))/(2 + n) + ((b^2*c + 6*a^2*d)*(a + b*x)^2)/(3 + n) - (4*a*d*(a + b*x)^3)/(4 + n) + (d*(a + b*x)^4)/(5 + n)))/b^5","A",1
352,1,109,102,0.1038495,"\int x (a+b x)^n \left(c+d x^2\right) \, dx","Integrate[x*(a + b*x)^n*(c + d*x^2),x]","\frac{(a+b x)^{n+1} \left(-6 a^3 d+6 a^2 b d (n+1) x-a b^2 \left(c \left(n^2+7 n+12\right)+3 d \left(n^2+3 n+2\right) x^2\right)+b^3 \left(n^2+4 n+3\right) x \left(c (n+4)+d (n+2) x^2\right)\right)}{b^4 (n+1) (n+2) (n+3) (n+4)}","-\frac{a \left(a^2 d+b^2 c\right) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{\left(3 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)}",1,"((a + b*x)^(1 + n)*(-6*a^3*d + 6*a^2*b*d*(1 + n)*x + b^3*(3 + 4*n + n^2)*x*(c*(4 + n) + d*(2 + n)*x^2) - a*b^2*(c*(12 + 7*n + n^2) + 3*d*(2 + 3*n + n^2)*x^2)))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))","A",1
353,1,65,70,0.0410392,"\int (a+b x)^n \left(c+d x^2\right) \, dx","Integrate[(a + b*x)^n*(c + d*x^2),x]","\frac{(a+b x)^{n+1} \left(2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left(c (n+3)+d (n+1) x^2\right)\right)}{b^3 (n+1) (n+2) (n+3)}","\frac{\left(a^2 d+b^2 c\right) (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)}",1,"((a + b*x)^(1 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)))/(b^3*(1 + n)*(2 + n)*(3 + n))","A",1
354,1,64,77,0.0396999,"\int \frac{(a+b x)^n \left(c+d x^2\right)}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^2))/x,x]","-\frac{(a+b x)^{n+1} \left(b^2 c (n+2) \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)+a d (a-b (n+1) x)\right)}{a b^2 (n+1) (n+2)}","-\frac{a d (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"-(((a + b*x)^(1 + n)*(a*d*(a - b*(1 + n)*x) + b^2*c*(2 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a*b^2*(1 + n)*(2 + n)))","A",1
355,1,199,232,0.1810201,"\int x^2 (a+b x)^n \left(c+d x^2\right)^2 \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^2)^2,x]","\frac{(a+b x)^{n+1} \left(\frac{\left(a^3 d+a b^2 c\right)^2}{n+1}+\frac{d (a+b x)^4 \left(15 a^2 d+2 b^2 c\right)}{n+5}-\frac{4 a d (a+b x)^3 \left(5 a^2 d+2 b^2 c\right)}{n+4}-\frac{2 a (a+b x) \left(a^2 d+b^2 c\right) \left(3 a^2 d+b^2 c\right)}{n+2}+\frac{(a+b x)^2 \left(15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right)}{n+3}+\frac{d^2 (a+b x)^6}{n+7}-\frac{6 a d^2 (a+b x)^5}{n+6}\right)}{b^7}","\frac{a^2 \left(a^2 d+b^2 c\right)^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{2 a \left(a^2 d+b^2 c\right) \left(3 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^7 (n+2)}-\frac{4 a d \left(5 a^2 d+2 b^2 c\right) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{d \left(15 a^2 d+2 b^2 c\right) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left(15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^2 (a+b x)^{n+7}}{b^7 (n+7)}",1,"((a + b*x)^(1 + n)*((a*b^2*c + a^3*d)^2/(1 + n) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x))/(2 + n) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^2)/(3 + n) - (4*a*d*(2*b^2*c + 5*a^2*d)*(a + b*x)^3)/(4 + n) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))/b^7","A",1
356,1,323,185,0.4982732,"\int x (a+b x)^n \left(c+d x^2\right)^2 \, dx","Integrate[x*(a + b*x)^n*(c + d*x^2)^2,x]","\frac{(a+b x)^{n+1} \left(4 (n+1) (a+b x) \left((n+5) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)-a d (n+2) (a+b x) \left(2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left(c (n+5)+d (n+3) x^2\right)\right)\right)-a (n+6) \left(4 (n+4) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left(c (n+3)+d (n+1) x^2\right)\right)-4 a d (n+1) (a+b x) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)+b^4 (n+1) (n+2) (n+3) (n+4) \left(c+d x^2\right)^2\right)+b^4 (n+1) (n+2) (n+3) (n+4) (n+5) (a+b x) \left(c+d x^2\right)^2\right)}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}","-\frac{a \left(a^2 d+b^2 c\right)^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac{\left(a^2 d+b^2 c\right) \left(5 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^6 (n+2)}-\frac{2 a d \left(5 a^2 d+3 b^2 c\right) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{2 d \left(5 a^2 d+b^2 c\right) (a+b x)^{n+4}}{b^6 (n+4)}-\frac{5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d^2 (a+b x)^{n+6}}{b^6 (n+6)}",1,"((a + b*x)^(1 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(a + b*x)*(c + d*x^2)^2 - a*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) + 4*(1 + n)*(a + b*x)*((b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))","A",1
357,1,160,140,0.1776443,"\int (a+b x)^n \left(c+d x^2\right)^2 \, dx","Integrate[(a + b*x)^n*(c + d*x^2)^2,x]","\frac{(a+b x)^{n+1} \left(\frac{4 \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left(c (n+3)+d (n+1) x^2\right)\right)}{b^4 (n+1) (n+2) (n+3)}-\frac{4 a d (a+b x) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)}{b^4 (n+2) (n+3) (n+4)}+\left(c+d x^2\right)^2\right)}{b (n+5)}","\frac{\left(a^2 d+b^2 c\right)^2 (a+b x)^{n+1}}{b^5 (n+1)}-\frac{4 a d \left(a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{2 d \left(3 a^2 d+b^2 c\right) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d^2 (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d^2 (a+b x)^{n+5}}{b^5 (n+5)}",1,"((a + b*x)^(1 + n)*((c + d*x^2)^2 + (4*(b^2*c + a^2*d)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)))/(b^4*(1 + n)*(2 + n)*(3 + n)) - (4*a*d*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)))/(b^4*(2 + n)*(3 + n)*(4 + n))))/(b*(5 + n))","A",1
358,1,132,148,0.1339988,"\int \frac{(a+b x)^n \left(c+d x^2\right)^2}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^2)^2)/x,x]","(a+b x)^{n+1} \left(\frac{d (a+b x) \left(3 a^2 d+2 b^2 c\right)}{b^4 (n+2)}-\frac{a d \left(a^2 d+2 b^2 c\right)}{b^4 (n+1)}+\frac{d^2 (a+b x)^3}{b^4 (n+4)}-\frac{3 a d^2 (a+b x)^2}{b^4 (n+3)}-\frac{c^2 \, _2F_1\left(1,n+1;n+2;\frac{a+b x}{a}\right)}{a n+a}\right)","-\frac{a d \left(a^2 d+2 b^2 c\right) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{d \left(3 a^2 d+2 b^2 c\right) (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d^2 (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"(a + b*x)^(1 + n)*(-((a*d*(2*b^2*c + a^2*d))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2*d)*(a + b*x))/(b^4*(2 + n)) - (3*a*d^2*(a + b*x)^2)/(b^4*(3 + n)) + (d^2*(a + b*x)^3)/(b^4*(4 + n)) - (c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))","A",1
359,1,302,343,0.2484848,"\int x^2 (a+b x)^n \left(c+d x^2\right)^3 \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^2)^3,x]","\frac{(a+b x)^{n+1} \left(\frac{d^2 (a+b x)^6 \left(28 a^2 d+3 b^2 c\right)}{n+7}-\frac{2 a d^2 (a+b x)^5 \left(28 a^2 d+9 b^2 c\right)}{n+6}-\frac{2 a (a+b x) \left(a^2 d+b^2 c\right)^2 \left(4 a^2 d+b^2 c\right)}{n+2}+\frac{a^2 \left(a^2 d+b^2 c\right)^3}{n+1}+\frac{d (a+b x)^4 \left(70 a^4 d^2+45 a^2 b^2 c d+3 b^4 c^2\right)}{n+5}-\frac{4 a d (a+b x)^3 \left(14 a^4 d^2+15 a^2 b^2 c d+3 b^4 c^2\right)}{n+4}+\frac{(a+b x)^2 \left(a^2 d+b^2 c\right) \left(28 a^4 d^2+17 a^2 b^2 c d+b^4 c^2\right)}{n+3}+\frac{d^3 (a+b x)^8}{n+9}-\frac{8 a d^3 (a+b x)^7}{n+8}\right)}{b^9}","-\frac{2 a d^2 \left(28 a^2 d+9 b^2 c\right) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{d^2 \left(28 a^2 d+3 b^2 c\right) (a+b x)^{n+7}}{b^9 (n+7)}+\frac{a^2 \left(a^2 d+b^2 c\right)^3 (a+b x)^{n+1}}{b^9 (n+1)}-\frac{2 a \left(a^2 d+b^2 c\right)^2 \left(4 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{\left(a^2 d+b^2 c\right) \left(28 a^4 d^2+17 a^2 b^2 c d+b^4 c^2\right) (a+b x)^{n+3}}{b^9 (n+3)}-\frac{4 a d \left(14 a^4 d^2+15 a^2 b^2 c d+3 b^4 c^2\right) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{d \left(70 a^4 d^2+45 a^2 b^2 c d+3 b^4 c^2\right) (a+b x)^{n+5}}{b^9 (n+5)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)}",1,"((a + b*x)^(1 + n)*((a^2*(b^2*c + a^2*d)^3)/(1 + n) - (2*a*(b^2*c + a^2*d)^2*(b^2*c + 4*a^2*d)*(a + b*x))/(2 + n) + ((b^2*c + a^2*d)*(b^4*c^2 + 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^2)/(3 + n) - (4*a*d*(3*b^4*c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^3)/(4 + n) + (d*(3*b^4*c^2 + 45*a^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^4)/(5 + n) - (2*a*d^2*(9*b^2*c + 28*a^2*d)*(a + b*x)^5)/(6 + n) + (d^2*(3*b^2*c + 28*a^2*d)*(a + b*x)^6)/(7 + n) - (8*a*d^3*(a + b*x)^7)/(8 + n) + (d^3*(a + b*x)^8)/(9 + n)))/b^9","A",1
360,1,709,282,1.4678519,"\int x (a+b x)^n \left(c+d x^2\right)^3 \, dx","Integrate[x*(a + b*x)^n*(c + d*x^2)^3,x]","\frac{(a+b x)^{n+1} \left(6 (n+1) (a+b x) \left((n+7) \left(a^2 d+b^2 c\right) \left(4 (n+5) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)-4 a d (n+2) (a+b x) \left(2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left(c (n+5)+d (n+3) x^2\right)\right)+b^4 (n+2) (n+3) (n+4) (n+5) \left(c+d x^2\right)^2\right)-a d (n+2) (a+b x) \left(4 (n+6) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left(c (n+5)+d (n+3) x^2\right)\right)-4 a d (n+3) (a+b x) \left(2 a^2 d-2 a b d (n+4) x+b^2 (n+5) \left(c (n+6)+d (n+4) x^2\right)\right)+b^4 (n+3) (n+4) (n+5) (n+6) \left(c+d x^2\right)^2\right)\right)-a (n+8) \left(6 (n+6) \left(a^2 d+b^2 c\right) \left(4 (n+4) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left(c (n+3)+d (n+1) x^2\right)\right)-4 a d (n+1) (a+b x) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)+b^4 (n+1) (n+2) (n+3) (n+4) \left(c+d x^2\right)^2\right)-6 a d (n+1) (a+b x) \left(4 (n+5) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)-4 a d (n+2) (a+b x) \left(2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left(c (n+5)+d (n+3) x^2\right)\right)+b^4 (n+2) (n+3) (n+4) (n+5) \left(c+d x^2\right)^2\right)+b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) \left(c+d x^2\right)^3\right)+b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (a+b x) \left(c+d x^2\right)^3\right)}{b^8 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)}","-\frac{5 a d^2 \left(7 a^2 d+3 b^2 c\right) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{3 d^2 \left(7 a^2 d+b^2 c\right) (a+b x)^{n+6}}{b^8 (n+6)}-\frac{a \left(a^2 d+b^2 c\right)^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left(a^2 d+b^2 c\right)^2 \left(7 a^2 d+b^2 c\right) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{3 a d \left(a^2 d+b^2 c\right) \left(7 a^2 d+3 b^2 c\right) (a+b x)^{n+3}}{b^8 (n+3)}+\frac{d \left(35 a^4 d^2+30 a^2 b^2 c d+3 b^4 c^2\right) (a+b x)^{n+4}}{b^8 (n+4)}-\frac{7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^3 (a+b x)^{n+8}}{b^8 (n+8)}",1,"((a + b*x)^(1 + n)*(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(a + b*x)*(c + d*x^2)^3 - a*(8 + n)*(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(c + d*x^2)^3 + 6*(b^2*c + a^2*d)*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) - 6*a*d*(1 + n)*(a + b*x)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))) + 6*(1 + n)*(a + b*x)*((b^2*c + a^2*d)*(7 + n)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2))) - a*d*(2 + n)*(a + b*x)*(b^4*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(6 + n)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)) - 4*a*d*(3 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(4 + n)*x + b^2*(5 + n)*(c*(6 + n) + d*(4 + n)*x^2))))))/(b^8*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n))","B",1
361,1,347,223,0.4984529,"\int (a+b x)^n \left(c+d x^2\right)^3 \, dx","Integrate[(a + b*x)^n*(c + d*x^2)^3,x]","\frac{(a+b x)^{n+1} \left(\frac{6 \left((n+6) \left(a^2 d+b^2 c\right) \left(4 (n+4) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left(c (n+3)+d (n+1) x^2\right)\right)-4 a d (n+1) (a+b x) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)+b^4 (n+1) (n+2) (n+3) (n+4) \left(c+d x^2\right)^2\right)-a d (n+1) (a+b x) \left(4 (n+5) \left(a^2 d+b^2 c\right) \left(2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left(c (n+4)+d (n+2) x^2\right)\right)-4 a d (n+2) (a+b x) \left(2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left(c (n+5)+d (n+3) x^2\right)\right)+b^4 (n+2) (n+3) (n+4) (n+5) \left(c+d x^2\right)^2\right)\right)}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)}+\left(c+d x^2\right)^3\right)}{b (n+7)}","-\frac{4 a d^2 \left(5 a^2 d+3 b^2 c\right) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{3 d^2 \left(5 a^2 d+b^2 c\right) (a+b x)^{n+5}}{b^7 (n+5)}+\frac{\left(a^2 d+b^2 c\right)^3 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{6 a d \left(a^2 d+b^2 c\right)^2 (a+b x)^{n+2}}{b^7 (n+2)}+\frac{3 d \left(a^2 d+b^2 c\right) \left(5 a^2 d+b^2 c\right) (a+b x)^{n+3}}{b^7 (n+3)}-\frac{6 a d^3 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^3 (a+b x)^{n+7}}{b^7 (n+7)}",1,"((a + b*x)^(1 + n)*((c + d*x^2)^3 + (6*((b^2*c + a^2*d)*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) - a*d*(1 + n)*(a + b*x)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))))/(b*(7 + n))","A",1
362,1,226,246,0.1968677,"\int \frac{(a+b x)^n \left(c+d x^2\right)^3}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^2)^3)/x,x]","(a+b x)^{n+1} \left(\frac{d^2 (a+b x)^3 \left(10 a^2 d+3 b^2 c\right)}{b^6 (n+4)}-\frac{a d^2 (a+b x)^2 \left(10 a^2 d+9 b^2 c\right)}{b^6 (n+3)}+\frac{d (a+b x) \left(5 a^4 d^2+9 a^2 b^2 c d+3 b^4 c^2\right)}{b^6 (n+2)}-\frac{a d \left(a^4 d^2+3 a^2 b^2 c d+3 b^4 c^2\right)}{b^6 (n+1)}+\frac{d^3 (a+b x)^5}{b^6 (n+6)}-\frac{5 a d^3 (a+b x)^4}{b^6 (n+5)}-\frac{c^3 \, _2F_1\left(1,n+1;n+2;\frac{a+b x}{a}\right)}{a n+a}\right)","-\frac{a d^2 \left(10 a^2 d+9 b^2 c\right) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{d^2 \left(10 a^2 d+3 b^2 c\right) (a+b x)^{n+4}}{b^6 (n+4)}-\frac{a d \left(a^4 d^2+3 a^2 b^2 c d+3 b^4 c^2\right) (a+b x)^{n+1}}{b^6 (n+1)}+\frac{d \left(5 a^4 d^2+9 a^2 b^2 c d+3 b^4 c^2\right) (a+b x)^{n+2}}{b^6 (n+2)}-\frac{5 a d^3 (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d^3 (a+b x)^{n+6}}{b^6 (n+6)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"(a + b*x)^(1 + n)*(-((a*d*(3*b^4*c^2 + 3*a^2*b^2*c*d + a^4*d^2))/(b^6*(1 + n))) + (d*(3*b^4*c^2 + 9*a^2*b^2*c*d + 5*a^4*d^2)*(a + b*x))/(b^6*(2 + n)) - (a*d^2*(9*b^2*c + 10*a^2*d)*(a + b*x)^2)/(b^6*(3 + n)) + (d^2*(3*b^2*c + 10*a^2*d)*(a + b*x)^3)/(b^6*(4 + n)) - (5*a*d^3*(a + b*x)^4)/(b^6*(5 + n)) + (d^3*(a + b*x)^5)/(b^6*(6 + n)) - (c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))","A",1
363,1,217,250,0.4964243,"\int \frac{x^4 (d+e x)^n}{a+c x^2} \, dx","Integrate[(x^4*(d + e*x)^n)/(a + c*x^2),x]","\frac{(d+e x)^{n+1} \left(\frac{2 \left(c d^2-a e^2\right)}{e^3 (n+1)}+\frac{(-a)^{3/2} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\sqrt{-a} a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{2 c (d+e x)^2}{e^3 (n+3)}-\frac{4 c d (d+e x)}{e^3 (n+2)}\right)}{2 c^2}","\frac{\left(c d^2-a e^2\right) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 c^2 (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 c^2 (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac{(d+e x)^{n+3}}{c e^3 (n+3)}",1,"((d + e*x)^(1 + n)*((2*(c*d^2 - a*e^2))/(e^3*(1 + n)) - (4*c*d*(d + e*x))/(e^3*(2 + n)) + (2*c*(d + e*x)^2)/(e^3*(3 + n)) + ((-a)^(3/2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (Sqrt[-a]*a*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))))/(2*c^2)","A",1
364,1,168,209,0.2068075,"\int \frac{x^3 (d+e x)^n}{a+c x^2} \, dx","Integrate[(x^3*(d + e*x)^n)/(a + c*x^2),x]","\frac{(d+e x)^{n+1} \left(\frac{a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}+\frac{a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}-\frac{2 \sqrt{c} (d-e (n+1) x)}{e^2 (n+2)}\right)}{2 c^{3/2} (n+1)}","\frac{a (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 c^{3/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{a (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 c^{3/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{d (d+e x)^{n+1}}{c e^2 (n+1)}+\frac{(d+e x)^{n+2}}{c e^2 (n+2)}",1,"((d + e*x)^(1 + n)*((-2*Sqrt[c]*(d - e*(1 + n)*x))/(e^2*(2 + n)) + (a*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) + (a*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e)))/(2*c^(3/2)*(1 + n))","A",1
365,1,170,194,0.1278879,"\int \frac{x^2 (d+e x)^n}{a+c x^2} \, dx","Integrate[(x^2*(d + e*x)^n)/(a + c*x^2),x]","\frac{(d+e x)^{n+1} \left(2 \left(a e^2+c d^2\right)+e \left(\sqrt{-a} \sqrt{c} d-a e\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)-e \left(\sqrt{-a} \sqrt{c} d+a e\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)\right)}{2 c e (n+1) \left(a e^2+c d^2\right)}","\frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 c (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{\sqrt{-a} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 c (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{(d+e x)^{n+1}}{c e (n+1)}",1,"((d + e*x)^(1 + n)*(2*(c*d^2 + a*e^2) + e*(Sqrt[-a]*Sqrt[c]*d - a*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] - e*(Sqrt[-a]*Sqrt[c]*d + a*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/(2*c*e*(c*d^2 + a*e^2)*(1 + n))","A",1
366,1,151,163,0.0635473,"\int \frac{x (d+e x)^n}{a+c x^2} \, dx","Integrate[(x*(d + e*x)^n)/(a + c*x^2),x]","-\frac{(d+e x)^{n+1} \left(\left(\sqrt{-a} e+\sqrt{c} d\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)+\left(\sqrt{c} d-\sqrt{-a} e\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)\right)}{2 \sqrt{c} (n+1) \left(a e^2+c d^2\right)}","-\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 \sqrt{c} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 \sqrt{c} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"-1/2*((d + e*x)^(1 + n)*((Sqrt[c]*d + Sqrt[-a]*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] + (Sqrt[c]*d - Sqrt[-a]*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/(Sqrt[c]*(c*d^2 + a*e^2)*(1 + n))","A",1
367,1,145,167,0.0701703,"\int \frac{(d+e x)^n}{a+c x^2} \, dx","Integrate[(d + e*x)^n/(a + c*x^2),x]","\frac{(d+e x)^{n+1} \left(\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}\right)}{2 \sqrt{-a} (n+1)}","\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 \sqrt{-a} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 \sqrt{-a} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"((d + e*x)^(1 + n)*(Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[c]*d - Sqrt[-a]*e) - Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[c]*d + Sqrt[-a]*e)))/(2*Sqrt[-a]*(1 + n))","A",1
368,1,189,207,0.1205447,"\int \frac{(d+e x)^n}{x \left(a+c x^2\right)} \, dx","Integrate[(d + e*x)^n/(x*(a + c*x^2)),x]","\frac{(d+e x)^{n+1} \left(-2 \left(a e^2+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{e x}{d}+1\right)+\left(\sqrt{-a} \sqrt{c} d e+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)+\left(c d^2-\sqrt{-a} \sqrt{c} d e\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)\right)}{2 a d (n+1) \left(a e^2+c d^2\right)}","\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 a (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 a (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{e x}{d}+1\right)}{a d (n+1)}",1,"((d + e*x)^(1 + n)*((c*d^2 + Sqrt[-a]*Sqrt[c]*d*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] + (c*d^2 - Sqrt[-a]*Sqrt[c]*d*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)] - 2*(c*d^2 + a*e^2)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (e*x)/d]))/(2*a*d*(c*d^2 + a*e^2)*(1 + n))","A",1
369,1,167,207,0.2508226,"\int \frac{(d+e x)^n}{x^2 \left(a+c x^2\right)} \, dx","Integrate[(d + e*x)^n/(x^2*(a + c*x^2)),x]","\frac{(d+e x)^{n+1} \left(-\frac{c \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d+a e}+\frac{c \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} \sqrt{c} d-a e}+\frac{2 e \, _2F_1\left(2,n+1;n+2;\frac{e x}{d}+1\right)}{d^2}\right)}{2 a (n+1)}","\frac{c (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 (-a)^{3/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{c (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 (-a)^{3/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{e (d+e x)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{e x}{d}+1\right)}{a d^2 (n+1)}",1,"((d + e*x)^(1 + n)*(-((c*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[c]*d + a*e)) + (c*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[-a]*Sqrt[c]*d - a*e) + (2*e*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (e*x)/d])/d^2))/(2*a*(1 + n))","A",1
370,1,413,332,0.8272438,"\int \frac{x^4 (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[(x^4*(d + e*x)^n)/(a + c*x^2)^2,x]","\frac{(d+e x)^{n+1} \left(\frac{a \left(\frac{\left(\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}-\frac{\left(-\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}\right)}{\sqrt{-a} (n+1) \left(a e^2+c d^2\right)}+\frac{2 a (a e+c d x)}{\left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{4 \sqrt{-a} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{4 \sqrt{-a} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{4}{e n+e}\right)}{4 c^2}","\frac{(d+e x)^{n+1} \left(3 \sqrt{-a} c d^2+a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 c^2 (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}-\frac{(d+e x)^{n+1} \left(3 \sqrt{-a} c d^2-a \sqrt{c} d e n+\sqrt{-a} a e^2 (n+3)\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 c^2 (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}+\frac{a (d+e x)^{n+1} (a e+c d x)}{2 c^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{n+1}}{c^2 e (n+1)}",1,"((d + e*x)^(1 + n)*(4/(e + e*n) + (2*a*(a*e + c*d*x))/((c*d^2 + a*e^2)*(a + c*x^2)) + (4*Sqrt[-a]*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (4*Sqrt[-a]*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (a*(((c*d^2 - a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) - ((c*d^2 - a*e^2*(-1 + n) - Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e)))/(Sqrt[-a]*(c*d^2 + a*e^2)*(1 + n))))/(4*c^2)","A",1
371,1,247,297,0.9179162,"\int \frac{x^3 (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[(x^3*(d + e*x)^n)/(a + c*x^2)^2,x]","-\frac{(d+e x)^{n+1} \left(\frac{\left(-\sqrt{-a} \sqrt{c} d e n+a e^2 (n+2)+2 c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\left(\sqrt{-a} \sqrt{c} d e n+a e^2 (n+2)+2 c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{2 a \sqrt{c} (e x-d)}{a+c x^2}\right)}{4 c^{3/2} \left(a e^2+c d^2\right)}","-\frac{(d+e x)^{n+1} \left(\sqrt{-a} \sqrt{c} d e n+a e^2 (n+2)+2 c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 c^{3/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{n+1} \left(\sqrt{-a} d e n-\frac{a e^2 (n+2)+2 c d^2}{\sqrt{c}}\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 c (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{a (d-e x) (d+e x)^{n+1}}{2 c \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"-1/4*((d + e*x)^(1 + n)*((2*a*Sqrt[c]*(-d + e*x))/(a + c*x^2) + ((2*c*d^2 - Sqrt[-a]*Sqrt[c]*d*e*n + a*e^2*(2 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + ((2*c*d^2 + Sqrt[-a]*Sqrt[c]*d*e*n + a*e^2*(2 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))))/(c^(3/2)*(c*d^2 + a*e^2))","A",1
372,1,403,306,0.6669689,"\int \frac{x^2 (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[(x^2*(d + e*x)^n)/(a + c*x^2)^2,x]","\frac{(d+e x)^{n+1} \left(\frac{a \left(\frac{\left(\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}-\frac{\left(-\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}\right)}{(-a)^{3/2} (n+1) \left(a e^2+c d^2\right)}-\frac{2 (a e+c d x)}{\left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{2 \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}-\frac{2 \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}\right)}{4 c}","\frac{(d+e x)^{n+1} \left(-\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 \sqrt{-a} c (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}-\frac{(d+e x)^{n+1} \left(\sqrt{-a} \sqrt{c} d e n+a e^2 (n+1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 \sqrt{-a} c (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}-\frac{(d+e x)^{n+1} (a e+c d x)}{2 c \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + n)*((-2*(a*e + c*d*x))/((c*d^2 + a*e^2)*(a + c*x^2)) + (2*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) - (2*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (a*(((c*d^2 - a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) - ((c*d^2 - a*e^2*(-1 + n) - Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e)))/((-a)^(3/2)*(c*d^2 + a*e^2)*(1 + n))))/(4*c)","A",1
373,1,230,279,0.3403124,"\int \frac{x (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[(x*(d + e*x)^n)/(a + c*x^2)^2,x]","\frac{(d+e x)^{n+1} \left(-\frac{\left(\sqrt{-a} c d e n-a \sqrt{c} e^2 n\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\left(\sqrt{-a} c d e n+a \sqrt{c} e^2 n\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{2 a c (d-e x)}{a+c x^2}\right)}{4 a c \left(a e^2+c d^2\right)}","\frac{e n \left(\sqrt{-a} e+\sqrt{c} d\right) (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 \sqrt{-a} \sqrt{c} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{e n \left(\sqrt{-a} \sqrt{c} d+a e\right) (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 a \sqrt{c} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}-\frac{(d-e x) (d+e x)^{n+1}}{2 \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + n)*((-2*a*c*(d - e*x))/(a + c*x^2) - ((Sqrt[-a]*c*d*e*n - a*Sqrt[c]*e^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + ((Sqrt[-a]*c*d*e*n + a*Sqrt[c]*e^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))))/(4*a*c*(c*d^2 + a*e^2))","A",1
374,1,253,304,0.3745617,"\int \frac{(d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^n/(a + c*x^2)^2,x]","\frac{(d+e x)^{n+1} \left(\frac{\left(\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{-a} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\left(\sqrt{-a} \sqrt{c} d e n+a e^2 (n-1)-c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{2 (a e+c d x)}{a+c x^2}\right)}{4 a \left(a e^2+c d^2\right)}","-\frac{(d+e x)^{n+1} \left(\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 (-a)^{3/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{n+1} \left(-\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 (-a)^{3/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}+\frac{(d+e x)^{n+1} (a e+c d x)}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + n)*((2*(a*e + c*d*x))/(a + c*x^2) + ((c*d^2 - a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + ((-(c*d^2) + a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n))))/(4*a*(c*d^2 + a*e^2))","A",1
375,1,391,489,0.8866002,"\int \frac{(d+e x)^n}{x \left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^n/(x*(a + c*x^2)^2),x]","\frac{(d+e x)^{n+1} \left(\frac{\sqrt{c} e n \left(\left(\sqrt{-a} c d^2-2 a \sqrt{c} d e+(-a)^{3/2} e^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)+\left(-\sqrt{-a} c d^2-2 a \sqrt{c} d e+\sqrt{-a} a e^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)\right)}{(n+1) \left(a e^2+c d^2\right)^2}+\frac{2 a c (d-e x)}{\left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{2 \sqrt{c} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{2 \sqrt{c} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{4 \, _2F_1\left(1,n+1;n+2;\frac{d+e x}{d}\right)}{d n+d}\right)}{4 a^2}","-\frac{\sqrt{c} e n \left(\sqrt{-a} \sqrt{c} d+a e\right) (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 a^2 (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 a^2 (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{\sqrt{c} (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 a^2 (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}-\frac{(d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{e x}{d}+1\right)}{a^2 d (n+1)}+\frac{\sqrt{c} e n \left(\sqrt{-a} e+\sqrt{c} d\right) (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 (-a)^{3/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{c (d-e x) (d+e x)^{n+1}}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)}",1,"((d + e*x)^(1 + n)*((2*a*c*(d - e*x))/((c*d^2 + a*e^2)*(a + c*x^2)) - (4*Hypergeometric2F1[1, 1 + n, 2 + n, (d + e*x)/d])/(d + d*n) + (2*Sqrt[c]*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((Sqrt[c]*d - Sqrt[-a]*e)*(1 + n)) + (2*Sqrt[c]*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (Sqrt[c]*e*n*((Sqrt[-a]*c*d^2 - 2*a*Sqrt[c]*d*e + (-a)^(3/2)*e^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] + (-(Sqrt[-a]*c*d^2) - 2*a*Sqrt[c]*d*e + Sqrt[-a]*a*e^2)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/((c*d^2 + a*e^2)^2*(1 + n))))/(4*a^2)","A",1
376,1,437,513,0.6675014,"\int \frac{(d+e x)^n}{x^2 \left(a+c x^2\right)^2} \, dx","Integrate[(d + e*x)^n/(x^2*(a + c*x^2)^2),x]","\frac{1}{4} (d+e x)^{n+1} \left(-\frac{2 c (a e+c d x)}{a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{4 e \, _2F_1\left(2,n+1;n+2;\frac{e x}{d}+1\right)}{a^2 d^2 (n+1)}+\frac{a c \left(\frac{\left(\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{\sqrt{c} d-\sqrt{-a} e}-\frac{\left(-\sqrt{-a} \sqrt{c} d e n-a e^2 (n-1)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{\sqrt{-a} e+\sqrt{c} d}\right)}{(-a)^{7/2} (n+1) \left(a e^2+c d^2\right)}+\frac{2 c \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{(-a)^{5/2} (n+1) \left(\sqrt{-a} e-\sqrt{c} d\right)}+\frac{2 c \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{(-a)^{5/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}\right)","-\frac{c (d+e x)^{n+1} (a e+c d x)}{2 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)}+\frac{e (d+e x)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{e x}{d}+1\right)}{a^2 d^2 (n+1)}-\frac{c (d+e x)^{n+1} \left(\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{4 (-a)^{5/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right)}+\frac{c (d+e x)^{n+1} \left(-\sqrt{-a} \sqrt{c} d e n+a e^2 (1-n)+c d^2\right) \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{4 (-a)^{5/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right)}-\frac{c (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right)}{2 (-a)^{5/2} (n+1) \left(\sqrt{c} d-\sqrt{-a} e\right)}+\frac{c (d+e x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right)}{2 (-a)^{5/2} (n+1) \left(\sqrt{-a} e+\sqrt{c} d\right)}",1,"((d + e*x)^(1 + n)*((-2*c*(a*e + c*d*x))/(a^2*(c*d^2 + a*e^2)*(a + c*x^2)) + (2*c*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/((-a)^(5/2)*(-(Sqrt[c]*d) + Sqrt[-a]*e)*(1 + n)) + (2*c*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/((-a)^(5/2)*(Sqrt[c]*d + Sqrt[-a]*e)*(1 + n)) + (a*c*(((c*d^2 - a*e^2*(-1 + n) + Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)])/(Sqrt[c]*d - Sqrt[-a]*e) - ((c*d^2 - a*e^2*(-1 + n) - Sqrt[-a]*Sqrt[c]*d*e*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)])/(Sqrt[c]*d + Sqrt[-a]*e)))/((-a)^(7/2)*(c*d^2 + a*e^2)*(1 + n)) + (4*e*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (e*x)/d])/(a^2*d^2*(1 + n))))/4","A",1
377,1,275,399,0.136008,"\int (g x)^m (d+e x)^n \left(a+c x^2\right)^2 \, dx","Integrate[(g*x)^m*(d + e*x)^n*(a + c*x^2)^2,x]","\frac{x (g x)^m (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} \left(a^2 e^4 \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)+2 a c d^2 e^2 \, _2F_1\left(m+1,-n-2;m+2;-\frac{e x}{d}\right)-4 a c d^2 e^2 \, _2F_1\left(m+1,-n-1;m+2;-\frac{e x}{d}\right)+2 a c d^2 e^2 \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)+c^2 d^4 \, _2F_1\left(m+1,-n-4;m+2;-\frac{e x}{d}\right)-4 c^2 d^4 \, _2F_1\left(m+1,-n-3;m+2;-\frac{e x}{d}\right)+6 c^2 d^4 \, _2F_1\left(m+1,-n-2;m+2;-\frac{e x}{d}\right)-4 c^2 d^4 \, _2F_1\left(m+1,-n-1;m+2;-\frac{e x}{d}\right)+c^2 d^4 \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)\right)}{e^4 (m+1)}","\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} \left(a^2 e^4 (m+n+2) (m+n+3) (m+n+4) (m+n+5)+c d^2 (m+1) (m+2) \left(2 a e^2 \left(m^2+m (2 n+9)+n^2+9 n+20\right)+c d^2 \left(m^2+7 m+12\right)\right)\right) \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)}{e^4 g (m+1) (m+n+2) (m+n+3) (m+n+4) (m+n+5)}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left(2 a e^2 \left(m^2+m (2 n+9)+n^2+9 n+20\right)+c d^2 \left(m^2+7 m+12\right)\right)}{e^4 g (m+n+2) (m+n+3) (m+n+4) (m+n+5)}+\frac{c (g x)^{m+2} (d+e x)^{n+1} \left(2 a e^2 \left(m^2+m (2 n+9)+n^2+9 n+20\right)+c d^2 \left(m^2+7 m+12\right)\right)}{e^3 g^2 (m+n+3) (m+n+4) (m+n+5)}-\frac{c^2 d (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e^2 g^3 (m+n+4) (m+n+5)}+\frac{c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}",1,"(x*(g*x)^m*(d + e*x)^n*(c^2*d^4*Hypergeometric2F1[1 + m, -4 - n, 2 + m, -((e*x)/d)] - 4*c^2*d^4*Hypergeometric2F1[1 + m, -3 - n, 2 + m, -((e*x)/d)] + 6*c^2*d^4*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((e*x)/d)] + 2*a*c*d^2*e^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((e*x)/d)] - 4*c^2*d^4*Hypergeometric2F1[1 + m, -1 - n, 2 + m, -((e*x)/d)] - 4*a*c*d^2*e^2*Hypergeometric2F1[1 + m, -1 - n, 2 + m, -((e*x)/d)] + c^2*d^4*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)] + 2*a*c*d^2*e^2*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)] + a^2*e^4*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)]))/(e^4*(1 + m)*(1 + (e*x)/d)^n)","A",1
378,1,113,164,0.0610355,"\int (g x)^m (d+e x)^n \left(a+c x^2\right) \, dx","Integrate[(g*x)^m*(d + e*x)^n*(a + c*x^2),x]","\frac{x (g x)^m (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} \left(\left(a e^2+c d^2\right) \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)+c d^2 \, _2F_1\left(m+1,-n-2;m+2;-\frac{e x}{d}\right)-2 c d^2 \, _2F_1\left(m+1,-n-1;m+2;-\frac{e x}{d}\right)\right)}{e^2 (m+1)}","\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} \left(a e^2 (m+n+2) (m+n+3)+c d^2 (m+1) (m+2)\right) \, _2F_1\left(m+1,-n;m+2;-\frac{e x}{d}\right)}{e^2 g (m+1) (m+n+2) (m+n+3)}-\frac{c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac{c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)}",1,"(x*(g*x)^m*(d + e*x)^n*(c*d^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((e*x)/d)] - 2*c*d^2*Hypergeometric2F1[1 + m, -1 - n, 2 + m, -((e*x)/d)] + (c*d^2 + a*e^2)*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)]))/(e^2*(1 + m)*(1 + (e*x)/d)^n)","A",1
379,0,0,148,0.1043465,"\int \frac{(g x)^m (d+e x)^n}{a+c x^2} \, dx","Integrate[((g*x)^m*(d + e*x)^n)/(a + c*x^2),x]","\int \frac{(g x)^m (d+e x)^n}{a+c x^2} \, dx","\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{2 a g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{2 a g (m+1)}",1,"Integrate[((g*x)^m*(d + e*x)^n)/(a + c*x^2), x]","F",-1
380,0,0,295,0.2032402,"\int \frac{(g x)^m (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","Integrate[((g*x)^m*(d + e*x)^n)/(a + c*x^2)^2,x]","\int \frac{(g x)^m (d+e x)^n}{\left(a+c x^2\right)^2} \, dx","\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,2;m+2;-\frac{e x}{d},-\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{4 a^2 g (m+1)}+\frac{(g x)^{m+1} (d+e x)^n \left(\frac{e x}{d}+1\right)^{-n} F_1\left(m+1;-n,2;m+2;-\frac{e x}{d},\frac{\sqrt{c} x}{\sqrt{-a}}\right)}{4 a^2 g (m+1)}",1,"Integrate[((g*x)^m*(d + e*x)^n)/(a + c*x^2)^2, x]","F",-1
381,1,112,125,0.0826248,"\int x^5 (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[x^5*(d + e*x)*(a + b*x^2)^p,x]","\frac{1}{14} \left(a+b x^2\right)^p \left(\frac{7 d \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+2 e x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 d \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}-\frac{a d \left(a+b x^2\right)^{p+2}}{b^3 (p+2)}+\frac{d \left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}+\frac{1}{7} e x^7 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((7*d*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (2*e*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/14","A",1
382,1,112,125,0.0659645,"\int x^4 (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[x^4*(d + e*x)*(a + b*x^2)^p,x]","\frac{1}{10} \left(a+b x^2\right)^p \left(\frac{5 e \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+2 d x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 e \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}-\frac{a e \left(a+b x^2\right)^{p+2}}{b^3 (p+2)}+\frac{e \left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}+\frac{1}{5} d x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((5*e*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (2*d*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/10","A",1
383,1,87,100,0.0719396,"\int x^3 (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[x^3*(d + e*x)*(a + b*x^2)^p,x]","\frac{1}{10} \left(a+b x^2\right)^p \left(2 e x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)-\frac{5 d \left(a+b x^2\right) \left(a-b (p+1) x^2\right)}{b^2 (p+1) (p+2)}\right)","-\frac{a d \left(a+b x^2\right)^{p+1}}{2 b^2 (p+1)}+\frac{d \left(a+b x^2\right)^{p+2}}{2 b^2 (p+2)}+\frac{1}{5} e x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((-5*d*(a + b*x^2)*(a - b*(1 + p)*x^2))/(b^2*(1 + p)*(2 + p)) + (2*e*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/10","A",1
384,1,87,100,0.0656006,"\int x^2 (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[x^2*(d + e*x)*(a + b*x^2)^p,x]","\frac{1}{6} \left(a+b x^2\right)^p \left(2 d x^3 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)-\frac{3 e \left(a+b x^2\right) \left(a-b (p+1) x^2\right)}{b^2 (p+1) (p+2)}\right)","-\frac{a e \left(a+b x^2\right)^{p+1}}{2 b^2 (p+1)}+\frac{e \left(a+b x^2\right)^{p+2}}{2 b^2 (p+2)}+\frac{1}{3} d x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((-3*e*(a + b*x^2)*(a - b*(1 + p)*x^2))/(b^2*(1 + p)*(2 + p)) + (2*d*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/6","A",1
385,1,71,75,0.0315003,"\int x (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[x*(d + e*x)*(a + b*x^2)^p,x]","\frac{1}{6} \left(a+b x^2\right)^p \left(\frac{3 d \left(a+b x^2\right)}{b (p+1)}+2 e x^3 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)\right)","\frac{d \left(a+b x^2\right)^{p+1}}{2 b (p+1)}+\frac{1}{3} e x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((3*d*(a + b*x^2))/(b*(1 + p)) + (2*e*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/6","A",1
386,1,98,70,0.071513,"\int (d+e x) \left(a+b x^2\right)^p \, dx","Integrate[(d + e*x)*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(2 b d (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+b e x^2 \left(\frac{b x^2}{a}+1\right)^p+a e \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)\right)}{2 b (p+1)}","d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+\frac{e \left(a+b x^2\right)^{p+1}}{2 b (p+1)}",1,"((a + b*x^2)^p*(b*e*x^2*(1 + (b*x^2)/a)^p + a*e*(-1 + (1 + (b*x^2)/a)^p) + 2*b*d*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)]))/(2*b*(1 + p)*(1 + (b*x^2)/a)^p)","A",1
387,1,88,88,0.0505883,"\int \frac{(d+e x) \left(a+b x^2\right)^p}{x} \, dx","Integrate[((d + e*x)*(a + b*x^2)^p)/x,x]","e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)-\frac{d \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}","e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)-\frac{d \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}",1,"(e*x*(a + b*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p - (d*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a*(1 + p))","A",1
388,1,91,91,0.0471625,"\int \frac{(d+e x) \left(a+b x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)*(a + b*x^2)^p)/x^2,x]","-\frac{d \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}-\frac{e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}","-\frac{d \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}-\frac{e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}",1,"-((d*(a + b*x^2)^p*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p)) - (e*(a + b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])/(2*a*(1 + p))","A",1
389,1,89,92,0.0458365,"\int \frac{(d+e x) \left(a+b x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)*(a + b*x^2)^p)/x^3,x]","\frac{1}{2} \left(a+b x^2\right)^p \left(\frac{b d \left(a+b x^2\right) \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)}{a^2 (p+1)}-\frac{2 e \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}\right)","\frac{b d \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (p+1)}-\frac{e \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}",1,"((a + b*x^2)^p*((-2*e*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) + (b*d*(a + b*x^2)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a])/(a^2*(1 + p))))/2","A",1
390,1,205,188,0.20272,"\int x^5 (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[x^5*(d + e*x)^2*(a + b*x^2)^p,x]","\frac{1}{14} \left(a+b x^2\right)^p \left(\frac{7 d^2 \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{7 e^2 \left(a+b x^2\right) \left(-6 a^3+6 a^2 b (p+1) x^2-3 a b^2 \left(p^2+3 p+2\right) x^4+b^3 \left(p^3+6 p^2+11 p+6\right) x^6\right)}{b^4 (p+1) (p+2) (p+3) (p+4)}+4 d e x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 \left(b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^4 (p+1)}-\frac{a \left(2 b d^2-3 a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^4 (p+2)}+\frac{\left(b d^2-3 a e^2\right) \left(a+b x^2\right)^{p+3}}{2 b^4 (p+3)}+\frac{e^2 \left(a+b x^2\right)^{p+4}}{2 b^4 (p+4)}+\frac{2}{7} d e x^7 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((7*d^2*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (7*e^2*(a + b*x^2)*(-6*a^3 + 6*a^2*b*(1 + p)*x^2 - 3*a*b^2*(2 + 3*p + p^2)*x^4 + b^3*(6 + 11*p + 6*p^2 + p^3)*x^6))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (4*d*e*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/14","A",1
391,1,156,177,0.159718,"\int x^4 (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[x^4*(d + e*x)^2*(a + b*x^2)^p,x]","\frac{1}{35} \left(a+b x^2\right)^p \left(\frac{35 d e \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+7 d^2 x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)+5 e^2 x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 d e \left(a+b x^2\right)^{p+1}}{b^3 (p+1)}-\frac{2 a d e \left(a+b x^2\right)^{p+2}}{b^3 (p+2)}+\frac{d e \left(a+b x^2\right)^{p+3}}{b^3 (p+3)}-\frac{x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(5 a e^2-b d^2 (2 p+7)\right) \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)}{5 b (2 p+7)}+\frac{e^2 x^5 \left(a+b x^2\right)^{p+1}}{b (2 p+7)}",1,"((a + b*x^2)^p*((35*d*e*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (7*d^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (5*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/35","A",1
392,1,152,149,0.1379457,"\int x^3 (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[x^3*(d + e*x)^2*(a + b*x^2)^p,x]","\frac{1}{10} \left(a+b x^2\right)^p \left(\frac{5 e^2 \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{5 d^2 \left(a+b x^2\right) \left(b (p+1) x^2-a\right)}{b^2 (p+1) (p+2)}+4 d e x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)\right)","-\frac{a \left(b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}+\frac{\left(b d^2-2 a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^3 (p+2)}+\frac{e^2 \left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}+\frac{2}{5} d e x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*((5*d^2*(a + b*x^2)*(-a + b*(1 + p)*x^2))/(b^2*(1 + p)*(2 + p)) + (5*e^2*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (4*d*e*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/10","A",1
393,1,139,152,0.1458808,"\int x^2 (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[x^2*(d + e*x)^2*(a + b*x^2)^p,x]","\frac{1}{15} \left(a+b x^2\right)^p \left(\frac{3 e \left(e \left(p^2+3 p+2\right) x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)-\frac{5 d \left(a+b x^2\right) \left(a-b (p+1) x^2\right)}{b^2}\right)}{(p+1) (p+2)}+5 d^2 x^3 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)\right)","-\frac{a d e \left(a+b x^2\right)^{p+1}}{b^2 (p+1)}+\frac{d e \left(a+b x^2\right)^{p+2}}{b^2 (p+2)}-\frac{x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a e^2-b d^2 (2 p+5)\right) \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)}{3 b (2 p+5)}+\frac{e^2 x^3 \left(a+b x^2\right)^{p+1}}{b (2 p+5)}",1,"((a + b*x^2)^p*((5*d^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (3*e*((-5*d*(a + b*x^2)*(a - b*(1 + p)*x^2))/b^2 + (e*(2 + 3*p + p^2)*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/((1 + p)*(2 + p))))/15","A",1
394,1,184,113,0.1817141,"\int x (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[x*(d + e*x)^2*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(-3 a^2 e^2 \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+3 b^2 x^2 \left(\frac{b x^2}{a}+1\right)^p \left(d^2 (p+2)+e^2 (p+1) x^2\right)+4 b^2 d e \left(p^2+3 p+2\right) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+3 a b \left(d^2 (p+2) \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+e^2 p x^2 \left(\frac{b x^2}{a}+1\right)^p\right)\right)}{6 b^2 (p+1) (p+2)}","\frac{\left(b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^2 (p+1)}+\frac{e^2 \left(a+b x^2\right)^{p+2}}{2 b^2 (p+2)}+\frac{2}{3} d e x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)",1,"((a + b*x^2)^p*(3*b^2*x^2*(1 + (b*x^2)/a)^p*(d^2*(2 + p) + e^2*(1 + p)*x^2) - 3*a^2*e^2*(-1 + (1 + (b*x^2)/a)^p) + 3*a*b*(e^2*p*x^2*(1 + (b*x^2)/a)^p + d^2*(2 + p)*(-1 + (1 + (b*x^2)/a)^p)) + 4*b^2*d*e*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)]))/(6*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)","A",1
395,1,133,133,0.1154101,"\int (d+e x)^2 \left(a+b x^2\right)^p \, dx","Integrate[(d + e*x)^2*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 b d^2 (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+e \left(3 d \left(b x^2 \left(\frac{b x^2}{a}+1\right)^p+a \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)\right)+b e (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)\right)\right)}{3 b (p+1)}","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2-b d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b (2 p+3)}+\frac{e (d+e x) \left(a+b x^2\right)^{p+1}}{b (2 p+3)}+\frac{d e (p+2) \left(a+b x^2\right)^{p+1}}{b (p+1) (2 p+3)}",1,"((a + b*x^2)^p*(3*b*d^2*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + e*(3*d*(b*x^2*(1 + (b*x^2)/a)^p + a*(-1 + (1 + (b*x^2)/a)^p)) + b*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])))/(3*b*(1 + p)*(1 + (b*x^2)/a)^p)","A",1
396,1,101,118,0.1009658,"\int \frac{(d+e x)^2 \left(a+b x^2\right)^p}{x} \, dx","Integrate[((d + e*x)^2*(a + b*x^2)^p)/x,x]","\frac{1}{2} \left(a+b x^2\right)^p \left(\frac{\left(a+b x^2\right) \left(a e^2-b d^2 \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)\right)}{a b (p+1)}+4 d e x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)","-\frac{d^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}+2 d e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+\frac{e^2 \left(a+b x^2\right)^{p+1}}{2 b (p+1)}",1,"((a + b*x^2)^p*((4*d*e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + ((a + b*x^2)*(a*e^2 - b*d^2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a]))/(a*b*(1 + p))))/2","A",1
397,1,134,127,0.0802257,"\int \frac{(d+e x)^2 \left(a+b x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)^2*(a + b*x^2)^p)/x^2,x]","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a d^2 (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)+e x \left(d \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)-a e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)\right)}{a (p+1) x}","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+b d^2 (2 p+1)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{a}-\frac{d^2 \left(a+b x^2\right)^{p+1}}{a x}-\frac{d e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{a (p+1)}",1,"-(((a + b*x^2)^p*(a*d^2*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)] + e*x*(-(a*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)]) + d*(a + b*x^2)*(1 + (b*x^2)/a)^p*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a])))/(a*(1 + p)*x*(1 + (b*x^2)/a)^p))","A",1
398,1,119,127,0.089239,"\int \frac{(d+e x)^2 \left(a+b x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)^2*(a + b*x^2)^p)/x^3,x]","\frac{1}{2} \left(a+b x^2\right)^p \left(-\frac{\left(a+b x^2\right) \left(a e^2 \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)-b d^2 \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)\right)}{a^2 (p+1)}-\frac{4 d e \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}\right)","-\frac{\left(a+b x^2\right)^{p+1} \left(a e^2+b d^2 p\right) \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (p+1)}-\frac{d^2 \left(a+b x^2\right)^{p+1}}{2 a x^2}-\frac{2 d e \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}",1,"((a + b*x^2)^p*((-4*d*e*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) - ((a + b*x^2)*(a*e^2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a] - b*d^2*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a]))/(a^2*(1 + p))))/2","A",1
399,1,249,247,0.2157149,"\int x^5 (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[x^5*(d + e*x)^3*(a + b*x^2)^p,x]","\frac{1}{126} \left(a+b x^2\right)^p \left(\frac{63 d^3 \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{189 d e^2 \left(a+b x^2\right) \left(-6 a^3+6 a^2 b (p+1) x^2-3 a b^2 \left(p^2+3 p+2\right) x^4+b^3 \left(p^3+6 p^2+11 p+6\right) x^6\right)}{b^4 (p+1) (p+2) (p+3) (p+4)}+54 d^2 e x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)+14 e^3 x^9 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{9}{2},-p;\frac{11}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 d \left(b d^2-3 a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^4 (p+1)}-\frac{a d \left(2 b d^2-9 a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^4 (p+2)}+\frac{d \left(b d^2-9 a e^2\right) \left(a+b x^2\right)^{p+3}}{2 b^4 (p+3)}+\frac{3 d e^2 \left(a+b x^2\right)^{p+4}}{2 b^4 (p+4)}-\frac{e x^7 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(7 a e^2-3 b d^2 (2 p+9)\right) \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)}{7 b (2 p+9)}+\frac{e^3 x^7 \left(a+b x^2\right)^{p+1}}{b (2 p+9)}",1,"((a + b*x^2)^p*((63*d^3*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (189*d*e^2*(a + b*x^2)*(-6*a^3 + 6*a^2*b*(1 + p)*x^2 - 3*a*b^2*(2 + 3*p + p^2)*x^4 + b^3*(6 + 11*p + 6*p^2 + p^3)*x^6))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (54*d^2*e*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (14*e^3*x^9*Hypergeometric2F1[9/2, -p, 11/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/126","A",1
400,1,249,249,0.1998598,"\int x^4 (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[x^4*(d + e*x)^3*(a + b*x^2)^p,x]","\frac{1}{70} \left(a+b x^2\right)^p \left(\frac{105 d^2 e \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{35 e^3 \left(a+b x^2\right) \left(-6 a^3+6 a^2 b (p+1) x^2-3 a b^2 \left(p^2+3 p+2\right) x^4+b^3 \left(p^3+6 p^2+11 p+6\right) x^6\right)}{b^4 (p+1) (p+2) (p+3) (p+4)}+14 d^3 x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)+30 d e^2 x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)","\frac{a^2 e \left(3 b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^4 (p+1)}-\frac{3 a e \left(2 b d^2-a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^4 (p+2)}+\frac{3 e \left(b d^2-a e^2\right) \left(a+b x^2\right)^{p+3}}{2 b^4 (p+3)}+\frac{e^3 \left(a+b x^2\right)^{p+4}}{2 b^4 (p+4)}-\frac{d x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(15 a e^2-b d^2 (2 p+7)\right) \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)}{5 b (2 p+7)}+\frac{3 d e^2 x^5 \left(a+b x^2\right)^{p+1}}{b (2 p+7)}",1,"((a + b*x^2)^p*((105*d^2*e*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (35*e^3*(a + b*x^2)*(-6*a^3 + 6*a^2*b*(1 + p)*x^2 - 3*a*b^2*(2 + 3*p + p^2)*x^4 + b^3*(6 + 11*p + 6*p^2 + p^3)*x^6))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (14*d^3*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (30*d*e^2*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/70","A",1
401,1,196,207,0.1476163,"\int x^3 (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[x^3*(d + e*x)^3*(a + b*x^2)^p,x]","\frac{1}{70} \left(a+b x^2\right)^p \left(\frac{105 d e^2 \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{35 d^3 \left(a+b x^2\right) \left(b (p+1) x^2-a\right)}{b^2 (p+1) (p+2)}+42 d^2 e x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)+10 e^3 x^7 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right)\right)","-\frac{a d \left(b d^2-3 a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}+\frac{d \left(b d^2-6 a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^3 (p+2)}+\frac{3 d e^2 \left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}-\frac{e x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(5 a e^2-3 b d^2 (2 p+7)\right) \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)}{5 b (2 p+7)}+\frac{e^3 x^5 \left(a+b x^2\right)^{p+1}}{b (2 p+7)}",1,"((a + b*x^2)^p*((35*d^3*(a + b*x^2)*(-a + b*(1 + p)*x^2))/(b^2*(1 + p)*(2 + p)) + (105*d*e^2*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (42*d^2*e*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (10*e^3*x^7*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/70","A",1
402,1,196,210,0.1424432,"\int x^2 (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[x^2*(d + e*x)^3*(a + b*x^2)^p,x]","\frac{1}{30} \left(a+b x^2\right)^p \left(\frac{15 e^3 \left(a+b x^2\right) \left(2 a^2-2 a b (p+1) x^2+b^2 \left(p^2+3 p+2\right) x^4\right)}{b^3 (p+1) (p+2) (p+3)}+\frac{45 d^2 e \left(a+b x^2\right) \left(b (p+1) x^2-a\right)}{b^2 (p+1) (p+2)}+10 d^3 x^3 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+18 d e^2 x^5 \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)\right)","-\frac{a e \left(3 b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^3 (p+1)}+\frac{e \left(3 b d^2-2 a e^2\right) \left(a+b x^2\right)^{p+2}}{2 b^3 (p+2)}+\frac{e^3 \left(a+b x^2\right)^{p+3}}{2 b^3 (p+3)}-\frac{d x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(9 a e^2-b d^2 (2 p+5)\right) \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)}{3 b (2 p+5)}+\frac{3 d e^2 x^3 \left(a+b x^2\right)^{p+1}}{b (2 p+5)}",1,"((a + b*x^2)^p*((45*d^2*e*(a + b*x^2)*(-a + b*(1 + p)*x^2))/(b^2*(1 + p)*(2 + p)) + (15*e^3*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (10*d^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (18*d*e^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/30","A",1
403,1,228,167,0.2305858,"\int x (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[x*(d + e*x)^3*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(5 d \left(-3 a^2 e^2 \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+b^2 x^2 \left(\frac{b x^2}{a}+1\right)^p \left(d^2 (p+2)+3 e^2 (p+1) x^2\right)+a b \left(d^2 (p+2) \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+3 e^2 p x^2 \left(\frac{b x^2}{a}+1\right)^p\right)\right)+10 b^2 d^2 e \left(p^2+3 p+2\right) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+2 b^2 e^3 \left(p^2+3 p+2\right) x^5 \, _2F_1\left(\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right)\right)}{10 b^2 (p+1) (p+2)}","\frac{d \left(b d^2-3 a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^2 (p+1)}+\frac{3 d e^2 \left(a+b x^2\right)^{p+2}}{2 b^2 (p+2)}-\frac{e x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2-b d^2 (2 p+5)\right) \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)}{b (2 p+5)}+\frac{e^3 x^3 \left(a+b x^2\right)^{p+1}}{b (2 p+5)}",1,"((a + b*x^2)^p*(5*d*(b^2*x^2*(1 + (b*x^2)/a)^p*(d^2*(2 + p) + 3*e^2*(1 + p)*x^2) - 3*a^2*e^2*(-1 + (1 + (b*x^2)/a)^p) + a*b*(3*e^2*p*x^2*(1 + (b*x^2)/a)^p + d^2*(2 + p)*(-1 + (1 + (b*x^2)/a)^p))) + 10*b^2*d^2*e*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)] + 2*b^2*e^3*(2 + 3*p + p^2)*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)]))/(10*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)","A",1
404,1,223,176,0.2127548,"\int (d+e x)^3 \left(a+b x^2\right)^p \, dx","Integrate[(d + e*x)^3*(a + b*x^2)^p,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(e \left(-a^2 e^2 \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+b^2 x^2 \left(\frac{b x^2}{a}+1\right)^p \left(3 d^2 (p+2)+e^2 (p+1) x^2\right)+2 b^2 d e \left(p^2+3 p+2\right) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)+a b \left(3 d^2 (p+2) \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)+e^2 p x^2 \left(\frac{b x^2}{a}+1\right)^p\right)\right)+2 b^2 d^3 \left(p^2+3 p+2\right) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{2 b^2 (p+1) (p+2)}","-\frac{e \left(a+b x^2\right)^{p+1} \left(a e^2-3 b d^2 (p+2)\right)}{2 b^2 (p+1) (p+2)}-\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a e^2-b d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b (2 p+3)}+\frac{3 d e^2 x \left(a+b x^2\right)^{p+1}}{b (2 p+3)}+\frac{e^3 x^2 \left(a+b x^2\right)^{p+1}}{2 b (p+2)}",1,"((a + b*x^2)^p*(2*b^2*d^3*(2 + 3*p + p^2)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + e*(b^2*x^2*(1 + (b*x^2)/a)^p*(3*d^2*(2 + p) + e^2*(1 + p)*x^2) - a^2*e^2*(-1 + (1 + (b*x^2)/a)^p) + a*b*(e^2*p*x^2*(1 + (b*x^2)/a)^p + 3*d^2*(2 + p)*(-1 + (1 + (b*x^2)/a)^p)) + 2*b^2*d*e*(2 + 3*p + p^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])))/(2*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)","A",1
405,1,170,171,0.1306545,"\int \frac{(d+e x)^3 \left(a+b x^2\right)^p}{x} \, dx","Integrate[((d + e*x)^3*(a + b*x^2)^p)/x,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(-3 b d^3 \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)+18 a b d^2 e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+a e^2 \left(9 d \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p+2 b e (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)\right)\right)}{6 a b (p+1)}","-\frac{d^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2-3 b d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b (2 p+3)}+\frac{3 d e^2 \left(a+b x^2\right)^{p+1}}{2 b (p+1)}+\frac{e^3 x \left(a+b x^2\right)^{p+1}}{b (2 p+3)}",1,"((a + b*x^2)^p*(18*a*b*d^2*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] - 3*b*d^3*(a + b*x^2)*(1 + (b*x^2)/a)^p*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a] + a*e^2*(9*d*(a + b*x^2)*(1 + (b*x^2)/a)^p + 2*b*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])))/(6*a*b*(1 + p)*(1 + (b*x^2)/a)^p)","A",1
406,1,154,159,0.1166217,"\int \frac{(d+e x)^3 \left(a+b x^2\right)^p}{x^2} \, dx","Integrate[((d + e*x)^3*(a + b*x^2)^p)/x^2,x]","\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(e x \left(\left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p \left(a e^2-3 b d^2 \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)\right)+6 a b d e (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)-2 a b d^3 (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)\right)}{2 a b (p+1) x}","-\frac{d^3 \left(a+b x^2\right)^{p+1}}{a x}+\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a e^2+b d^2 (2 p+1)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{a}-\frac{3 d^2 e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a (p+1)}+\frac{e^3 \left(a+b x^2\right)^{p+1}}{2 b (p+1)}",1,"((a + b*x^2)^p*(-2*a*b*d^3*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)] + e*x*(6*a*b*d*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + (a + b*x^2)*(1 + (b*x^2)/a)^p*(a*e^2 - 3*b*d^2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a]))))/(2*a*b*(1 + p)*x*(1 + (b*x^2)/a)^p)","A",1
407,1,174,168,0.1169245,"\int \frac{(d+e x)^3 \left(a+b x^2\right)^p}{x^3} \, dx","Integrate[((d + e*x)^3*(a + b*x^2)^p)/x^3,x]","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(x \left(d \left(a+b x^2\right) \left(\frac{b x^2}{a}+1\right)^p \left(3 a e^2 \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)-b d^2 \, _2F_1\left(2,p+1;p+2;\frac{b x^2}{a}+1\right)\right)-2 a^2 e^3 (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)+6 a^2 d^2 e (p+1) \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)\right)}{2 a^2 (p+1) x}","-\frac{d \left(a+b x^2\right)^{p+1} \left(3 a e^2+b d^2 p\right) \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 (p+1)}-\frac{d^3 \left(a+b x^2\right)^{p+1}}{2 a x^2}+\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+3 b d^2 (2 p+1)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{a}-\frac{3 d^2 e \left(a+b x^2\right)^{p+1}}{a x}",1,"-1/2*((a + b*x^2)^p*(6*a^2*d^2*e*(1 + p)*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)] + x*(-2*a^2*e^3*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + d*(a + b*x^2)*(1 + (b*x^2)/a)^p*(3*a*e^2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^2)/a] - b*d^2*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + (b*x^2)/a]))))/(a^2*(1 + p)*x*(1 + (b*x^2)/a)^p)","A",1
408,0,0,199,0.6494987,"\int \frac{x^4 \left(a+b x^2\right)^p}{d+e x} \, dx","Integrate[(x^4*(a + b*x^2)^p)/(d + e*x),x]","\int \frac{x^4 \left(a+b x^2\right)^p}{d+e x} \, dx","\frac{x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{5}{2};-p,1;\frac{7}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{5 d}+\frac{\left(b d^2-a e^2\right) \left(a+b x^2\right)^{p+1}}{2 b^2 e^3 (p+1)}+\frac{\left(a+b x^2\right)^{p+2}}{2 b^2 e (p+2)}-\frac{d^4 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e^3 (p+1) \left(a e^2+b d^2\right)}",1,"Integrate[(x^4*(a + b*x^2)^p)/(d + e*x), x]","F",-1
409,1,260,163,0.3291657,"\int \frac{x^3 \left(a+b x^2\right)^p}{d+e x} \, dx","Integrate[(x^3*(a + b*x^2)^p)/(d + e*x),x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(\frac{b x^2}{a}+1\right)^{-p} \left(6 b d^2 (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+e \left(2 b e (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right)-3 d \left(b x^2 \left(\frac{b x^2}{a}+1\right)^p+a \left(\left(\frac{b x^2}{a}+1\right)^p-1\right)\right)\right)\right)}{b (p+1)}-\frac{3 d^3 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}\right)}{6 e^4}","-\frac{e x^5 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{5}{2};-p,1;\frac{7}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{5 d^2}+\frac{d^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e^2 (p+1) \left(a e^2+b d^2\right)}-\frac{d \left(a+b x^2\right)^{p+1}}{2 b e^2 (p+1)}",1,"((a + b*x^2)^p*((-3*d^3*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) + (e*(6*b*d^2*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + e*(-3*d*(b*x^2*(1 + (b*x^2)/a)^p + a*(-1 + (1 + (b*x^2)/a)^p)) + 2*b*e*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])))/(b*(1 + p)*(1 + (b*x^2)/a)^p)))/(6*e^4)","A",0
410,1,227,161,0.2385783,"\int \frac{x^2 \left(a+b x^2\right)^p}{d+e x} \, dx","Integrate[(x^2*(a + b*x^2)^p)/(d + e*x),x]","\frac{\left(a+b x^2\right)^p \left(b d^2 (p+1) \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)-2 b d e p (p+1) x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)-a e^2 p \left(\frac{b x^2}{a}+1\right)^{-p}+a e^2 p+b e^2 p x^2\right)}{2 b e^3 p (p+1)}","\frac{x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,1;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d}-\frac{d^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e (p+1) \left(a e^2+b d^2\right)}+\frac{\left(a+b x^2\right)^{p+1}}{2 b e (p+1)}",1,"((a + b*x^2)^p*(a*e^2*p + b*e^2*p*x^2 - (a*e^2*p)/(1 + (b*x^2)/a)^p + (b*d^2*(1 + p)*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (2*b*d*e*p*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/(2*b*e^3*p*(1 + p))","A",0
411,1,172,173,0.1856764,"\int \frac{x \left(a+b x^2\right)^p}{d+e x} \, dx","Integrate[(x*(a + b*x^2)^p)/(d + e*x),x]","\frac{\left(a+b x^2\right)^p \left(2 e x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)-\frac{d \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}\right)}{2 e^2}","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e}+\frac{d \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)}+\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e}",1,"((a + b*x^2)^p*(-((d*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + (2*e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/(2*e^2)","A",0
412,1,131,125,0.0462624,"\int \frac{\left(a+b x^2\right)^p}{d+e x} \, dx","Integrate[(a + b*x^2)^p/(d + e*x),x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{2 e p}","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d}-\frac{e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(2*e*p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)","A",0
413,1,170,176,0.17821,"\int \frac{\left(a+b x^2\right)^p}{x (d+e x)} \, dx","Integrate[(a + b*x^2)^p/(x*(d + e*x)),x]","\frac{\left(a+b x^2\right)^p \left(\left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)-\left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)\right)}{2 d p}","-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^2}+\frac{e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d (p+1) \left(a e^2+b d^2\right)}-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a d (p+1)}",1,"((a + b*x^2)^p*(-(AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))]/(1 + a/(b*x^2))^p))/(2*d*p)","A",0
414,1,214,178,0.3488063,"\int \frac{\left(a+b x^2\right)^p}{x^2 (d+e x)} \, dx","Integrate[(a + b*x^2)^p/(x^2*(d + e*x)),x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-\frac{2 d \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}-\frac{e \left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)}{p}\right)}{2 d^2}","-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};-p,1;\frac{1}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d x}-\frac{e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d^2 (p+1) \left(a e^2+b d^2\right)}+\frac{e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a d^2 (p+1)}",1,"((a + b*x^2)^p*((e*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (2*d*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) - (e*Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))])/(p*(1 + a/(b*x^2))^p)))/(2*d^2)","A",0
415,1,256,213,0.2788144,"\int \frac{\left(a+b x^2\right)^p}{x^3 (d+e x)} \, dx","Integrate[(a + b*x^2)^p/(x^3*(d + e*x)),x]","\frac{\left(a+b x^2\right)^p \left(-\frac{e^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}+\left(\frac{a}{b x^2}+1\right)^{-p} \left(\frac{d^2 \, _2F_1\left(1-p,-p;2-p;-\frac{a}{b x^2}\right)}{(p-1) x^2}+\frac{e^2 \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)}{p}\right)+\frac{2 d e \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}\right)}{2 d^3}","-\frac{\left(a+b x^2\right)^{p+1} \left(a e^2+b d^2 p\right) \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a^2 d^3 (p+1)}+\frac{e \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};-p,1;\frac{1}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^2 x}+\frac{e^4 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d^3 (p+1) \left(a e^2+b d^2\right)}-\frac{\left(a+b x^2\right)^{p+1}}{2 a d x^2}",1,"((a + b*x^2)^p*(-((e^2*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + (2*d*e*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) + ((d^2*Hypergeometric2F1[1 - p, -p, 2 - p, -(a/(b*x^2))])/((-1 + p)*x^2) + (e^2*Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))])/p)/(1 + a/(b*x^2))^p))/(2*d^3)","A",0
416,0,0,392,0.7052523,"\int \frac{x^4 \left(a+b x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^4*(a + b*x^2)^p)/(d + e*x)^2,x]","\int \frac{x^4 \left(a+b x^2\right)^p}{(d+e x)^2} \, dx","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a^2 e^4-2 a b d^2 e^2 (3 p+4)-2 b^2 d^4 \left(2 p^2+7 p+6\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{b e^4 (2 p+3) \left(a e^2+b d^2\right)}-\frac{2 d^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(2 a e^2+b d^2 (p+2)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e^4 \left(a e^2+b d^2\right)}-\frac{d^4 \left(a+b x^2\right)^{p+1}}{e^3 (d+e x) \left(a e^2+b d^2\right)}+\frac{d^3 \left(a+b x^2\right)^{p+1} \left(2 a e^2+b d^2 (p+2)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{e^3 (p+1) \left(a e^2+b d^2\right)^2}-\frac{d (3 p+4) \left(a+b x^2\right)^{p+1}}{b e^3 (p+1) (2 p+3)}+\frac{(d+e x) \left(a+b x^2\right)^{p+1}}{b e^3 (2 p+3)}",1,"Integrate[(x^4*(a + b*x^2)^p)/(d + e*x)^2, x]","F",-1
417,1,343,321,0.6871767,"\int \frac{x^3 \left(a+b x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^3*(a + b*x^2)^p)/(d + e*x)^2,x]","\frac{\left(a+b x^2\right)^p \left(-\frac{2 d^3 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}+\frac{3 d^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}+e \left(\frac{e \left(-a \left(\frac{b x^2}{a}+1\right)^{-p}+a+b x^2\right)}{b p+b}-4 d x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)\right)}{2 e^4}","\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a e^2+b d^2 (2 p+3)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e^3 \left(a e^2+b d^2\right)}-\frac{d^2 \left(a+b x^2\right)^{p+1} \left(3 a e^2+b d^2 (2 p+3)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e^2 (p+1) \left(a e^2+b d^2\right)^2}-\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(2 a e^2+b d^2 (2 p+3)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e^3 \left(a e^2+b d^2\right)}+\frac{d^3 \left(a+b x^2\right)^{p+1}}{e^2 (d+e x) \left(a e^2+b d^2\right)}+\frac{\left(a+b x^2\right)^{p+1}}{2 b e^2 (p+1)}",1,"((a + b*x^2)^p*((-2*d^3*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) + (3*d^2*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) + e*((e*(a + b*x^2 - a/(1 + (b*x^2)/a)^p))/(b + b*p) - (4*d*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p)))/(2*e^4)","A",0
418,1,300,281,0.3338682,"\int \frac{x^2 \left(a+b x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x^2*(a + b*x^2)^p)/(d + e*x)^2,x]","\frac{\left(a+b x^2\right)^p \left(\frac{d^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}-\frac{d \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}+e x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{e^3}","-\frac{2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+b d^2 (p+1)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e^2 \left(a e^2+b d^2\right)}+\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+2 b d^2 (p+1)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e^2 \left(a e^2+b d^2\right)}+\frac{d \left(a+b x^2\right)^{p+1} \left(a e^2+b d^2 (p+1)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{e (p+1) \left(a e^2+b d^2\right)^2}-\frac{d^2 \left(a+b x^2\right)^{p+1}}{e (d+e x) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((d^2*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) - (d*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) + (e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/e^3","A",0
419,1,223,273,0.1834928,"\int \frac{x \left(a+b x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(x*(a + b*x^2)^p)/(d + e*x)^2,x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} \left((2 p-1) (d+e x) F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)-2 d p F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)\right)}{2 e^2 p (2 p-1) (d+e x)}","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+b d^2 (2 p+1)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d e \left(a e^2+b d^2\right)}-\frac{b d (2 p+1) x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e \left(a e^2+b d^2\right)}-\frac{\left(a+b x^2\right)^{p+1} \left(a e^2+b d^2 (2 p+1)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)^2}+\frac{d \left(a+b x^2\right)^{p+1}}{(d+e x) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*(-2*d*p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)] + (-1 + 2*p)*(d + e*x)*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]))/(2*e^2*p*(-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x))","A",0
420,1,141,244,0.0863473,"\int \frac{\left(a+b x^2\right)^p}{(d+e x)^2} \, dx","Integrate[(a + b*x^2)^p/(d + e*x)^2,x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{e (2 p-1) (d+e x)}","-\frac{2 b p x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{a e^2+b d^2}+\frac{b (2 p+1) x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{a e^2+b d^2}-\frac{b d e \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{(p+1) \left(a e^2+b d^2\right)^2}+\frac{e^2 x \left(a+b x^2\right)^{p+1}}{\left(d^2-e^2 x^2\right) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(e*(-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x))","A",0
421,1,303,368,0.3474586,"\int \frac{\left(a+b x^2\right)^p}{x (d+e x)^2} \, dx","Integrate[(a + b*x^2)^p/(x*(d + e*x)^2),x]","\frac{\left(a+b x^2\right)^p \left(\frac{\left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)-\left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-\frac{2 d \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}\right)}{2 d^2}","-\frac{e^3 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d^5}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3}+\frac{e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d^2 (p+1) \left(a e^2+b d^2\right)}+\frac{b e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{(p+1) \left(a e^2+b d^2\right)^2}-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a d^2 (p+1)}",1,"((a + b*x^2)^p*((-2*d*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) + (-(AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))]/(1 + a/(b*x^2))^p)/p))/(2*d^2)","A",0
422,1,342,421,0.4570325,"\int \frac{\left(a+b x^2\right)^p}{x^2 (d+e x)^2} \, dx","Integrate[(a + b*x^2)^p/(x^2*(d + e*x)^2),x]","\frac{\left(a+b x^2\right)^p \left(\frac{d e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}+\frac{e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-\frac{d \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}-\frac{e \left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)}{p}\right)}{d^3}","\frac{e^4 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d^6}+\frac{2 e^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4}+\frac{e^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4}+\frac{e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{a d^3 (p+1)}-\frac{b e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{d (p+1) \left(a e^2+b d^2\right)^2}-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{d^2 x}-\frac{e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{d^3 (p+1) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((d*e*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) + (e*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (d*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) - (e*Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))])/(p*(1 + a/(b*x^2))^p)))/d^3","A",0
423,1,462,449,0.9327192,"\int \frac{x^4 \left(a+b x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^4*(a + b*x^2)^p)/(d + e*x)^3,x]","\frac{\left(a+b x^2\right)^p \left(\frac{d^4 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(p-1) (d+e x)^2}-\frac{8 d^3 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}+\frac{6 d^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-6 d e x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)+\frac{a e^2}{b p+b}+\frac{e^2 x^2}{p+1}\right)}{2 e^5}","\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(6 a^2 e^4+3 a b d^2 e^2 (3 p+4)+b^2 d^4 \left(2 p^2+7 p+6\right)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e^4 \left(a e^2+b d^2\right)^2}-\frac{d x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a^2 e^4+2 a b d^2 e^2 (4 p+5)+b^2 d^4 \left(2 p^2+7 p+6\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e^4 \left(a e^2+b d^2\right)^2}-\frac{d^2 \left(a+b x^2\right)^{p+1} \left(6 a^2 e^4+3 a b d^2 e^2 (3 p+4)+b^2 d^4 \left(2 p^2+7 p+6\right)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e^3 (p+1) \left(a e^2+b d^2\right)^3}-\frac{d^4 \left(a+b x^2\right)^{p+1}}{2 e^3 (d+e x)^2 \left(a e^2+b d^2\right)}+\frac{d^3 \left(a+b x^2\right)^{p+1} \left(4 a e^2+b d^2 (p+3)\right)}{e^3 (d+e x) \left(a e^2+b d^2\right)^2}+\frac{\left(a+b x^2\right)^{p+1}}{2 b e^3 (p+1)}",1,"((a + b*x^2)^p*((a*e^2)/(b + b*p) + (e^2*x^2)/(1 + p) - (8*d^3*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) + (d^4*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2) + (6*d^2*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (6*d*e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/(2*e^5)","A",0
424,1,436,416,0.7418709,"\int \frac{x^3 \left(a+b x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^3*(a + b*x^2)^p)/(d + e*x)^3,x]","\frac{\left(a+b x^2\right)^p \left(-\frac{d^3 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(p-1) (d+e x)^2}+\frac{6 d^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}-\frac{3 d \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}+2 e x \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)\right)}{2 e^4}","-\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left(2 p^2+5 p+3\right)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e^3 \left(a e^2+b d^2\right)^2}+\frac{d \left(a+b x^2\right)^{p+1} \left(3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left(2 p^2+5 p+3\right)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e^2 (p+1) \left(a e^2+b d^2\right)^3}+\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a^2 e^4+a b d^2 e^2 (6 p+5)+b^2 d^4 \left(2 p^2+5 p+3\right)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e^3 \left(a e^2+b d^2\right)^2}-\frac{d^2 \left(a+b x^2\right)^{p+1} \left(3 a e^2+b d^2 (p+2)\right)}{e^2 (d+e x) \left(a e^2+b d^2\right)^2}+\frac{d^3 \left(a+b x^2\right)^{p+1}}{2 e^2 (d+e x)^2 \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((6*d^2*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) - (d^3*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2) - (3*d*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) + (2*e*x*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/(2*e^4)","A",0
425,1,290,396,0.3440591,"\int \frac{x^2 \left(a+b x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x^2*(a + b*x^2)^p)/(d + e*x)^3,x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} \left(\frac{d^2 F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(p-1) (d+e x)^2}-\frac{4 d F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}+\frac{F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}\right)}{2 e^3}","\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a^2 e^4+a b d^2 e^2 (5 p+2)+b^2 d^4 \left(2 p^2+3 p+1\right)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d e^2 \left(a e^2+b d^2\right)^2}-\frac{\left(a+b x^2\right)^{p+1} \left(a^2 e^4+a b d^2 e^2 (5 p+2)+b^2 d^4 \left(2 p^2+3 p+1\right)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 e (p+1) \left(a e^2+b d^2\right)^3}-\frac{b d (2 p+1) x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(2 a e^2+b d^2 (p+1)\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e^2 \left(a e^2+b d^2\right)^2}+\frac{d \left(a+b x^2\right)^{p+1} \left(2 a e^2+b d^2 (p+1)\right)}{e (d+e x) \left(a e^2+b d^2\right)^2}-\frac{d^2 \left(a+b x^2\right)^{p+1}}{2 e (d+e x)^2 \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((-4*d*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*(d + e*x)) + (d^2*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + p)*(d + e*x)^2) + AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]/p))/(2*e^3*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)","A",0
426,1,229,336,0.2082979,"\int \frac{x \left(a+b x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(x*(a + b*x^2)^p)/(d + e*x)^3,x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} \left(2 (p-1) (d+e x) F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)+d (1-2 p) F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)\right)}{2 e^2 (p-1) (2 p-1) (d+e x)^2}","-\frac{b p x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(3 a e^2+b d^2 (2 p+1)\right) F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{e \left(a e^2+b d^2\right)^2}+\frac{b (2 p+1) x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \left(a e^2+b d^2 p\right) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right)}{e \left(a e^2+b d^2\right)^2}+\frac{b d p \left(a+b x^2\right)^{p+1} \left(3 a e^2+b d^2 (2 p+1)\right) \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)^3}-\frac{\left(a+b x^2\right)^{p+1} \left(a e^2+b d^2 p\right)}{(d+e x) \left(a e^2+b d^2\right)^2}+\frac{d \left(a+b x^2\right)^{p+1}}{2 (d+e x)^2 \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*(2*(-1 + p)*(d + e*x)*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)] + d*(1 - 2*p)*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]))/(2*e^2*(-1 + p)*(-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2)","A",0
427,1,142,322,0.1119526,"\int \frac{\left(a+b x^2\right)^p}{(d+e x)^3} \, dx","Integrate[(a + b*x^2)^p/(d + e*x)^3,x]","\frac{\left(a+b x^2\right)^p \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{2 e (p-1) (d+e x)^2}","\frac{e^2 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5}+\frac{x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3}-\frac{3 b^2 d^2 e \left(a+b x^2\right)^{p+1} \, _2F_1\left(3,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)^3}+\frac{b e \left(a+b x^2\right)^{p+1} \left(2 a e^2+b d^2 (p+1)\right) \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{4 (p+1) \left(a e^2+b d^2\right)^3}-\frac{d^2 e \left(a+b x^2\right)^{p+1}}{4 \left(d^2-e^2 x^2\right)^2 \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(2*e*(-1 + p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2)","A",0
428,1,434,700,0.7157666,"\int \frac{\left(a+b x^2\right)^p}{x (d+e x)^3} \, dx","Integrate[(a + b*x^2)^p/(x*(d + e*x)^3),x]","\frac{\left(a+b x^2\right)^p \left(-\frac{d^2 \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(p-1) (d+e x)^2}+\frac{\left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)-\left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-\frac{2 d \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}\right)}{2 d^3}","-\frac{e^3 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d^6}-\frac{e^3 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^6}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4}-\frac{e x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4}+\frac{3 b^2 d e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(3,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)^3}-\frac{\left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a d^3 (p+1)}-\frac{b e^2 \left(a+b x^2\right)^{p+1} \left(2 a e^2+b d^2 (p+1)\right) \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{4 d (p+1) \left(a e^2+b d^2\right)^3}+\frac{b e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{d (p+1) \left(a e^2+b d^2\right)^2}+\frac{d e^2 \left(a+b x^2\right)^{p+1}}{4 \left(d^2-e^2 x^2\right)^2 \left(a e^2+b d^2\right)}+\frac{e^2 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d^3 (p+1) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((-2*d*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) - (d^2*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2) + (-(AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)]/(((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p)) + Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))]/(1 + a/(b*x^2))^p)/p))/(2*d^3)","A",0
429,1,478,754,0.7658391,"\int \frac{\left(a+b x^2\right)^p}{x^2 (d+e x)^3} \, dx","Integrate[(a + b*x^2)^p/(x^2*(d + e*x)^3),x]","\frac{\left(a+b x^2\right)^p \left(\frac{d^2 e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(2-2 p;-p,-p;3-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(p-1) (d+e x)^2}+\frac{4 d e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{(2 p-1) (d+e x)}+\frac{3 e \left(\frac{e \left(x-\sqrt{-\frac{a}{b}}\right)}{d+e x}\right)^{-p} \left(\frac{e \left(\sqrt{-\frac{a}{b}}+x\right)}{d+e x}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right)}{p}-\frac{2 d \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{x}-\frac{3 e \left(\frac{a}{b x^2}+1\right)^{-p} \, _2F_1\left(-p,-p;1-p;-\frac{a}{b x^2}\right)}{p}\right)}{2 d^4}","\frac{2 e^4 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{3 d^7}+\frac{e^4 x^3 \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{3}{2};-p,3;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^7}+\frac{3 e^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5}+\frac{2 e^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5}+\frac{e^2 x \left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,3;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5}-\frac{3 b^2 e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(3,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 (p+1) \left(a e^2+b d^2\right)^3}+\frac{3 e \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b x^2}{a}+1\right)}{2 a d^4 (p+1)}-\frac{\left(a+b x^2\right)^p \left(\frac{b x^2}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right)}{d^3 x}+\frac{b e^3 \left(a+b x^2\right)^{p+1} \left(2 a e^2+b d^2 (p+1)\right) \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{4 d^2 (p+1) \left(a e^2+b d^2\right)^3}-\frac{2 b e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(2,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{d^2 (p+1) \left(a e^2+b d^2\right)^2}-\frac{e^3 \left(a+b x^2\right)^{p+1}}{4 \left(d^2-e^2 x^2\right)^2 \left(a e^2+b d^2\right)}-\frac{3 e^3 \left(a+b x^2\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{e^2 \left(b x^2+a\right)}{b d^2+a e^2}\right)}{2 d^4 (p+1) \left(a e^2+b d^2\right)}",1,"((a + b*x^2)^p*((4*d*e*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + 2*p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)) + (d^2*e*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/((-1 + p)*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p*(d + e*x)^2) + (3*e*AppellF1[-2*p, -p, -p, 1 - 2*p, (d - Sqrt[-(a/b)]*e)/(d + e*x), (d + Sqrt[-(a/b)]*e)/(d + e*x)])/(p*((e*(-Sqrt[-(a/b)] + x))/(d + e*x))^p*((e*(Sqrt[-(a/b)] + x))/(d + e*x))^p) - (2*d*Hypergeometric2F1[-1/2, -p, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^p) - (3*e*Hypergeometric2F1[-p, -p, 1 - p, -(a/(b*x^2))])/(p*(1 + a/(b*x^2))^p)))/(2*d^4)","A",0
430,1,182,276,0.2060812,"\int (g x)^m (d+e x)^3 \left(a+c x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)^3*(a + c*x^2)^p,x]","x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(\frac{d^3 \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)}{m+1}+e x \left(\frac{3 d^2 \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)}{m+2}+e x \left(\frac{3 d \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{c x^2}{a}\right)}{m+3}+\frac{e x \, _2F_1\left(\frac{m+4}{2},-p;\frac{m+6}{2};-\frac{c x^2}{a}\right)}{m+4}\right)\right)\right)","-\frac{e (g x)^{m+2} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(a e^2 (m+2)-3 c d^2 (m+2 p+4)\right) \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)}{c g^2 (m+2) (m+2 p+4)}-\frac{d (g x)^{m+1} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(3 a e^2 (m+1)-c d^2 (m+2 p+3)\right) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)}{c g (m+1) (m+2 p+3)}+\frac{3 d e^2 (g x)^{m+1} \left(a+c x^2\right)^{p+1}}{c g (m+2 p+3)}+\frac{e^3 (g x)^{m+2} \left(a+c x^2\right)^{p+1}}{c g^2 (m+2 p+4)}",1,"(x*(g*x)^m*(a + c*x^2)^p*((d^3*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)])/(1 + m) + e*x*((3*d^2*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)])/(2 + m) + e*x*((3*d*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((c*x^2)/a)])/(3 + m) + (e*x*Hypergeometric2F1[(4 + m)/2, -p, (6 + m)/2, -((c*x^2)/a)])/(4 + m)))))/(1 + (c*x^2)/a)^p","A",1
431,1,158,205,0.1042322,"\int (g x)^m (d+e x)^2 \left(a+c x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)^2*(a + c*x^2)^p,x]","\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)+e (m+1) x \left(2 d (m+3) \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)+e (m+2) x \, _2F_1\left(\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{c x^2}{a}\right)\right)\right)}{(m+1) (m+2) (m+3)}","-\frac{(g x)^{m+1} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(a e^2 (m+1)-c d^2 (m+2 p+3)\right) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)}{c g (m+1) (m+2 p+3)}+\frac{2 d e (g x)^{m+2} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)}{g^2 (m+2)}+\frac{e^2 (g x)^{m+1} \left(a+c x^2\right)^{p+1}}{c g (m+2 p+3)}",1,"(x*(g*x)^m*(a + c*x^2)^p*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)] + e*(1 + m)*x*(2*d*(3 + m)*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)] + e*(2 + m)*x*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((c*x^2)/a)])))/((1 + m)*(2 + m)*(3 + m)*(1 + (c*x^2)/a)^p)","A",1
432,1,106,135,0.0329989,"\int (g x)^m (d+e x) \left(a+c x^2\right)^p \, dx","Integrate[(g*x)^m*(d + e*x)*(a + c*x^2)^p,x]","\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d (m+2) \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)+e (m+1) x \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)\right)}{(m+1) (m+2)}","\frac{d (g x)^{m+1} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)}{g (m+1)}+\frac{e (g x)^{m+2} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+2}{2},-p;\frac{m+4}{2};-\frac{c x^2}{a}\right)}{g^2 (m+2)}",1,"(x*(g*x)^m*(a + c*x^2)^p*(d*(2 + m)*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)] + e*(1 + m)*x*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)]))/((1 + m)*(2 + m)*(1 + (c*x^2)/a)^p)","A",1
433,1,64,66,0.0096096,"\int (g x)^m \left(a+c x^2\right)^p \, dx","Integrate[(g*x)^m*(a + c*x^2)^p,x]","\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+1}{2}+1;-\frac{c x^2}{a}\right)}{m+1}","\frac{(g x)^{m+1} \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \, _2F_1\left(\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{c x^2}{a}\right)}{g (m+1)}",1,"(x*(g*x)^m*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, -((c*x^2)/a)])/((1 + m)*(1 + (c*x^2)/a)^p)","A",1
434,0,0,157,0.0900012,"\int \frac{(g x)^m \left(a+c x^2\right)^p}{d+e x} \, dx","Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x),x]","\int \frac{(g x)^m \left(a+c x^2\right)^p}{d+e x} \, dx","\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};-p,1;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d (m+1)}-\frac{e x^2 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+2}{2};-p,1;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^2 (m+2)}",1,"Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x), x]","F",-1
435,0,0,238,0.0893705,"\int \frac{(g x)^m \left(a+c x^2\right)^p}{(d+e x)^2} \, dx","Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^2,x]","\int \frac{(g x)^m \left(a+c x^2\right)^p}{(d+e x)^2} \, dx","\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};-p,2;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^2 (m+1)}+\frac{e^2 x^3 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+3}{2};-p,2;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4 (m+3)}-\frac{2 e x^2 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+2}{2};-p,2;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3 (m+2)}",1,"Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^2, x]","F",-1
436,0,0,321,0.2179349,"\int \frac{(g x)^m \left(a+c x^2\right)^p}{(d+e x)^3} \, dx","Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^3,x]","\int \frac{(g x)^m \left(a+c x^2\right)^p}{(d+e x)^3} \, dx","-\frac{e^3 x^4 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+4}{2};-p,3;\frac{m+6}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^6 (m+4)}+\frac{3 e^2 x^3 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+3}{2};-p,3;\frac{m+5}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^5 (m+3)}-\frac{3 e x^2 (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+2}{2};-p,3;\frac{m+4}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^4 (m+2)}+\frac{x (g x)^m \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};-p,3;\frac{m+3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)}{d^3 (m+1)}",1,"Integrate[((g*x)^m*(a + c*x^2)^p)/(d + e*x)^3, x]","F",-1
437,1,304,345,1.5933193,"\int \frac{x^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{d+e x} \, dx","Integrate[(x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{3 \sqrt{c d} \sqrt{c d^2-a e^2} \left(5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\sqrt{c} \sqrt{d} \sqrt{e} \left(-15 a^3 e^6+a^2 c d e^4 (10 e x-17 d)+a c^2 d^2 e^2 \left(-25 d^2+12 d e x-8 e^2 x^2\right)+c^3 d^3 \left(105 d^3-70 d^2 e x+56 d e^2 x^2-48 e^3 x^3\right)\right)\right)}{192 c^{7/2} d^{7/2} e^{9/2}}","-\frac{\left(-15 a^3 e^6-2 c d e x \left(-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right)-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left(c d^2-a e^2\right) \left(5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac{1}{24} x^2 \left(\frac{a}{c d}-\frac{7 d}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}+\frac{x^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-15*a^3*e^6 + a^2*c*d*e^4*(-17*d + 10*e*x) + a*c^2*d^2*e^2*(-25*d^2 + 12*d*e*x - 8*e^2*x^2) + c^3*d^3*(105*d^3 - 70*d^2*e*x + 56*d*e^2*x^2 - 48*e^3*x^3))) + (3*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(192*c^(7/2)*d^(7/2)*e^(9/2))","A",1
438,1,245,251,0.8246704,"\int \frac{x^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{d+e x} \, dx","Integrate[(x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{c} \sqrt{d} \sqrt{e} \left(-3 a^2 e^4+2 a c d e^2 (e x-2 d)+c^2 d^2 \left(15 d^2-10 d e x+8 e^2 x^2\right)\right)-\frac{3 \sqrt{c d} \sqrt{c d^2-a e^2} \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{24 c^{5/2} d^{5/2} e^{7/2}}","-\frac{\left(c d^2-a e^2\right) \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{5/2} d^{5/2} e^{7/2}}+\frac{\left(\left(5 c d^2-3 a e^2\right) \left(a e^2+3 c d^2\right)-2 c d e x \left(5 c d^2-a e^2\right)\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{24 c^2 d^2 e^3}+\frac{x^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-3*a^2*e^4 + 2*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(15*d^2 - 10*d*e*x + 8*e^2*x^2)) - (3*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(24*c^(5/2)*d^(5/2)*e^(7/2))","A",1
439,1,197,207,0.6607568,"\int \frac{x \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{d+e x} \, dx","Integrate[(x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{c d} \sqrt{c d^2-a e^2} \left(a e^2+3 c d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\sqrt{c} \sqrt{d} \sqrt{e} \left(a e^2+c d (2 e x-3 d)\right)\right)}{4 c^{3/2} d^{3/2} e^{5/2}}","\frac{\left(c d^2-a e^2\right) \left(a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 c d e (d+e x)}-\frac{1}{4} \left(\frac{a}{c d}+\frac{3 d}{e^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e^2 + c*d*(-3*d + 2*e*x)) + (Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(3*c*d^2 + a*e^2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(4*c^(3/2)*d^(3/2)*e^(5/2))","A",1
440,1,155,131,0.7524086,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{d+e x} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{e}-\frac{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{(c d)^{3/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{e^{3/2}}","\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e}-\frac{\left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 \sqrt{c} \sqrt{d} e^{3/2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e] - (c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/((c*d)^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/e^(3/2)","A",1
441,1,210,168,0.1819683,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{x (d+e x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x*(d + e*x)),x]","-\frac{2 \sqrt{a e+c d x} \left(\sqrt{a} \sqrt{c} e \sqrt{d+e x} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)-\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}}","\frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{\sqrt{d}}",1,"(-2*Sqrt[a*e + c*d*x]*(-(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]) + Sqrt[a]*Sqrt[c]*e*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
442,1,117,137,0.1386252,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{x^2 (d+e x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^2*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\left(a e^2-c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{a} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{\sqrt{d}}{x}\right)}{d^{3/2}}","-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{d x}-\frac{\left(c d^2-a e^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 \sqrt{a} d^{3/2} \sqrt{e}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[d]/x) + ((-(c*d^2) + a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/d^(3/2)","A",1
443,1,162,202,0.1604975,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{x^3 (d+e x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^3*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\left(-3 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x} \sqrt{a e+c d x}}+\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left(a e (3 e x-2 d)-c d^2 x\right)}{x^2}\right)}{4 a^{3/2} d^{5/2} e^{3/2}}","\frac{\left(c d^2-a e^2\right) \left(3 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left(\frac{c}{a e}-\frac{3 e}{d^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 x}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 d x^2}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-(c*d^2*x) + a*e*(-2*d + 3*e*x)))/x^2 + ((c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(4*a^(3/2)*d^(5/2)*e^(3/2))","A",1
444,1,210,286,0.2390217,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{x^4 (d+e x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left(a^2 e^2 \left(-8 d^2+10 d e x-15 e^2 x^2\right)-2 a c d^2 e x (d-2 e x)+3 c^2 d^4 x^2\right)}{x^3}-\frac{3 \left(-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x} \sqrt{a e+c d x}}\right)}{24 a^{5/2} d^{7/2} e^{5/2}}","\frac{\left(3 c d^2-5 a e^2\right) \left(3 a e^2+c d^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left(c d^2-a e^2\right) \left(5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left(\frac{c}{a e}-\frac{5 e}{d^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 x^2}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 d x^3}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(d - 2*e*x) + a^2*e^2*(-8*d^2 + 10*d*e*x - 15*e^2*x^2)))/x^3 - (3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*a^(5/2)*d^(7/2)*e^(5/2))","A",1
445,1,273,389,0.3616185,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{x^5 (d+e x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^5*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left(a^3 e^3 \left(-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right)+a^2 c d^2 e^2 x \left(-8 d^2+12 d e x-25 e^2 x^2\right)+a c^2 d^4 e x^2 (10 d-17 e x)-15 c^3 d^6 x^3\right)}{x^4}+\frac{3 \left(-35 a^4 e^8+20 a^3 c d^2 e^6+6 a^2 c^2 d^4 e^4+4 a c^3 d^6 e^2+5 c^4 d^8\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x} \sqrt{a e+c d x}}\right)}{192 a^{7/2} d^{9/2} e^{7/2}}","\frac{\left(-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac{\left(-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac{\left(c d^2-a e^2\right) \left(35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 d x^4}-\frac{\left(\frac{c}{a e}-\frac{7 e}{d^2}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{24 x^3}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(10*d - 17*e*x) + a^2*c*d^2*e^2*x*(-8*d^2 + 12*d*e*x - 25*e^2*x^2) + a^3*e^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/x^4 + (3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*a^(7/2)*d^(9/2)*e^(7/2))","A",1
446,1,425,449,2.2880009,"\int \frac{x^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{c} \sqrt{d} \sqrt{e} \left(-105 a^5 e^{10}+5 a^4 c d e^8 (11 d+14 e x)+2 a^3 c^2 d^2 e^6 \left(27 d^2-16 d e x-28 e^2 x^2\right)+6 a^2 c^3 d^3 e^4 \left(13 d^3-6 d^2 e x+4 d e^2 x^2+8 e^3 x^3\right)+a c^4 d^4 e^2 \left(-525 d^4+336 d^3 e x-264 d^2 e^2 x^2+224 d e^3 x^3+1664 e^4 x^4\right)+c^5 d^5 \left(315 d^5-210 d^4 e x+168 d^3 e^2 x^2-144 d^2 e^3 x^3+128 d e^4 x^4+1280 e^5 x^5\right)\right)-\frac{15 \sqrt{c d} \left(c d^2-a e^2\right)^{5/2} \left(7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{7680 c^{9/2} d^{9/2} e^{11/2}}","-\frac{\left(-35 a^3 e^6-6 c d e x \left(-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right)-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{960 c^3 d^3 e^4}-\frac{\left(7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 c^{9/2} d^{9/2} e^{11/2}}+\frac{\left(-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 c^4 d^4 e^5}+\frac{1}{20} x^2 \left(\frac{a}{c d}-\frac{3 d}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}+\frac{x^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{6 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 + 5*a^4*c*d*e^8*(11*d + 14*e*x) + 2*a^3*c^2*d^2*e^6*(27*d^2 - 16*d*e*x - 28*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(13*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 + 8*e^3*x^3) + a*c^4*d^4*e^2*(-525*d^4 + 336*d^3*e*x - 264*d^2*e^2*x^2 + 224*d*e^3*x^3 + 1664*e^4*x^4) + c^5*d^5*(315*d^5 - 210*d^4*e*x + 168*d^3*e^2*x^2 - 144*d^2*e^3*x^3 + 128*d*e^4*x^4 + 1280*e^5*x^5)) - (15*Sqrt[c*d]*(c*d^2 - a*e^2)^(5/2)*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7680*c^(9/2)*d^(9/2)*e^(11/2))","A",1
447,1,497,352,2.7708674,"\int \frac{x^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\frac{5 \left(3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \left(8 c^3 d^3 e^3 \sqrt{c d} \sqrt{c d^2-a e^2} (a e+c d x)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}-c d \left(c d^2-a e^2\right) \left(-3 c^{5/2} d^{5/2} \sqrt{e} \left(c d^2-a e^2\right)^2 \sqrt{a e+c d x} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-2 e^2 (c d)^{5/2} \sqrt{c d^2-a e^2} (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+3 e (c d)^{5/2} \left(c d^2-a e^2\right)^{3/2} (a e+c d x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}\right)\right)}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-48 c^4 d^4 e^3 (d+e x) \left(5 a e^2+7 c d^2\right) (a e+c d x)^3}{384 c^5 d^5 e^4 (a e+c d x)}+x (d+e x) (a e+c d x)^2\right)}{5 c d e}","\frac{\left(-15 a^2 e^4-6 c d e x \left(7 c d^2-3 a e^2\right)-12 a c d^2 e^2+35 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{240 c^2 d^2 e^3}+\frac{\left(3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left(3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac{x^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(x*(a*e + c*d*x)^2*(d + e*x) + (-48*c^4*d^4*e^3*(7*c*d^2 + 5*a*e^2)*(a*e + c*d*x)^3*(d + e*x) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(8*c^3*d^3*Sqrt[c*d]*e^3*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - c*d*(c*d^2 - a*e^2)*(3*(c*d)^(5/2)*e*(c*d^2 - a*e^2)^(3/2)*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 2*(c*d)^(5/2)*e^2*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 3*c^(5/2)*d^(5/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])))/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]))/(384*c^5*d^5*e^4*(a*e + c*d*x))))/(5*c*d*e)","A",1
448,1,276,295,1.2397349,"\int \frac{x \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(x*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{c} \sqrt{d} \sqrt{e} \left(-9 a^3 e^6+3 a^2 c d e^4 (3 d+2 e x)+a c^2 d^2 e^2 \left(-31 d^2+20 d e x+72 e^2 x^2\right)+c^3 d^3 \left(15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right)\right)-\frac{3 \sqrt{c d} \left(c d^2-a e^2\right)^{5/2} \left(3 a e^2+5 c d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{192 c^{5/2} d^{5/2} e^{7/2}}","-\frac{\left(3 a e^2+5 c d^2\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{5/2} d^{5/2} e^{7/2}}+\frac{\left(3 a e^2+5 c d^2\right) \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 c^2 d^2 e^3}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4 c d e (d+e x)}-\frac{1}{24} \left(\frac{3 a}{c d}+\frac{5 d}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-9*a^3*e^6 + 3*a^2*c*d*e^4*(3*d + 2*e*x) + a*c^2*d^2*e^2*(-31*d^2 + 20*d*e*x + 72*e^2*x^2) + c^3*d^3*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3)) - (3*Sqrt[c*d]*(c*d^2 - a*e^2)^(5/2)*(5*c*d^2 + 3*a*e^2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(192*c^(5/2)*d^(5/2)*e^(7/2))","A",1
449,1,264,201,0.6748785,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x),x]","\frac{\sqrt{c} \sqrt{d} \left(3 \left(c d^2-a e^2\right)^{7/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(-3 a^3 e^5-a^2 c d e^3 (8 d+17 e x)+a c^2 d^2 e \left(3 d^2-10 d e x-22 e^2 x^2\right)+c^3 d^3 x \left(3 d^2-2 d e x-8 e^2 x^2\right)\right)\right)}{24 e^{5/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 e}+\frac{1}{8} \left(\frac{a}{c d}-\frac{d}{e^2}\right) \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}",1,"(Sqrt[c]*Sqrt[d]*(-(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(-3*a^3*e^5 - a^2*c*d*e^3*(8*d + 17*e*x) + a*c^2*d^2*e*(3*d^2 - 10*d*e*x - 22*e^2*x^2) + c^3*d^3*x*(3*d^2 - 2*d*e*x - 8*e^2*x^2))) + 3*(c*d^2 - a*e^2)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(24*(c*d)^(5/2)*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
450,1,275,251,0.8532877,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(-\frac{8 a^{3/2} \sqrt{d} e^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x}}-\frac{\sqrt{c} \sqrt{d} \left(-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\sqrt{e} \sqrt{a e+c d x} \left(5 a e^2+c d (d+2 e x)\right)\right)}{4 e^{3/2} \sqrt{a e+c d x}}","-a^{3/2} \sqrt{d} e^{3/2} \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)-\frac{\left(-3 a^2 e^4-6 a c d^2 e^2+c^2 d^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 \sqrt{c} \sqrt{d} e^{3/2}}+\frac{\left(5 a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(5*a*e^2 + c*d*(d + 2*e*x)) - (Sqrt[c]*Sqrt[d]*(c^2*d^4 - 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (8*a^(3/2)*Sqrt[d]*e^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(4*e^(3/2)*Sqrt[a*e + c*d*x])","A",1
451,1,263,240,1.2007252,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^2 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{c} d \sqrt{c d} \left(3 a e^2+c d^2\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{\sqrt{a} \sqrt{e} \left(a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x}}+\frac{\sqrt{d} \sqrt{a e+c d x} (c d x-a e)}{x}\right)}{\sqrt{d} \sqrt{a e+c d x}}","-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (a e-c d x)}{x}+\frac{\sqrt{c} \sqrt{d} \left(3 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 \sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \left(a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 \sqrt{d}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[d]*(-(a*e) + c*d*x)*Sqrt[a*e + c*d*x])/x + (Sqrt[c]*d*Sqrt[c*d]*(c*d^2 + 3*a*e^2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (Sqrt[a]*Sqrt[e]*(3*c*d^2 + a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(Sqrt[d]*Sqrt[a*e + c*d*x])","A",1
452,1,285,256,2.4462905,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^3 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^3*(d + e*x)),x]","\frac{\sqrt{a e+c d x} \left(-\frac{\sqrt{d+e x} \left(-a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{a}}+\frac{8 e (c d)^{5/2} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{c^{3/2}}-\frac{\sqrt{d} \sqrt{e} (d+e x) \sqrt{a e+c d x} \left(a e (2 d+e x)+5 c d^2 x\right)}{x^2}\right)}{4 d^{3/2} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\left(-a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 \sqrt{a} d^{3/2} \sqrt{e}}+c^{3/2} d^{3/2} \sqrt{e} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)-\frac{\left(x \left(a e^2+5 c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 d x^2}",1,"(Sqrt[a*e + c*d*x]*(-((Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)*(5*c*d^2*x + a*e*(2*d + e*x)))/x^2) + (8*(c*d)^(5/2)*e*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/c^(3/2) - ((3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[a]))/(4*d^(3/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
453,1,188,211,0.2770324,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^4 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^4*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{3 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x} \sqrt{a e+c d x}}-\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left(a^2 e^2 \left(8 d^2+2 d e x-3 e^2 x^2\right)+2 a c d^2 e x (7 d+4 e x)+3 c^2 d^4 x^2\right)}{x^3}\right)}{24 a^{3/2} d^{5/2} e^{3/2}}","\frac{\left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left(\frac{c}{a e}-\frac{e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 x^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 d x^3}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^2*d^4*x^2 + 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(8*d^2 + 2*d*e*x - 3*e^2*x^2)))/x^3) + (3*(c*d^2 - a*e^2)^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*a^(3/2)*d^(5/2)*e^(3/2))","A",1
454,1,253,295,0.3031424,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^5 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^5*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{x \left(5 a e^2+3 c d^2\right) \left(\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a^2 e^2 \left(8 d^2+2 d e x-3 e^2 x^2\right)+2 a c d^2 e x (7 d+4 e x)+3 c^2 d^4 x^2\right)-3 x^3 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)\right)}{a^{3/2} d^{5/2} e^{3/2} \sqrt{d+e x} \sqrt{a e+c d x}}-48 (d+e x) (a e+c d x)^2\right)}{192 a d e x^4}","-\frac{\left(5 a e^2+3 c d^2\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left(5 a e^2+3 c d^2\right) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 d x^4}-\frac{\left(\frac{3 c}{a e}-\frac{5 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 x^3}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-48*(a*e + c*d*x)^2*(d + e*x) + ((3*c*d^2 + 5*a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(3*c^2*d^4*x^2 + 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(8*d^2 + 2*d*e*x - 3*e^2*x^2)) - 3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(3/2)*d^(5/2)*e^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*a*d*e*x^4)","A",1
455,1,310,395,0.4867394,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^6 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{5 x^2 \left(7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \left(\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a^2 e^2 \left(-8 d^2-2 d e x+3 e^2 x^2\right)-2 a c d^2 e x (7 d+4 e x)-3 c^2 d^4 x^2\right)+3 x^3 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)\right)}{a^{5/2} d^{7/2} e^{5/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{48 x (d+e x) \left(7 a e^2+5 c d^2\right) (a e+c d x)^2}{a d e}-384 (d+e x) (a e+c d x)^2\right)}{1920 a d e x^5}","\frac{\left(-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac{\left(7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left(7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 d x^5}-\frac{\left(\frac{3 c}{a e}-\frac{7 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{40 x^4}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-384*(a*e + c*d*x)^2*(d + e*x) + (48*(5*c*d^2 + 7*a*e^2)*x*(a*e + c*d*x)^2*(d + e*x))/(a*d*e) + (5*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*x^2*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(-8*d^2 - 2*d*e*x + 3*e^2*x^2)) + 3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*d^(7/2)*e^(5/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a*d*e*x^5)","A",1
456,1,380,498,0.7749553,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{x^7 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]","-\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{16 x^2 (d+e x) \left(63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right) (a e+c d x)^2}{a^2 d^2 e^2}+\frac{5 x^3 \left(21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right) \left(\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a^2 e^2 \left(-8 d^2-2 d e x+3 e^2 x^2\right)-2 a c d^2 e x (7 d+4 e x)-3 c^2 d^4 x^2\right)+3 x^3 \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)\right)}{a^{7/2} d^{9/2} e^{7/2} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{128 x (d+e x) \left(9 a e^2+7 c d^2\right) (a e+c d x)^2}{a d e}+1280 (d+e x) (a e+c d x)^2\right)}{7680 a d e x^6}","\frac{\left(-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac{\left(-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac{\left(-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac{\left(21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{6 d x^6}-\frac{\left(\frac{c}{a e}-\frac{3 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{20 x^5}",1,"-1/7680*(Sqrt[(a*e + c*d*x)*(d + e*x)]*(1280*(a*e + c*d*x)^2*(d + e*x) - (128*(7*c*d^2 + 9*a*e^2)*x*(a*e + c*d*x)^2*(d + e*x))/(a*d*e) + (16*(35*c^2*d^4 + 54*a*c*d^2*e^2 + 63*a^2*e^4)*x^2*(a*e + c*d*x)^2*(d + e*x))/(a^2*d^2*e^2) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*x^3*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(-8*d^2 - 2*d*e*x + 3*e^2*x^2)) + 3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(7/2)*d^(9/2)*e^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(a*d*e*x^6)","A",1
457,1,681,574,3.6081295,"\int \frac{x^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{105 \sqrt{c} \sqrt{d} \left(15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right) \left(c d^2-a e^2\right)^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\frac{\sqrt{e} \left(1575 a^8 e^{15}-525 a^7 c d e^{13} (7 d-e x)+35 a^6 c^2 d^2 e^{11} \left(29 d^2-37 d e x-6 e^2 x^2\right)+5 a^5 c^3 d^3 e^9 \left(185 d^3+93 d^2 e x+100 d e^2 x^2+24 e^3 x^3\right)+5 a^4 c^4 d^4 e^7 \left(265 d^4+65 d^3 e x-30 d^2 e^2 x^2-56 d e^3 x^3-16 e^4 x^4\right)+a^3 c^5 d^5 e^5 \left(-11193 d^5+8359 d^4 e x-6088 d^3 e^2 x^2+5040 d^2 e^3 x^3+139200 d e^4 x^4+104320 e^5 x^5\right)+a^2 c^6 d^6 e^3 \left(11445 d^6-18669 d^5 e x+12962 d^4 e^2 x^2-10544 d^3 e^3 x^3+9120 d^2 e^4 x^4+350080 d e^5 x^5+272640 e^6 x^6\right)+a c^7 d^7 e \left(-3465 d^7+13755 d^6 e x-9324 d^5 e^2 x^2+7512 d^4 e^3 x^3-6464 d^3 e^4 x^4+5760 d^2 e^5 x^5+299520 d e^6 x^6+240640 e^7 x^7\right)+c^8 d^8 x \left(-3465 d^7+2310 d^6 e x-1848 d^5 e^2 x^2+1584 d^4 e^3 x^3-1408 d^3 e^4 x^4+1280 d^2 e^5 x^5+87040 d e^6 x^6+71680 e^7 x^7\right)\right)}{a e+c d x}\right)}{573440 c^5 d^5 e^{13/2}}","-\frac{\left(-105 a^3 e^6-10 c d e x \left(-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right)-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4480 c^3 d^3 e^4}+\frac{3 \left(15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{32768 c^{11/2} d^{11/2} e^{13/2}}-\frac{3 \left(15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right) \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac{\left(15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right) \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2048 c^4 d^4 e^5}+\frac{1}{112} x^2 \left(\frac{5 a}{c d}-\frac{11 d}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}+\frac{x^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{8 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*(1575*a^8*e^15 - 525*a^7*c*d*e^13*(7*d - e*x) + 35*a^6*c^2*d^2*e^11*(29*d^2 - 37*d*e*x - 6*e^2*x^2) + 5*a^5*c^3*d^3*e^9*(185*d^3 + 93*d^2*e*x + 100*d*e^2*x^2 + 24*e^3*x^3) + 5*a^4*c^4*d^4*e^7*(265*d^4 + 65*d^3*e*x - 30*d^2*e^2*x^2 - 56*d*e^3*x^3 - 16*e^4*x^4) + a^3*c^5*d^5*e^5*(-11193*d^5 + 8359*d^4*e*x - 6088*d^3*e^2*x^2 + 5040*d^2*e^3*x^3 + 139200*d*e^4*x^4 + 104320*e^5*x^5) + a^2*c^6*d^6*e^3*(11445*d^6 - 18669*d^5*e*x + 12962*d^4*e^2*x^2 - 10544*d^3*e^3*x^3 + 9120*d^2*e^4*x^4 + 350080*d*e^5*x^5 + 272640*e^6*x^6) + c^8*d^8*x*(-3465*d^7 + 2310*d^6*e*x - 1848*d^5*e^2*x^2 + 1584*d^4*e^3*x^3 - 1408*d^3*e^4*x^4 + 1280*d^2*e^5*x^5 + 87040*d*e^6*x^6 + 71680*e^7*x^7) + a*c^7*d^7*e*(-3465*d^7 + 13755*d^6*e*x - 9324*d^5*e^2*x^2 + 7512*d^4*e^3*x^3 - 6464*d^3*e^4*x^4 + 5760*d^2*e^5*x^5 + 299520*d*e^6*x^6 + 240640*e^7*x^7)))/(a*e + c*d*x) + (105*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(9/2)*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(573440*c^5*d^5*e^(13/2))","A",1
458,1,562,452,5.7059569,"\int \frac{x^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x),x]","\frac{((d+e x) (a e+c d x))^{3/2} \left(\frac{7 \sqrt{c d} \left(5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right) \left(15 \sqrt{e} \sqrt{c d} \left(c d^2-a e^2\right)^{11/2} (a e+c d x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}-15 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^6 \sqrt{a e+c d x} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-10 e^{3/2} \sqrt{c d} \left(c d^2-a e^2\right)^{9/2} (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+8 e^{5/2} \sqrt{c d} \left(c d^2-a e^2\right)^{7/2} (a e+c d x)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+16 e^{7/2} \sqrt{c d} \left(c d^2-a e^2\right)^{3/2} (a e+c d x)^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \left(c d (11 d+8 e x)-3 a e^2\right)\right)}{15360 c^3 d^3 e^{9/2} \left(c d^2-a e^2\right)^{5/2} \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{3/2} (a e+c d x)^2}-\frac{(d+e x) \left(7 a e^2+9 c d^2\right) (a e+c d x)^2}{12 c d e}+x (d+e x) (a e+c d x)^2\right)}{7 c d e}","\frac{\left(-35 a^2 e^4-10 c d e x \left(9 c d^2-5 a e^2\right)-20 a c d^2 e^2+63 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{840 c^2 d^2 e^3}-\frac{\left(5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac{\left(5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right) \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac{\left(5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right) \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{384 c^3 d^3 e^4}+\frac{x^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 e}",1,"(((a*e + c*d*x)*(d + e*x))^(3/2)*(-1/12*((9*c*d^2 + 7*a*e^2)*(a*e + c*d*x)^2*(d + e*x))/(c*d*e) + x*(a*e + c*d*x)^2*(d + e*x) + (7*Sqrt[c*d]*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(15*Sqrt[c*d]*Sqrt[e]*(c*d^2 - a*e^2)^(11/2)*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 10*Sqrt[c*d]*e^(3/2)*(c*d^2 - a*e^2)^(9/2)*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 8*Sqrt[c*d]*e^(5/2)*(c*d^2 - a*e^2)^(7/2)*(a*e + c*d*x)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 16*Sqrt[c*d]*e^(7/2)*(c*d^2 - a*e^2)^(3/2)*(a*e + c*d*x)^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*(-3*a*e^2 + c*d*(11*d + 8*e*x)) - 15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^6*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(15360*c^3*d^3*e^(9/2)*(c*d^2 - a*e^2)^(5/2)*(a*e + c*d*x)^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/2))))/(7*c*d*e)","A",1
459,1,506,381,2.7350868,"\int \frac{x \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(x*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x),x]","\frac{(a e+c d x) ((d+e x) (a e+c d x))^{5/2} \left(7-\frac{7 \sqrt{c d} \sqrt{c d^2-a e^2} \left(5 a e^2+7 c d^2\right) \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{3/2} \left(15 \sqrt{e} \sqrt{c d} \left(c d^2-a e^2\right)^{11/2} (a e+c d x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}-15 \sqrt{c} \sqrt{d} \left(c d^2-a e^2\right)^6 \sqrt{a e+c d x} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-10 e^{3/2} \sqrt{c d} \left(c d^2-a e^2\right)^{9/2} (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+8 e^{5/2} \sqrt{c d} \left(c d^2-a e^2\right)^{7/2} (a e+c d x)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+16 e^{7/2} \sqrt{c d} \left(c d^2-a e^2\right)^{3/2} (a e+c d x)^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \left(c d (11 d+8 e x)-3 a e^2\right)\right)}{1280 c^5 d^5 e^{7/2} (d+e x)^4 (a e+c d x)^4}\right)}{42 c d e}","\frac{\left(5 a e^2+7 c d^2\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left(5 a e^2+7 c d^2\right) \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac{\left(5 a e^2+7 c d^2\right) \left(c d^2-a e^2\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{192 c^2 d^2 e^3}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{6 c d e (d+e x)}-\frac{1}{60} \left(\frac{5 a}{c d}+\frac{7 d}{e^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}",1,"((a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(7 - (7*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(7*c*d^2 + 5*a*e^2)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/2)*(15*Sqrt[c*d]*Sqrt[e]*(c*d^2 - a*e^2)^(11/2)*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 10*Sqrt[c*d]*e^(3/2)*(c*d^2 - a*e^2)^(9/2)*(a*e + c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 8*Sqrt[c*d]*e^(5/2)*(c*d^2 - a*e^2)^(7/2)*(a*e + c*d*x)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 16*Sqrt[c*d]*e^(7/2)*(c*d^2 - a*e^2)^(3/2)*(a*e + c*d*x)^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*(-3*a*e^2 + c*d*(11*d + 8*e*x)) - 15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^6*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(1280*c^5*d^5*e^(7/2)*(a*e + c*d*x)^4*(d + e*x)^4)))/(42*c*d*e)","A",1
460,1,384,274,1.2174011,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{d+e x} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x),x]","\frac{\sqrt{c d} \left(\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left(-15 a^5 e^9+5 a^4 c d e^7 (14 d-e x)+2 a^3 c^2 d^2 e^5 \left(64 d^2+268 d e x+129 e^2 x^2\right)+2 a^2 c^3 d^3 e^3 \left(-35 d^3+87 d^2 e x+489 d e^2 x^2+292 e^3 x^3\right)+a c^4 d^4 e \left(15 d^4-80 d^3 e x+54 d^2 e^2 x^2+688 d e^3 x^3+464 e^4 x^4\right)+c^5 d^5 x \left(15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right)\right)-15 \left(c d^2-a e^2\right)^{11/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{640 c^{7/2} d^{7/2} e^{7/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{3 \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{3 \left(c d^2-a e^2\right)^3 \left(a e^2+c d^2+2 c d e x\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 e}+\frac{1}{16} \left(\frac{a}{c d}-\frac{d}{e^2}\right) \left(a e^2+c d^2+2 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}",1,"(Sqrt[c*d]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[e]*(d + e*x)*(-15*a^5*e^9 + 5*a^4*c*d*e^7*(14*d - e*x) + 2*a^3*c^2*d^2*e^5*(64*d^2 + 268*d*e*x + 129*e^2*x^2) + 2*a^2*c^3*d^3*e^3*(-35*d^3 + 87*d^2*e*x + 489*d*e^2*x^2 + 292*e^3*x^3) + c^5*d^5*x*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4) + a*c^4*d^4*e*(15*d^4 - 80*d^3*e*x + 54*d^2*e^2*x^2 + 688*d*e^3*x^3 + 464*e^4*x^4)) - 15*(c*d^2 - a*e^2)^(11/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(640*c^(7/2)*d^(7/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
461,1,390,394,2.0065126,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(-\frac{384 a^{5/2} c d^{5/2} e^5 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x}}+\sqrt{e} \sqrt{a e+c d x} \left(15 a^3 e^6+a^2 c d e^4 (337 d+118 e x)+a c^2 d^2 e^2 \left(57 d^2+244 d e x+136 e^2 x^2\right)+c^3 \left(-9 d^6+6 d^5 e x+72 d^4 e^2 x^2+48 d^3 e^3 x^3\right)\right)+\frac{3 \sqrt{c} \sqrt{d} \left(-5 a^4 e^8+60 a^3 c d^2 e^6+90 a^2 c^2 d^4 e^4-20 a c^3 d^6 e^2+3 c^4 d^8\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{192 c d e^{5/2} \sqrt{a e+c d x}}","-a^{5/2} d^{3/2} e^{5/2} \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)-\frac{\left(-5 a^3 e^6-83 a^2 c d^2 e^4-11 a c^2 d^4 e^2+2 c d e x \left(c d^2-5 a e^2\right) \left(a e^2+3 c d^2\right)+3 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 c d e^2}+\frac{\left(-5 a^4 e^8+60 a^3 c d^2 e^6+90 a^2 c^2 d^4 e^4-20 a c^3 d^6 e^2+3 c^4 d^8\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left(11 a e^2+3 c d^2+6 c d e x\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 e}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(15*a^3*e^6 + a^2*c*d*e^4*(337*d + 118*e*x) + a*c^2*d^2*e^2*(57*d^2 + 244*d*e*x + 136*e^2*x^2) + c^3*(-9*d^6 + 6*d^5*e*x + 72*d^4*e^2*x^2 + 48*d^3*e^3*x^3)) + (3*Sqrt[c]*Sqrt[d]*(3*c^4*d^8 - 20*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 - 5*a^4*e^8)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (384*a^(5/2)*c*d^(5/2)*e^5*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(192*c*d*e^(5/2)*Sqrt[a*e + c*d*x])","A",1
462,1,350,352,2.0727078,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^2 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(-\frac{24 a^{3/2} \sqrt{d} e^3 \left(3 a e^2+5 c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x}}+\frac{\sqrt{e} \sqrt{a e+c d x} \left(3 a^2 e^3 (11 e x-8 d)+2 a c d e^2 x (34 d+13 e x)+c^2 d^2 x \left(3 d^2+14 d e x+8 e^2 x^2\right)\right)}{x}-\frac{3 \sqrt{c} \sqrt{d} \left(-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right)}{24 e^{3/2} \sqrt{a e+c d x}}","-\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left(3 a e^2+5 c d^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)+\frac{\left(19 a^2 e^4+2 c d e x \left(7 a e^2+c d^2\right)+28 a c d^2 e^2+c^2 d^4\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 e}-\frac{\left(-5 a^3 e^6-45 a^2 c d^2 e^4-15 a c^2 d^4 e^2+c^3 d^6\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{(3 a e-c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 x}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*Sqrt[a*e + c*d*x]*(3*a^2*e^3*(-8*d + 11*e*x) + 2*a*c*d*e^2*x*(34*d + 13*e*x) + c^2*d^2*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2)))/x - (3*Sqrt[c]*Sqrt[d]*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (24*a^(3/2)*Sqrt[d]*e^3*(5*c*d^2 + 3*a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(24*e^(3/2)*Sqrt[a*e + c*d*x])","A",1
463,1,334,339,2.2453755,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^3 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{d} \sqrt{a e+c d x} \left(-a^2 e^2 (2 d+5 e x)-9 a c d e x (d-e x)+c^2 d^2 x^2 (5 d+2 e x)\right)}{x^2}+\frac{3 \sqrt{c} d \sqrt{c d} \left(5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{\sqrt{e} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{3 \sqrt{a} \sqrt{e} \left(a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{d+e x}}\right)}{4 \sqrt{d} \sqrt{a e+c d x}}","\frac{3 \sqrt{c} \sqrt{d} \left(5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 \sqrt{e}}-\frac{3 \sqrt{a} \sqrt{e} \left(a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 \sqrt{d}}-\frac{(a e-c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 x^2}-\frac{3 \left(a e \left(a e^2+3 c d^2\right)-c d x \left(3 a e^2+c d^2\right)\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 x}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[d]*Sqrt[a*e + c*d*x]*(-9*a*c*d*e*x*(d - e*x) + c^2*d^2*x^2*(5*d + 2*e*x) - a^2*e^2*(2*d + 5*e*x)))/x^2 + (3*Sqrt[c]*d*Sqrt[c*d]*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (3*Sqrt[a]*Sqrt[e]*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(4*Sqrt[d]*Sqrt[a*e + c*d*x])","A",1
464,1,357,371,3.0383551,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^4 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)),x]","\frac{\sqrt{a e+c d x} \left(-\frac{\sqrt{d} \sqrt{e} (d+e x) \sqrt{a e+c d x} \left(a^2 e^2 \left(8 d^2+14 d e x+3 e^2 x^2\right)+2 a c d^2 e x (13 d+34 e x)+3 c^2 d^3 x^2 (11 d-8 e x)\right)}{x^3}-\frac{3 \sqrt{d+e x} \left(-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{a}}+\frac{24 e (c d)^{5/2} \sqrt{c d^2-a e^2} \left(5 a e^2+3 c d^2\right) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{c^{3/2}}\right)}{24 d^{3/2} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\left(-a^2 e^4-2 c d e x \left(a e^2+7 c d^2\right)+12 a c d^2 e^2+5 c^2 d^4\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 d x}-\frac{\left(-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 \sqrt{a} d^{3/2} \sqrt{e}}+\frac{1}{2} c^{3/2} d^{3/2} \sqrt{e} \left(5 a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)-\frac{\left(3 x \left(a e^2+3 c d^2\right)+4 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{12 d x^3}",1,"(Sqrt[a*e + c*d*x]*(-((Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)*(3*c^2*d^3*x^2*(11*d - 8*e*x) + 2*a*c*d^2*e*x*(13*d + 34*e*x) + a^2*e^2*(8*d^2 + 14*d*e*x + 3*e^2*x^2)))/x^3) + (24*(c*d)^(5/2)*e*Sqrt[c*d^2 - a*e^2]*(3*c*d^2 + 5*a*e^2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/c^(3/2) - (3*(5*c^3*d^6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[a]))/(24*d^(3/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
465,1,404,404,3.4823621,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^5 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^5*(d + e*x)),x]","\frac{\sqrt{a e+c d x} \left(-\frac{\sqrt{d} \sqrt{e} (d+e x) \sqrt{a e+c d x} \left(3 a^3 e^3 \left(16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right)+a^2 c d^2 e^2 x \left(136 d^2+244 d e x+57 e^2 x^2\right)+a c^2 d^4 e x^2 (118 d+337 e x)+15 c^3 d^6 x^3\right)}{a x^4}+\frac{3 \sqrt{d+e x} \left(-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{a^{3/2}}+384 c^{3/2} d^4 e^3 \sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)}{192 d^{5/2} e^{3/2} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\left(x \left(-3 a^3 e^6+11 a^2 c d^2 e^4+83 a c^2 d^4 e^2+5 c^3 d^6\right)+2 a d e \left(5 c d^2-a e^2\right) \left(3 a e^2+c d^2\right)\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 a d^2 e x^2}+\frac{\left(-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)-\frac{\left(x \left(3 a e^2+11 c d^2\right)+6 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 d x^4}",1,"(Sqrt[a*e + c*d*x]*(-((Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)*(15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(118*d + 337*e*x) + a^2*c*d^2*e^2*x*(136*d^2 + 244*d*e*x + 57*e^2*x^2) + 3*a^3*e^3*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 - 3*e^3*x^3)))/(a*x^4)) + 384*c^(3/2)*d^4*Sqrt[c*d]*e^3*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])] + (3*(5*c^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/a^(3/2)))/(192*d^(5/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
466,1,295,289,0.9388191,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^6 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x]","\frac{((d+e x) (a e+c d x))^{3/2} \left(\frac{5 \left(c d^2-a e^2\right) \left(\frac{x \left(c d^2-a e^2\right) \left(\frac{x \left(a e^2-c d^2\right) \left(3 x^2 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a e (2 d+5 e x)-3 c d^2 x\right)\right)}{a^{5/2} \sqrt{d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt{a e+c d x}\right)}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right)}{64 d x^4 (d+e x)^{3/2} (a e+c d x)^{3/2}}-\frac{2 (d+e x) (a e+c d x)}{x^5}\right)}{10 d}","-\frac{3 \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{256 a^{5/2} d^{7/2} e^{5/2}}+\frac{3 \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 d x^5}-\frac{\left(\frac{c}{a e}-\frac{e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{16 x^4}",1,"(((a*e + c*d*x)*(d + e*x))^(3/2)*((-2*(a*e + c*d*x)*(d + e*x))/x^5 + (5*(c*d^2 - a*e^2)*(-16*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d))/(64*d*x^4*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(10*d)","A",1
467,1,344,386,0.994476,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^7 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]","\frac{((d+e x) (a e+c d x))^{3/2} \left(-\frac{\left(7 a e^2+5 c d^2\right) \left(5 x \left(c d^2-a e^2\right) \left(\frac{x \left(c d^2-a e^2\right) \left(\frac{x \left(a e^2-c d^2\right) \left(3 x^2 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a e (2 d+5 e x)-3 c d^2 x\right)\right)}{a^{5/2} \sqrt{d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt{a e+c d x}\right)}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right)-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right)}{1280 d^2 x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}-\frac{(d+e x) (a e+c d x)^2}{x^6}\right)}{6 a d e}","\frac{\left(7 a e^2+5 c d^2\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left(7 a e^2+5 c d^2\right) \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac{\left(7 a e^2+5 c d^2\right) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{6 d x^6}-\frac{\left(\frac{5 c}{a e}-\frac{7 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{60 x^5}",1,"(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^6) - ((5*c*d^2 + 7*a*e^2)*(-128*d*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*x*(-16*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d)))/(1280*d^2*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(6*a*d*e)","A",1
468,1,408,500,0.7466458,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^8 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]","\frac{((d+e x) (a e+c d x))^{3/2} \left(\frac{7 \left(9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \left(5 x \left(c d^2-a e^2\right) \left(\frac{x \left(c d^2-a e^2\right) \left(\frac{x \left(a e^2-c d^2\right) \left(3 x^2 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a e (2 d+5 e x)-3 c d^2 x\right)\right)}{a^{5/2} \sqrt{d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt{a e+c d x}\right)}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right)-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right)}{15360 a d^3 e x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac{(d+e x) \left(9 a e^2+7 c d^2\right) (a e+c d x)^2}{12 a d e x^6}-\frac{(d+e x) (a e+c d x)^2}{x^7}\right)}{7 a d e}","\frac{\left(-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac{\left(9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left(9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac{\left(9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 d x^7}-\frac{\left(\frac{5 c}{a e}-\frac{9 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{84 x^6}",1,"(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^7) + ((7*c*d^2 + 9*a*e^2)*(a*e + c*d*x)^2*(d + e*x))/(12*a*d*e*x^6) + (7*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(-128*d*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*x*(-16*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d)))/(15360*a*d^3*e*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7*a*d*e)","A",1
469,1,512,628,1.4556149,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{x^9 (d+e x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^9*(d + e*x)),x]","\frac{((d+e x) (a e+c d x))^{3/2} \left(-\frac{(d+e x) \left(33 a^2 e^4+34 a c d^2 e^2+21 c^2 d^4\right) (a e+c d x)^2}{56 a^2 d^2 e^2 x^6}+\frac{\left(33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right) \left(128 a^{5/2} d^{5/2} e^{5/2} (d+e x)^{5/2} (a e+c d x)^{5/2}+5 x \left(c d^2-a e^2\right) \left(16 a^{5/2} d^{3/2} e^{5/2} (d+e x)^{5/2} (a e+c d x)^{3/2}+x \left(c d^2-a e^2\right) \left(8 a^{5/2} \sqrt{d} e^{5/2} (d+e x)^{5/2} \sqrt{a e+c d x}+x \left(c d^2-a e^2\right) \left(3 x^2 \left(c d^2-a e^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left(a e (2 d+5 e x)-3 c d^2 x\right)\right)\right)\right)\right)}{10240 a^{9/2} d^{11/2} e^{9/2} x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac{(d+e x) \left(11 a e^2+9 c d^2\right) (a e+c d x)^2}{14 a d e x^7}-\frac{(d+e x) (a e+c d x)^2}{x^8}\right)}{8 a d e}","\frac{\left(-33 a^2 e^4+10 a c d^2 e^2+15 c^2 d^4\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{448 a^2 d^3 e^2 x^6}-\frac{\left(-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{3 \left(33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right) \left(c d^2-a e^2\right)^5 \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{32768 a^{11/2} d^{13/2} e^{11/2}}-\frac{3 \left(33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right) \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+2 a d e\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16384 a^5 d^6 e^5 x^2}+\frac{\left(33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\right) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+2 a d e\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2048 a^4 d^5 e^4 x^4}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{8 d x^8}-\frac{\left(\frac{5 c}{a e}-\frac{11 e}{d^2}\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{112 x^7}",1,"(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^8) + ((9*c*d^2 + 11*a*e^2)*(a*e + c*d*x)^2*(d + e*x))/(14*a*d*e*x^7) - ((21*c^2*d^4 + 34*a*c*d^2*e^2 + 33*a^2*e^4)*(a*e + c*d*x)^2*(d + e*x))/(56*a^2*d^2*e^2*x^6) + ((15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*(128*a^(5/2)*d^(5/2)*e^(5/2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*x*(16*a^(5/2)*d^(3/2)*e^(5/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + (c*d^2 - a*e^2)*x*(8*a^(5/2)*Sqrt[d]*e^(5/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + (c*d^2 - a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])))))/(10240*a^(9/2)*d^(11/2)*e^(9/2)*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(8*a*d*e)","A",1
470,1,331,271,0.5143775,"\int \frac{x^3}{(d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[x^3/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{3 \sqrt{c d} \sqrt{c d^2-a e^2} \left(-a^3 e^6-a^2 c d^2 e^4-3 a c^2 d^4 e^2+5 c^3 d^6\right) \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)+c^{3/2} d^{3/2} \sqrt{e} \left(3 a^3 e^5 (d+e x)+a^2 c d e^3 \left(4 d^2+5 d e x+e^2 x^2\right)-a c^2 d^2 e \left(15 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right)+c^3 d^4 x \left(-15 d^2-5 d e x+2 e^2 x^2\right)\right)}{4 c^{7/2} d^{7/2} e^{7/2} \left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{3 \left(a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{5/2} d^{5/2} e^{7/2}}-\frac{3 \left(a e^2+3 c d^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 c^2 d^2 e^3}+\frac{(d+e x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 c d e^3}-\frac{2 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^3 (d+e x) \left(c d^2-a e^2\right)}",1,"(c^(3/2)*d^(3/2)*Sqrt[e]*(3*a^3*e^5*(d + e*x) + a^2*c*d*e^3*(4*d^2 + 5*d*e*x + e^2*x^2) + c^3*d^4*x*(-15*d^2 - 5*d*e*x + 2*e^2*x^2) - a*c^2*d^2*e*(15*d^3 + d^2*e*x - 4*d*e^2*x^2 + 2*e^3*x^3)) + 3*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(4*c^(7/2)*d^(7/2)*e^(7/2)*(c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
471,1,255,195,0.364098,"\int \frac{x^2}{(d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[x^2/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{c^{3/2} d^{3/2} \sqrt{e} \left(-a^2 e^3 (d+e x)+a c d e \left(3 d^2-e^2 x^2\right)+c^2 d^3 x (3 d+e x)\right)-\sqrt{c d} \sqrt{c d^2-a e^2} \left(-a^2 e^4-2 a c d^2 e^2+3 c^2 d^4\right) \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)}{c^{5/2} d^{5/2} e^{5/2} \left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","-\frac{\left(a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 c^{3/2} d^{3/2} e^{5/2}}+\frac{2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e^2 (d+e x) \left(c d^2-a e^2\right)}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d e^2}",1,"(c^(3/2)*d^(3/2)*Sqrt[e]*(-(a^2*e^3*(d + e*x)) + c^2*d^3*x*(3*d + e*x) + a*c*d*e*(3*d^2 - e^2*x^2)) - Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(3*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(c^(5/2)*d^(5/2)*e^(5/2)*(c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
472,1,189,139,0.4163827,"\int \frac{x}{(d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[x/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{c d} \left(c d^2-a e^2\right)^{3/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)-2 c^{3/2} d^{5/2} \sqrt{e} (a e+c d x)}{c^{3/2} d^{3/2} e^{3/2} \left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{\tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{\sqrt{c} \sqrt{d} e^{3/2}}-\frac{2 d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{e (d+e x) \left(c d^2-a e^2\right)}",1,"(-2*c^(3/2)*d^(5/2)*Sqrt[e]*(a*e + c*d*x) + 2*Sqrt[c*d]*(c*d^2 - a*e^2)^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(c^(3/2)*d^(3/2)*e^(3/2)*(c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
473,1,42,52,0.0155299,"\int \frac{1}{(d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 (a e+c d x)}{\left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{(d+e x) \left(c d^2-a e^2\right)}",1,"(2*(a*e + c*d*x))/((c*d^2 - a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
474,1,131,143,0.1260254,"\int \frac{1}{x (d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/(x*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \left(-\frac{\sqrt{d} e^{3/2} (a e+c d x)}{c d^2-a e^2}-\frac{\sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{\sqrt{a}}\right)}{d^{3/2} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 e (a e+c d x)}{d \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{\sqrt{a} d^{3/2} \sqrt{e}}",1,"(2*(-((Sqrt[d]*e^(3/2)*(a*e + c*d*x))/(c*d^2 - a*e^2)) - (Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[a]))/(d^(3/2)*Sqrt[e]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
475,1,201,229,0.1234043,"\int \frac{1}{x^2 (d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/(x^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{x \sqrt{d+e x} \left(-3 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right) \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \left(a^2 e^3 (d+3 e x)-a c d e \left(d^2-3 e^2 x^2\right)-c^2 d^3 x (d+e x)\right)}{a^{3/2} d^{5/2} e^{3/2} x \left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(3 a e^2+c d^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left(c d^2-3 a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{a d^2 e x \left(c d^2-a e^2\right)}-\frac{2 e (a e+c d x)}{d x \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(Sqrt[a]*Sqrt[d]*Sqrt[e]*(-(c^2*d^3*x*(d + e*x)) + a^2*e^3*(d + 3*e*x) - a*c*d*e*(d^2 - 3*e^2*x^2)) + (c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*x*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(a^(3/2)*d^(5/2)*e^(3/2)*(c*d^2 - a*e^2)*x*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
476,1,283,329,0.1886187,"\int \frac{1}{x^3 (d+e x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[1/(x^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left(a^3 e^4 \left(2 d^2-5 d e x-15 e^2 x^2\right)-a^2 c d e^2 \left(2 d^3-4 d^2 e x+d e^2 x^2+15 e^3 x^3\right)+a c^2 d^3 e x \left(d^2+5 d e x+4 e^2 x^2\right)+3 c^3 d^5 x^2 (d+e x)\right)-3 x^2 \sqrt{d+e x} \left(-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right) \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)}{4 a^{5/2} d^{7/2} e^{5/2} x^2 \left(c d^2-a e^2\right) \sqrt{(d+e x) (a e+c d x)}}","\frac{\left(3 c d^2-5 a e^2\right) \left(3 a e^2+c d^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 a^2 d^3 e^2 x \left(c d^2-a e^2\right)}-\frac{3 \left(5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left(c d^2-5 a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 a d^2 e x^2 \left(c d^2-a e^2\right)}-\frac{2 e (a e+c d x)}{d x^2 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^3*d^5*x^2*(d + e*x) + a^3*e^4*(2*d^2 - 5*d*e*x - 15*e^2*x^2) + a*c^2*d^3*e*x*(d^2 + 5*d*e*x + 4*e^2*x^2) - a^2*c*d*e^2*(2*d^3 - 4*d^2*e*x + d*e^2*x^2 + 15*e^3*x^3)) - 3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*x^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(4*a^(5/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)*x^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
477,1,296,515,5.6318155,"\int \frac{x^5}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{\frac{2 (d+e x)^2 (a e+c d x)^2 \left(\frac{24 a^5 e^9}{c^3 \left(c d^2-a e^2\right)^3 (a e+c d x)}-\frac{3 \left(7 a e^2+11 c d^2\right)}{c^3}+\frac{8 d^8}{(d+e x)^2 \left(c d^2-a e^2\right)^2}+\frac{40 \left(3 a d^7 e^2-2 c d^9\right)}{(d+e x) \left(c d^2-a e^2\right)^3}+\frac{6 d e x}{c^2}\right)}{3 d^3 e^4}+\frac{5 (d+e x)^{3/2} \left(3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) (a e+c d x)^{3/2} \log \left(2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right)}{c^{7/2} d^{7/2} e^{9/2}}}{8 ((d+e x) (a e+c d x))^{3/2}}","\frac{5 \left(3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac{2 x^2 \left(a d e \left(c d^2-a e^2\right) \left(-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right)+x \left(c d^2-a e^2\right) \left(-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right)\right)}{3 c d e^2 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\left(-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left(-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right)-190 a c^3 d^6 e^2+105 c^4 d^8\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left(c d^2-a e^2\right)^3}-\frac{2 d x^4 \left(c d x \left(c d^2-a e^2\right)+a e \left(c d^2-a e^2\right)\right)}{3 e \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"((2*(a*e + c*d*x)^2*(d + e*x)^2*((-3*(11*c*d^2 + 7*a*e^2))/c^3 + (6*d*e*x)/c^2 + (24*a^5*e^9)/(c^3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)) + (8*d^8)/((c*d^2 - a*e^2)^2*(d + e*x)^2) + (40*(-2*c*d^9 + 3*a*d^7*e^2))/((c*d^2 - a*e^2)^3*(d + e*x))))/(3*d^3*e^4) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(7/2)*d^(7/2)*e^(9/2)))/(8*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
478,1,387,438,1.3415281,"\int \frac{x^4}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[x^4/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{(a e+c d x) \left(-\frac{a e \left(3 a e^2-c d^2\right) \left(a^2 e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)+2 a c d^2 e x (2 d+3 e x)-c^2 d^4 x^2\right)}{c d \left(c d^2-a e^2\right)^3}-\frac{\left(3 a e^2+5 c d^2\right) \sqrt{a e+c d x} \left(c^{3/2} d^{7/2} \sqrt{e} \left(c d^2-a e^2\right) \sqrt{a e+c d x}-(d+e x) \left(2 c^{3/2} d^{5/2} \sqrt{e} \left(2 c d^2-3 a e^2\right) \sqrt{a e+c d x}-3 \sqrt{c d} \left(c d^2-a e^2\right)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right)\right)\right)}{c^{5/2} d^{5/2} e^{5/2} \left(c d^2-a e^2\right)^2}+3 x^3\right)}{3 c d e ((d+e x) (a e+c d x))^{3/2}}","-\frac{2 x \left(a d e \left(c d^2-a e^2\right) \left(-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right)+x \left(c d^2-a e^2\right) \left(-3 a^3 e^6-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right)\right)}{3 c d e^2 \left(c d^2-a e^2\right)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{\left(-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^2 d^2 e^3 \left(c d^2-a e^2\right)^3}-\frac{\left(3 a e^2+5 c d^2\right) \tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac{2 d x^3 \left(c d x \left(c d^2-a e^2\right)+a e \left(c d^2-a e^2\right)\right)}{3 e \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"((a*e + c*d*x)*(3*x^3 - (a*e*(-(c*d^2) + 3*a*e^2)*(-(c^2*d^4*x^2) + 2*a*c*d^2*e*x*(2*d + 3*e*x) + a^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2)))/(c*d*(c*d^2 - a*e^2)^3) - ((5*c*d^2 + 3*a*e^2)*Sqrt[a*e + c*d*x]*(c^(3/2)*d^(7/2)*Sqrt[e]*(c*d^2 - a*e^2)*Sqrt[a*e + c*d*x] - (d + e*x)*(2*c^(3/2)*d^(5/2)*Sqrt[e]*(2*c*d^2 - 3*a*e^2)*Sqrt[a*e + c*d*x] - 3*Sqrt[c*d]*(c*d^2 - a*e^2)^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])))/(c^(5/2)*d^(5/2)*e^(5/2)*(c*d^2 - a*e^2)^2)))/(3*c*d*e*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
479,1,1443,297,4.6396379,"\int \frac{x^3}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[x^3/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{a^3 e^3 (a e+c d x)^2 \left(\frac{c d (d+e x)}{c d^2-a e^2}\right)^{5/2} \left(\frac{56 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^5}{a^3 e \left(c d^2-a e^2\right)^2}-\frac{280 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^4}{a^2 \left(c d^2-a e^2\right)^2}+\frac{96 \, _4F_3\left(\frac{1}{2},2,2,\frac{7}{2};1,1,\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^4}{a^3 e^2 \left(a e^2-c d^2\right)}+\frac{294 \sin ^{-1}\left(\sqrt{\frac{e (a e+c d x)}{a e^2-c d^2}}\right) (a e+c d x)^3}{a^3 e^3 \left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{5/2}}+\frac{392 e \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^3}{a \left(c d^2-a e^2\right)^2}-\frac{288 \, _4F_3\left(\frac{1}{2},2,2,\frac{7}{2};1,1,\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^3}{a^2 \left(a e^3-c d^2 e\right)}-\frac{1575 \sin ^{-1}\left(\sqrt{\frac{e (a e+c d x)}{a e^2-c d^2}}\right) (a e+c d x)^2}{a^2 e^2 \left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{5/2}}-\frac{168 e^2 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^2}{\left(c d^2-a e^2\right)^2}+\frac{288 \, _4F_3\left(\frac{1}{2},2,2,\frac{7}{2};1,1,\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)^2}{a \left(a e^2-c d^2\right)}+\frac{196 c d (d+e x) (a e+c d x)^2}{a^3 e^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\frac{3780 \sin ^{-1}\left(\sqrt{\frac{e (a e+c d x)}{a e^2-c d^2}}\right) (a e+c d x)}{a e \left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{5/2}}-\frac{96 e \, _4F_3\left(\frac{1}{2},2,2,\frac{7}{2};1,1,\frac{9}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right) (a e+c d x)}{a e^2-c d^2}-\frac{294 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}{a^3 e^5}-\frac{1050 c d (d+e x) (a e+c d x)}{a^2 e^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{1995 \sin ^{-1}\left(\sqrt{\frac{e (a e+c d x)}{a e^2-c d^2}}\right)}{\left(\frac{e (a e+c d x)}{a e^2-c d^2}\right)^{5/2}}-56 \left(\frac{c d x}{a e}+1\right)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+\frac{1575 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}{a^2 e^4}+336 \left(\frac{c d x}{a e}+1\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}-504 \left(\frac{c d x}{a e}+1\right) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+1568 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+\frac{2520 c d (d+e x)}{a e^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{3780 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}{a e^3 (a e+c d x)}-\frac{1330 c d (d+e x)}{e \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}+\frac{1995 \left(c d^2-a e^2\right)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}{e^2 (a e+c d x)^2}\right)}{252 c^4 d^4 ((a e+c d x) (d+e x))^{5/2}}","-\frac{2 \left(x \left(-3 a^3 e^6-a^2 c d^2 e^4-7 a c^2 d^4 e^2+3 c^3 d^6\right)+a d e \left(c d^2-3 a e^2\right) \left(a e^2+3 c d^2\right)\right)}{3 c d e^2 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{\tanh ^{-1}\left(\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{c^{3/2} d^{3/2} e^{5/2}}-\frac{2 d x^2 \left(c d x \left(c d^2-a e^2\right)+a e \left(c d^2-a e^2\right)\right)}{3 e \left(c d^2-a e^2\right)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(a^3*e^3*(a*e + c*d*x)^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(5/2)*((2520*c*d*(d + e*x))/(a*e^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (1330*c*d*(d + e*x))/(e*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (1050*c*d*(a*e + c*d*x)*(d + e*x))/(a^2*e^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) + (196*c*d*(a*e + c*d*x)^2*(d + e*x))/(a^3*e^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) + 1568*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + (1575*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a^2*e^4) + (1995*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e^2*(a*e + c*d*x)^2) - (3780*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a*e^3*(a*e + c*d*x)) - (294*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a^3*e^5) - 504*(1 + (c*d*x)/(a*e))*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 336*(1 + (c*d*x)/(a*e))^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 56*(1 + (c*d*x)/(a*e))^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - (1995*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2) + (3780*(a*e + c*d*x)*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a*e*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) - (1575*(a*e + c*d*x)^2*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a^2*e^2*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) + (294*(a*e + c*d*x)^3*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a^3*e^3*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) - (168*e^2*(a*e + c*d*x)^2*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d^2 - a*e^2)^2 + (392*e*(a*e + c*d*x)^3*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a*(c*d^2 - a*e^2)^2) - (280*(a*e + c*d*x)^4*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^2*(c*d^2 - a*e^2)^2) + (56*(a*e + c*d*x)^5*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^3*e*(c*d^2 - a*e^2)^2) - (96*e*(a*e + c*d*x)*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(-(c*d^2) + a*e^2) + (288*(a*e + c*d*x)^2*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a*(-(c*d^2) + a*e^2)) - (288*(a*e + c*d*x)^3*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^2*(-(c*d^2*e) + a*e^3)) + (96*(a*e + c*d*x)^4*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^3*e^2*(-(c*d^2) + a*e^2))))/(252*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2))","C",0
480,1,99,126,0.0653112,"\int \frac{x^2}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{-2 a^2 e^2 \left(8 d^2+12 d e x+3 e^2 x^2\right)-4 a c d^2 e x (2 d+3 e x)+2 c^2 d^4 x^2}{3 (d+e x) \left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)}}","\frac{2 x^2}{3 (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{8 a e \left(x \left(a e^2+c d^2\right)+2 a d e\right)}{3 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*c^2*d^4*x^2 - 4*a*c*d^2*e*x*(2*d + 3*e*x) - 2*a^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2))/(3*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
481,1,100,138,0.0351381,"\int \frac{x}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[x/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{2 \left(a^2 e^3 (2 d+3 e x)+2 a c d e \left(3 d^2+5 d e x+3 e^2 x^2\right)+c^2 d^3 x (3 d+2 e x)\right)}{3 (d+e x) \left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \left(3 a e^2+c d^2\right) \left(a e^2+c d^2+2 c d e x\right)}{3 e \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 d}{3 e (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*(c^2*d^3*x*(3*d + 2*e*x) + a^2*e^3*(2*d + 3*e*x) + 2*a*c*d*e*(3*d^2 + 5*d*e*x + 3*e^2*x^2)))/(3*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
482,1,95,121,0.0323119,"\int \frac{1}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{2 a^2 e^4-4 a c d e^2 (3 d+2 e x)-2 c^2 d^2 \left(3 d^2+12 d e x+8 e^2 x^2\right)}{3 (d+e x) \left(c d^2-a e^2\right)^3 \sqrt{(d+e x) (a e+c d x)}}","\frac{2}{3 (d+e x) \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{8 c d \left(a e^2+c d^2+2 c d e x\right)}{3 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*a^2*e^4 - 4*a*c*d*e^2*(3*d + 2*e*x) - 2*c^2*d^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2))/(3*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
483,1,262,271,0.414857,"\int \frac{1}{x (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/(x*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{2 \left(-\frac{(d+e x) (a e+c d x)^{3/2} \left(\sqrt{a} \sqrt{d} \sqrt{e} \left(3 a^2 e^5-8 a c d^2 e^3-3 c^2 d^4 e\right) \sqrt{a e+c d x}+3 \sqrt{d+e x} \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)\right)}{3 \sqrt{a} d^{5/2} \sqrt{e} \left(c d^2-a e^2\right)^2}+\frac{\left(a e^3+3 c d^2 e\right) (a e+c d x)^2}{3 c d^3-3 a d e^2}+c d (a e+c d x)\right)}{a e \left(c d^2-a e^2\right) ((d+e x) (a e+c d x))^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{a^{3/2} d^{5/2} e^{3/2}}+\frac{2 \left(-3 a^3 e^6+7 a^2 c d^2 e^4+a c^2 d^4 e^2+c d e x \left(3 c d^2-a e^2\right) \left(3 a e^2+c d^2\right)+3 c^3 d^6\right)}{3 a d^2 e \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 e (a e+c d x)}{3 d \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(c*d*(a*e + c*d*x) + ((3*c*d^2*e + a*e^3)*(a*e + c*d*x)^2)/(3*c*d^3 - 3*a*d*e^2) - ((a*e + c*d*x)^(3/2)*(d + e*x)*(Sqrt[a]*Sqrt[d]*Sqrt[e]*(-3*c^2*d^4*e - 8*a*c*d^2*e^3 + 3*a^2*e^5)*Sqrt[a*e + c*d*x] + 3*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(3*Sqrt[a]*d^(5/2)*Sqrt[e]*(c*d^2 - a*e^2)^2)))/(a*e*(c*d^2 - a*e^2)*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
484,1,370,394,0.569016,"\int \frac{1}{x^2 (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{(a e+c d x) \left(3 a^{3/2} d^{5/2} e^{3/2} \left(a e^2-c d^2\right)^3+\sqrt{a} d^{3/2} \sqrt{e} x \left(a e^2-c d^2\right) \left(5 a^2 e^5-6 a c d^2 e^3+9 c^2 d^4 e\right) (a e+c d x)+x (d+e x) \sqrt{a e+c d x} \left(\sqrt{a} \sqrt{d} \sqrt{e} \left(15 a^3 e^7-31 a^2 c d^2 e^5+9 a c^2 d^4 e^3-9 c^3 d^6 e\right) \sqrt{a e+c d x}+3 \sqrt{d+e x} \left(5 a e^2+3 c d^2\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)\right)+3 \sqrt{a} c d^{7/2} \sqrt{e} x \left(c d^2-a e^2\right)^2 \left(a e^2-3 c d^2\right)\right)}{3 a^{5/2} d^{7/2} e^{5/2} x \left(c d^2-a e^2\right)^3 ((d+e x) (a e+c d x))^{3/2}}","\frac{\left(5 a e^2+3 c d^2\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{2 a^{5/2} d^{7/2} e^{5/2}}+\frac{2 \left(-5 a^3 e^6+c d e x \left(-5 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right)+9 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right)}{3 a d^2 e x \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\left(-15 a^3 e^6+31 a^2 c d^2 e^4-9 a c^2 d^4 e^2+9 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 a^2 d^3 e^2 x \left(c d^2-a e^2\right)^3}-\frac{2 e (a e+c d x)}{3 d x \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"((a*e + c*d*x)*(3*a^(3/2)*d^(5/2)*e^(3/2)*(-(c*d^2) + a*e^2)^3 + 3*Sqrt[a]*c*d^(7/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*(-3*c*d^2 + a*e^2)*x + Sqrt[a]*d^(3/2)*Sqrt[e]*(-(c*d^2) + a*e^2)*(9*c^2*d^4*e - 6*a*c*d^2*e^3 + 5*a^2*e^5)*x*(a*e + c*d*x) + x*Sqrt[a*e + c*d*x]*(d + e*x)*(Sqrt[a]*Sqrt[d]*Sqrt[e]*(-9*c^3*d^6*e + 9*a*c^2*d^4*e^3 - 31*a^2*c*d^2*e^5 + 15*a^3*e^7)*Sqrt[a*e + c*d*x] + 3*(c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])))/(3*a^(5/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)^3*x*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
485,1,467,522,0.9345064,"\int \frac{1}{x^3 (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{(a e+c d x) \left(6 a^{5/2} d^{7/2} e^{5/2} \left(a e^2-c d^2\right)^3+x \left(3 a^{3/2} d^{5/2} e^{3/2} \left(7 a e^2+5 c d^2\right) \left(c d^2-a e^2\right)^3-3 \sqrt{a} d^{5/2} \sqrt{e} x \left(7 a^2 c d e^4-15 c^3 d^5\right) \left(c d^2-a e^2\right)^2-\sqrt{a} d^{3/2} \sqrt{e} x \left(a e^2-c d^2\right) \left(35 a^3 e^7-33 a^2 c d^2 e^5-15 a c^2 d^4 e^3+45 c^3 d^6 e\right) (a e+c d x)-x (d+e x) \sqrt{a e+c d x} \left(15 \sqrt{d+e x} \left(7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+\sqrt{a} \sqrt{d} \sqrt{e} \left(105 a^4 e^9-190 a^3 c d^2 e^7+36 a^2 c^2 d^4 e^5+30 a c^3 d^6 e^3-45 c^4 d^8 e\right) \sqrt{a e+c d x}\right)\right)\right)}{12 a^{7/2} d^{9/2} e^{7/2} x^2 \left(c d^2-a e^2\right)^3 ((d+e x) (a e+c d x))^{3/2}}","-\frac{5 \left(7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{8 a^{7/2} d^{9/2} e^{7/2}}+\frac{2 \left(-7 a^3 e^6+c d e x \left(-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right)+11 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right)}{3 a d^2 e x^2 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\left(-35 a^3 e^6+61 a^2 c d^2 e^4-9 a c^2 d^4 e^2+15 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{6 a^2 d^3 e^2 x^2 \left(c d^2-a e^2\right)^3}+\frac{\left(-105 a^4 e^8+190 a^3 c d^2 e^6-36 a^2 c^2 d^4 e^4-30 a c^3 d^6 e^2+45 c^4 d^8\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 a^3 d^4 e^3 x \left(c d^2-a e^2\right)^3}-\frac{2 e (a e+c d x)}{3 d x^2 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"((a*e + c*d*x)*(6*a^(5/2)*d^(7/2)*e^(5/2)*(-(c*d^2) + a*e^2)^3 + x*(3*a^(3/2)*d^(5/2)*e^(3/2)*(c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2) - 3*Sqrt[a]*d^(5/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*(-15*c^3*d^5 + 7*a^2*c*d*e^4)*x - Sqrt[a]*d^(3/2)*Sqrt[e]*(-(c*d^2) + a*e^2)*(45*c^3*d^6*e - 15*a*c^2*d^4*e^3 - 33*a^2*c*d^2*e^5 + 35*a^3*e^7)*x*(a*e + c*d*x) - x*Sqrt[a*e + c*d*x]*(d + e*x)*(Sqrt[a]*Sqrt[d]*Sqrt[e]*(-45*c^4*d^8*e + 30*a*c^3*d^6*e^3 + 36*a^2*c^2*d^4*e^5 - 190*a^3*c*d^2*e^7 + 105*a^4*e^9)*Sqrt[a*e + c*d*x] + 15*(c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))))/(12*a^(7/2)*d^(9/2)*e^(7/2)*(c*d^2 - a*e^2)^3*x^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
486,1,593,664,1.3949345,"\int \frac{1}{x^4 (d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[1/(x^4*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","\frac{(a e+c d x) \left(24 a^{7/2} d^{9/2} e^{7/2} \left(a e^2-c d^2\right)^3+x \left(6 a^{5/2} d^{7/2} e^{5/2} \left(c d^2-a e^2\right)^3 \left(9 a e^2+7 c d^2\right)+3 a^{3/2} d^{5/2} e^{3/2} x \left(a e^2-c d^2\right)^3 \left(63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right)+x^2 \left(9 \sqrt{a} c d^{7/2} \sqrt{e} \left(c d^2-a e^2\right)^2 \left(21 a^3 e^6+3 a^2 c d^2 e^4-5 a c^2 d^4 e^2-35 c^3 d^6\right)+3 \sqrt{a} d^{3/2} \sqrt{e} \left(a e^2-c d^2\right) \left(105 a^4 e^9-84 a^3 c d^2 e^7-42 a^2 c^2 d^4 e^5-20 a c^3 d^6 e^3+105 c^4 d^8 e\right) (a e+c d x)+(d+e x) \sqrt{a e+c d x} \left(45 \sqrt{d+e x} \left(21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right) \left(c d^2-a e^2\right)^3 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right)+3 \sqrt{a} \sqrt{d} \sqrt{e} \left(315 a^5 e^{11}-525 a^4 c d^2 e^9+78 a^3 c^2 d^4 e^7+54 a^2 c^3 d^6 e^5+55 a c^4 d^8 e^3-105 c^5 d^{10} e\right) \sqrt{a e+c d x}\right)\right)\right)\right)}{72 a^{9/2} d^{11/2} e^{9/2} x^3 \left(c d^2-a e^2\right)^3 ((d+e x) (a e+c d x))^{3/2}}","\frac{2 \left(-9 a^3 e^6+c d e x \left(-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right)+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right)}{3 a d^2 e x^3 \left(c d^2-a e^2\right)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\left(-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left(c d^2-a e^2\right)^3}+\frac{5 \left(21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right) \tanh ^{-1}\left(\frac{x \left(a e^2+c d^2\right)+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}\right)}{16 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left(-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left(c d^2-a e^2\right)^3}-\frac{\left(-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left(c d^2-a e^2\right)^3}-\frac{2 e (a e+c d x)}{3 d x^3 \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"((a*e + c*d*x)*(24*a^(7/2)*d^(9/2)*e^(7/2)*(-(c*d^2) + a*e^2)^3 + x*(6*a^(5/2)*d^(7/2)*e^(5/2)*(c*d^2 - a*e^2)^3*(7*c*d^2 + 9*a*e^2) + 3*a^(3/2)*d^(5/2)*e^(3/2)*(-(c*d^2) + a*e^2)^3*(35*c^2*d^4 + 54*a*c*d^2*e^2 + 63*a^2*e^4)*x + x^2*(9*Sqrt[a]*c*d^(7/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*(-35*c^3*d^6 - 5*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 21*a^3*e^6) + 3*Sqrt[a]*d^(3/2)*Sqrt[e]*(-(c*d^2) + a*e^2)*(105*c^4*d^8*e - 20*a*c^3*d^6*e^3 - 42*a^2*c^2*d^4*e^5 - 84*a^3*c*d^2*e^7 + 105*a^4*e^9)*(a*e + c*d*x) + Sqrt[a*e + c*d*x]*(d + e*x)*(3*Sqrt[a]*Sqrt[d]*Sqrt[e]*(-105*c^5*d^10*e + 55*a*c^4*d^8*e^3 + 54*a^2*c^3*d^6*e^5 + 78*a^3*c^2*d^4*e^7 - 525*a^4*c*d^2*e^9 + 315*a^5*e^11)*Sqrt[a*e + c*d*x] + 45*(c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])))))/(72*a^(9/2)*d^(11/2)*e^(9/2)*(c*d^2 - a*e^2)^3*x^3*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
487,1,235,259,0.1233968,"\int \frac{x^2}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 \left(a^4 e^6 \left(8 d^2+20 d e x+15 e^2 x^2\right)+4 a^3 c d e^4 \left(20 d^3+53 d^2 e x+45 d e^2 x^2+15 e^3 x^3\right)+2 a^2 c^2 d^2 e^2 \left(20 d^4+110 d^3 e x+189 d^2 e^2 x^2+110 d e^3 x^3+20 e^4 x^4\right)+4 a c^3 d^4 e x \left(15 d^3+45 d^2 e x+53 d e^2 x^2+20 e^3 x^3\right)+c^4 d^6 x^2 \left(15 d^2+20 d e x+8 e^2 x^2\right)\right)}{15 (d+e x) \left(c d^2-a e^2\right)^5 ((d+e x) (a e+c d x))^{3/2}}","\frac{8 \left(5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right) \left(a e^2+c d^2+2 c d e x\right)}{15 e \left(c d^2-a e^2\right)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{8 \left(x \left(-2 a^3 e^6+a^2 c d^2 e^4+c^3 d^6\right)+a d e \left(c d^2-a e^2\right) \left(3 a e^2+c d^2\right)\right)}{15 e \left(c d^2-a e^2\right)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}+\frac{2 x^2}{5 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(c^4*d^6*x^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + a^4*e^6*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 4*a^3*c*d*e^4*(20*d^3 + 53*d^2*e*x + 45*d*e^2*x^2 + 15*e^3*x^3) + 4*a*c^3*d^4*e*x*(15*d^3 + 45*d^2*e*x + 53*d*e^2*x^2 + 20*e^3*x^3) + 2*a^2*c^2*d^2*e^2*(20*d^4 + 110*d^3*e*x + 189*d^2*e^2*x^2 + 110*d*e^3*x^3 + 20*e^4*x^4)))/(15*(c*d^2 - a*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
488,1,433,341,0.1949097,"\int \frac{x^2}{(d+e x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{7/2}} \, dx","Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)),x]","-\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-a^6 e^{10} \left(8 d^2+28 d e x+35 e^2 x^2\right)+2 a^5 c d e^8 \left(112 d^3+382 d^2 e x+455 d e^2 x^2+140 e^3 x^3\right)+5 a^4 c^2 d^2 e^6 \left(336 d^4+1288 d^3 e x+1859 d^2 e^2 x^2+1288 d e^3 x^3+336 e^4 x^4\right)+20 a^3 c^3 d^3 e^4 \left(56 d^5+406 d^4 e x+1001 d^3 e^2 x^2+1084 d^2 e^3 x^3+560 d e^4 x^4+112 e^5 x^5\right)+a^2 c^4 d^4 e^2 \left(56 d^6+2996 d^5 e x+13195 d^4 e^2 x^2+24080 d^3 e^3 x^3+20320 d^2 e^4 x^4+7616 d e^5 x^5+896 e^6 x^6\right)+2 a c^5 d^6 e x \left(70 d^5+1295 d^4 e x+4060 d^3 e^2 x^2+5600 d^2 e^3 x^3+3616 d e^4 x^4+896 e^5 x^5\right)+3 c^6 d^8 x^2 \left(35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right)\right)}{105 (d+e x)^4 \left(c d^2-a e^2\right)^7 (a e+c d x)^3}","-\frac{128 c d \left(7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right) \left(a e^2+c d^2+2 c d e x\right)}{105 \left(c d^2-a e^2\right)^7 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{16 \left(7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right) \left(a e^2+c d^2+2 c d e x\right)}{105 e \left(c d^2-a e^2\right)^5 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}-\frac{8 \left(x \left(3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right)+2 a d e \left(2 a e^2+c d^2\right)\right)}{35 e \left(c d^2-a e^2\right)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}+\frac{2 x^2}{7 (d+e x) \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}",1,"(-2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(a^6*e^10*(8*d^2 + 28*d*e*x + 35*e^2*x^2)) + 2*a^5*c*d*e^8*(112*d^3 + 382*d^2*e*x + 455*d*e^2*x^2 + 140*e^3*x^3) + 3*c^6*d^8*x^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448*d*e^3*x^3 + 128*e^4*x^4) + 5*a^4*c^2*d^2*e^6*(336*d^4 + 1288*d^3*e*x + 1859*d^2*e^2*x^2 + 1288*d*e^3*x^3 + 336*e^4*x^4) + 20*a^3*c^3*d^3*e^4*(56*d^5 + 406*d^4*e*x + 1001*d^3*e^2*x^2 + 1084*d^2*e^3*x^3 + 560*d*e^4*x^4 + 112*e^5*x^5) + 2*a*c^5*d^6*e*x*(70*d^5 + 1295*d^4*e*x + 4060*d^3*e^2*x^2 + 5600*d^2*e^3*x^3 + 3616*d*e^4*x^4 + 896*e^5*x^5) + a^2*c^4*d^4*e^2*(56*d^6 + 2996*d^5*e*x + 13195*d^4*e^2*x^2 + 24080*d^3*e^3*x^3 + 20320*d^2*e^4*x^4 + 7616*d*e^5*x^5 + 896*e^6*x^6)))/(105*(c*d^2 - a*e^2)^7*(a*e + c*d*x)^3*(d + e*x)^4)","A",1
489,1,221,170,0.9202117,"\int x^3 \sqrt{1+x} \sqrt{1-x+x^2} \, dx","Integrate[x^3*Sqrt[1 + x]*Sqrt[1 - x + x^2],x]","\frac{2 \left(x \sqrt{x+1} \left(5 x^5-5 x^4+5 x^3+3 x^2-3 x+3\right)+\sqrt{-\frac{6 i}{\sqrt{3}+3 i}} \left(\sqrt{3}+3 i\right) (x+1) \sqrt{\frac{\left(\sqrt{3}-3 i\right) x+\sqrt{3}+3 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{\frac{\left(\sqrt{3}+3 i\right) x+\sqrt{3}-3 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)\right)}{55 \sqrt{x^2-x+1}}","\frac{6}{55} \sqrt{x+1} \sqrt{x^2-x+1} x+\frac{2}{11} \sqrt{x+1} \sqrt{x^2-x+1} x^4-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{55 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(2*(x*Sqrt[1 + x]*(3 - 3*x + 3*x^2 + 5*x^3 - 5*x^4 + 5*x^5) + Sqrt[(-6*I)/(3*I + Sqrt[3])]*(3*I + Sqrt[3])*(1 + x)*Sqrt[(3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[(-3*I + Sqrt[3] + (3*I + Sqrt[3])*x)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/(55*Sqrt[1 - x + x^2])","C",1
490,1,23,23,0.0323158,"\int x^2 \sqrt{1+x} \sqrt{1-x+x^2} \, dx","Integrate[x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2],x]","\frac{2}{9} (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}","\frac{2}{9} (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}",1,"(2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2))/9","A",1
491,1,347,294,0.5084598,"\int x \sqrt{1+x} \sqrt{1-x+x^2} \, dx","Integrate[x*Sqrt[1 + x]*Sqrt[1 - x + x^2],x]","\frac{\sqrt{x+1} \left(4 \sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}} \left(x^2-x+1\right) x^2+3 \sqrt{2} \left(\sqrt{3}-i\right) \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)-3 \sqrt{2} \left(\sqrt{3}-3 i\right) \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)\right)}{14 \sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{2}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{6 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left(x+\sqrt{3}+1\right)}+\frac{2 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(Sqrt[1 + x]*(4*x^2*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*(1 - x + x^2) - 3*Sqrt[2]*(-3*I + Sqrt[3])*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])] + 3*Sqrt[2]*(-I + Sqrt[3])*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/(14*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
492,1,169,144,0.4809886,"\int \sqrt{1+x} \sqrt{1-x+x^2} \, dx","Integrate[Sqrt[1 + x]*Sqrt[1 - x + x^2],x]","\frac{2 x \sqrt{x+1} \left(x^2-x+1\right)+\frac{i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{5 \sqrt{x^2-x+1}}","\frac{2}{5} x \sqrt{x^2-x+1} \sqrt{x+1}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (x+1)^{3/2} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{5 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(2*x*Sqrt[1 + x]*(1 - x + x^2) + (I*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(5*Sqrt[1 - x + x^2])","C",1
493,1,197,66,0.4072801,"\int \frac{\sqrt{1+x} \sqrt{1-x+x^2}}{x} \, dx","Integrate[(Sqrt[1 + x]*Sqrt[1 - x + x^2])/x,x]","\frac{\sqrt{x+1} \left(2 \left(x^2-x+1\right)+\frac{3 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \Pi \left(\frac{3}{2}-\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right)}{3 \sqrt{x^2-x+1}}","\frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left(\sqrt{x^3+1}\right)}{3 \sqrt{x^3+1}}",1,"(Sqrt[1 + x]*(2*(1 - x + x^2) + ((3*I)*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticPi[3/2 - (I/2)*Sqrt[3], I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(3*Sqrt[1 - x + x^2])","C",1
494,1,349,287,0.3994707,"\int \frac{\sqrt{1+x} \sqrt{1-x+x^2}}{x^2} \, dx","Integrate[(Sqrt[1 + x]*Sqrt[1 - x + x^2])/x^2,x]","-\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{x}+\frac{3 \sqrt{1+\frac{2 i (x+1)}{\sqrt{3}-3 i}} \sqrt{1-\frac{2 i (x+1)}{\sqrt{3}+3 i}} \left(\frac{\left(\sqrt{3}-i\right) \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x+1} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}-\frac{\left(\sqrt{3}-3 i\right) \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x+1} E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right)}{2 \sqrt{2} \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{(x+1)^2-3 (x+1)+3}}","-\frac{\sqrt{x^2-x+1} \sqrt{x+1}}{x}+\frac{3 \sqrt{x^2-x+1} \sqrt{x+1}}{x+\sqrt{3}+1}+\frac{\sqrt{2} 3^{3/4} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (x+1)^{3/2} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (x+1)^{3/2} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"-((Sqrt[1 + x]*Sqrt[1 - x + x^2])/x) + (3*Sqrt[1 + ((2*I)*(1 + x))/(-3*I + Sqrt[3])]*Sqrt[1 - ((2*I)*(1 + x))/(3*I + Sqrt[3])]*(-(((-3*I + Sqrt[3])*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 + x]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]) + ((-I + Sqrt[3])*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 + x]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(2*Sqrt[2]*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[3 - 3*(1 + x) + (1 + x)^2])","C",1
495,1,185,146,0.4120728,"\int \frac{\sqrt{1+x} \sqrt{1-x+x^2}}{x^3} \, dx","Integrate[(Sqrt[1 + x]*Sqrt[1 - x + x^2])/x^3,x]","\frac{\sqrt{x+1} \left(-\frac{2 \left(x^2-x+1\right)}{x^2}-\frac{3 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right)}{4 \sqrt{x^2-x+1}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}-\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{2 x^2}",1,"(Sqrt[1 + x]*((-2*(1 - x + x^2))/x^2 - ((3*I)*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(4*Sqrt[1 - x + x^2])","C",1
496,1,235,201,0.883079,"\int x^3 (1+x)^{3/2} \left(1-x+x^2\right)^{3/2} \, dx","Integrate[x^3*(1 + x)^(3/2)*(1 - x + x^2)^(3/2),x]","\frac{2 \left(x \sqrt{x+1} \left(55 x^8-55 x^7+55 x^6+100 x^5-100 x^4+100 x^3+27 x^2-27 x+27\right)-\frac{9 i \sqrt{6} (x+1) \sqrt{\frac{\left(\sqrt{3}-3 i\right) x+\sqrt{3}+3 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{\frac{\left(\sqrt{3}+3 i\right) x+\sqrt{3}-3 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}\right)}{935 \sqrt{x^2-x+1}}","\frac{54}{935} \sqrt{x+1} \sqrt{x^2-x+1} x+\frac{18}{187} \sqrt{x+1} \sqrt{x^2-x+1} x^4-\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{935 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}+\frac{2}{17} \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right) x^4",1,"(2*(x*Sqrt[1 + x]*(27 - 27*x + 27*x^2 + 100*x^3 - 100*x^4 + 100*x^5 + 55*x^6 - 55*x^7 + 55*x^8) - ((9*I)*Sqrt[6]*(1 + x)*Sqrt[(3*I + Sqrt[3] + (-3*I + Sqrt[3])*x)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[(-3*I + Sqrt[3] + (3*I + Sqrt[3])*x)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])]))/(935*Sqrt[1 - x + x^2])","C",1
497,1,23,23,0.0422803,"\int x^2 (1+x)^{3/2} \left(1-x+x^2\right)^{3/2} \, dx","Integrate[x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2),x]","\frac{2}{15} (x+1)^{5/2} \left(x^2-x+1\right)^{5/2}","\frac{2}{15} (x+1)^{5/2} \left(x^2-x+1\right)^{5/2}",1,"(2*(1 + x)^(5/2)*(1 - x + x^2)^(5/2))/15","A",1
498,1,244,325,0.4914769,"\int x (1+x)^{3/2} \left(1-x+x^2\right)^{3/2} \, dx","Integrate[x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2),x]","\frac{\sqrt{x+1} \left(4 x^2 \left(x^2-x+1\right) \left(7 x^3+16\right)-\frac{27 \sqrt{2} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \left(\left(\sqrt{3}-3 i\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)-\left(\sqrt{3}-i\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)\right)}{\sqrt{-\frac{i (x+1)}{-2 i x+\sqrt{3}+i}}}\right)}{182 \sqrt{x^2-x+1}}","\frac{18}{91} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{54 \sqrt{x+1} \sqrt{x^2-x+1}}{91 \left(x+\sqrt{3}+1\right)}+\frac{2}{13} \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right) x^2+\frac{18 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{91 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{91 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(Sqrt[1 + x]*(4*x^2*(1 - x + x^2)*(16 + 7*x^3) - (27*Sqrt[2]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*((-3*I + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])] - (-I + Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/Sqrt[((-I)*(1 + x))/(I + Sqrt[3] - (2*I)*x)]))/(182*Sqrt[1 - x + x^2])","C",0
499,1,176,173,0.6355506,"\int (1+x)^{3/2} \left(1-x+x^2\right)^{3/2} \, dx","Integrate[(1 + x)^(3/2)*(1 - x + x^2)^(3/2),x]","\frac{2 x \sqrt{x+1} \left(x^2-x+1\right) \left(5 x^3+14\right)+\frac{9 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{55 \sqrt{x^2-x+1}}","\frac{18}{55} x \sqrt{x^2-x+1} \sqrt{x+1}+\frac{2}{11} x \sqrt{x^2-x+1} \left(x^3+1\right) \sqrt{x+1}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (x+1)^{3/2} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{55 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(2*x*Sqrt[1 + x]*(1 - x + x^2)*(14 + 5*x^3) + ((9*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(55*Sqrt[1 - x + x^2])","C",1
500,1,201,94,0.2956552,"\int \frac{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}}{x} \, dx","Integrate[((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x,x]","\frac{\sqrt{x+1} \left(\frac{2}{9} \left(x^2-x+1\right) \left(x^3+4\right)+\frac{i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \Pi \left(\frac{3}{2}-\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right)}{\sqrt{x^2-x+1}}","\frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}+\frac{2}{9} \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)-\frac{2 \sqrt{x+1} \sqrt{x^2-x+1} \tanh ^{-1}\left(\sqrt{x^3+1}\right)}{3 \sqrt{x^3+1}}",1,"(Sqrt[1 + x]*((2*(1 - x + x^2)*(4 + x^3))/9 + (I*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticPi[3/2 - (I/2)*Sqrt[3], I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/Sqrt[1 - x + x^2]","C",1
501,1,244,323,0.4929433,"\int \frac{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}}{x^2} \, dx","Integrate[((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^2,x]","\frac{\sqrt{x+1} \left(\frac{4 \left(x^2-x+1\right) \left(2 x^3-7\right)}{x}-\frac{27 \sqrt{2} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \left(\left(\sqrt{3}-3 i\right) E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)-\left(\sqrt{3}-i\right) F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)\right)}{\sqrt{-\frac{i (x+1)}{-2 i x+\sqrt{3}+i}}}\right)}{28 \sqrt{x^2-x+1}}","\frac{9}{7} \sqrt{x+1} \sqrt{x^2-x+1} x^2+\frac{27 \sqrt{x+1} \sqrt{x^2-x+1}}{7 \left(x+\sqrt{3}+1\right)}-\frac{\sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}{x}+\frac{9 \sqrt{2} 3^{3/4} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{7 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}-\frac{27 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1)^{3/2} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{14 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(Sqrt[1 + x]*((4*(1 - x + x^2)*(-7 + 2*x^3))/x - (27*Sqrt[2]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*((-3*I + Sqrt[3])*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])] - (-I + Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]))/Sqrt[((-I)*(1 + x))/(I + Sqrt[3] - (2*I)*x)]))/(28*Sqrt[1 - x + x^2])","C",0
502,1,192,175,0.3497862,"\int \frac{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}}{x^3} \, dx","Integrate[((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^3,x]","\frac{\sqrt{x+1} \left(\frac{2 \left(x^2-x+1\right) \left(4 x^3-5\right)}{x^2}-\frac{27 i \sqrt{2} \sqrt{\frac{-2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{-\frac{i (x+1)}{3 i+\sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i (x+1)}{\sqrt{3}+3 i}}}\right)}{20 \sqrt{x^2-x+1}}","\frac{9}{10} x \sqrt{x^2-x+1} \sqrt{x+1}-\frac{\sqrt{x^2-x+1} \left(x^3+1\right) \sqrt{x+1}}{2 x^2}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \sqrt{x^2-x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (x+1)^{3/2} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{10 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(x^3+1\right)}",1,"(Sqrt[1 + x]*((2*(1 - x + x^2)*(-5 + 4*x^3))/x^2 - ((27*I)*Sqrt[2]*Sqrt[(I + Sqrt[3] - (2*I)*x)/(3*I + Sqrt[3])]*Sqrt[(-I + Sqrt[3] + (2*I)*x)/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[((-I)*(1 + x))/(3*I + Sqrt[3])]))/(20*Sqrt[1 - x + x^2])","C",1
503,1,169,142,0.6050466,"\int \frac{x^3}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[x^3/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\frac{6 x \sqrt{x+1} \left(x^2-x+1\right)-\frac{2 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{15 \sqrt{x^2-x+1}}","\frac{2 x \left(x^3+1\right)}{5 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{4 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{5 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}",1,"(6*x*Sqrt[1 + x]*(1 - x + x^2) - ((2*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(15*Sqrt[1 - x + x^2])","C",1
504,1,23,23,0.02441,"\int \frac{x^2}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[x^2/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}","\frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1}",1,"(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3","A",1
505,1,375,253,0.9959099,"\int \frac{x}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[x/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\frac{(x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{6 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{2 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}+\frac{2 \left(x^3+1\right)}{\sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}",1,"((1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(6*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
506,1,148,110,0.1423711,"\int \frac{1}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[1/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\frac{i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{\frac{2}{3}-\frac{4 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}",1,"(I*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[2/3 - (4*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/(Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
507,1,2463,42,12.8006188,"\int \frac{1}{x \sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[1/(x*Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\text{Result too large to show}","-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left(\sqrt{x^3+1}\right)}{3 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"2*(((-I)*(1 + x)*Sqrt[1 - 6/((3 - I*Sqrt[3])*(1 + x))]*Sqrt[1 - 6/((3 + I*Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[3])]/Sqrt[1 + x]], (3 - I*Sqrt[3])/(3 + I*Sqrt[3])])/(Sqrt[6]*Sqrt[-(3 - I*Sqrt[3])^(-1)]*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) + (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) - (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]))","C",0
508,1,400,282,0.8322423,"\int \frac{1}{x^2 \sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[1/(x^2*Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","-\frac{\sqrt{x+1} \sqrt{x^2-x+1}}{x}+\frac{(x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{\sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{x^3+1}{x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{x^3+1}{\sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}",1,"-((Sqrt[1 + x]*Sqrt[1 - x + x^2])/x) + ((1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(12*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
509,1,171,146,0.6497144,"\int \frac{1}{x^3 \sqrt{1+x} \sqrt{1-x+x^2}} \, dx","Integrate[1/(x^3*Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]","\frac{-\frac{6 \sqrt{x+1} \left(x^2-x+1\right)}{x^2}-\frac{i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{12 \sqrt{x^2-x+1}}","\frac{-x^3-1}{2 x^2 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{\sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}",1,"((-6*Sqrt[1 + x]*(1 - x + x^2))/x^2 - (I*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(12*Sqrt[1 - x + x^2])","C",1
510,1,161,137,0.5695859,"\int \frac{x^3}{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[x^3/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","\frac{-\frac{6 x}{\sqrt{x+1}}+\frac{2 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{9 \sqrt{x^2-x+1}}","\frac{4 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{2 x}{3 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"((-6*x)/Sqrt[1 + x] + ((2*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(9*Sqrt[1 - x + x^2])","C",1
511,1,23,23,0.030247,"\int \frac{x^2}{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[x^2/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","-\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}","-\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"-2/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2])","A",1
512,1,402,282,0.7070209,"\int \frac{x}{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[x/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","\frac{2 x^2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{(x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{18 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{2 x^2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}+\frac{\sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{2 \left(x^3+1\right)}{3 \sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}",1,"(2*x^2)/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) - ((1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(18*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
513,1,216,137,0.3602096,"\int \frac{1}{(1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[1/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","\sqrt{(x+1)^2-3 (x+1)+3} \left(\frac{2 (x+1)^{3/2}}{9 \left((x+1)^2-3 (x+1)+3\right)}-\frac{2}{9 \sqrt{x+1}}\right)+\frac{i \sqrt{\frac{2}{3}} (x+1) \sqrt{1-\frac{6}{\left(3-i \sqrt{3}\right) (x+1)}} \sqrt{1-\frac{6}{\left(3+i \sqrt{3}\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6}{3-i \sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3-i \sqrt{3}}{3+i \sqrt{3}}\right)}{3 \sqrt{-\frac{1}{3-i \sqrt{3}}} \sqrt{(x+1)^2-3 (x+1)+3}}","\frac{2 x}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}",1,"Sqrt[3 - 3*(1 + x) + (1 + x)^2]*(-2/(9*Sqrt[1 + x]) + (2*(1 + x)^(3/2))/(9*(3 - 3*(1 + x) + (1 + x)^2))) + ((I/3)*Sqrt[2/3]*(1 + x)*Sqrt[1 - 6/((3 - I*Sqrt[3])*(1 + x))]*Sqrt[1 - 6/((3 + I*Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[3])]/Sqrt[1 + x]], (3 - I*Sqrt[3])/(3 + I*Sqrt[3])])/(Sqrt[-(3 - I*Sqrt[3])^(-1)]*Sqrt[3 - 3*(1 + x) + (1 + x)^2])","C",1
514,1,2511,66,6.0770951,"\int \frac{1}{x (1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[1/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","\text{Result too large to show}","\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left(\sqrt{x^3+1}\right)}{3 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"Sqrt[1 + x]*Sqrt[1 - x + x^2]*(2/(9*(1 + x)) - (2*(-2 + x))/(9*(1 - x + x^2))) + 2*(((-I)*(1 + x)*Sqrt[1 - 6/((3 - I*Sqrt[3])*(1 + x))]*Sqrt[1 - 6/((3 + I*Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[3])]/Sqrt[1 + x]], (3 - I*Sqrt[3])/(3 + I*Sqrt[3])])/(Sqrt[6]*Sqrt[-(3 - I*Sqrt[3])^(-1)]*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) + (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) - (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]))","C",0
515,1,409,316,0.7708497,"\int \frac{1}{x^2 (1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[1/(x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","-\frac{5 x^3+3}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 (x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{36 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{2}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2\ 3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{5 \left(x^3+1\right)}{3 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{5 \left(x^3+1\right)}{3 \sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}",1,"-1/3*(3 + 5*x^3)/(x*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + (5*(1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(36*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
516,1,170,170,0.3947726,"\int \frac{1}{x^3 (1+x)^{3/2} \left(1-x+x^2\right)^{3/2}} \, dx","Integrate[1/(x^3*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]","\frac{-\frac{6 \left(7 x^3+3\right)}{x^2 \sqrt{x+1}}-\frac{7 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{36 \sqrt{x^2-x+1}}","\frac{2}{3 x^2 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{7 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{6 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{7 \left(x^3+1\right)}{6 x^2 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"((-6*(3 + 7*x^3))/(x^2*Sqrt[1 + x]) - ((7*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(36*Sqrt[1 - x + x^2])","C",1
517,1,178,168,0.4980196,"\int \frac{x^3}{(1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[x^3/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","\frac{\frac{6 x \left(2 x^3-1\right)}{(x+1)^{3/2} \left(x^2-x+1\right)}+\frac{2 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{81 \sqrt{x^2-x+1}}","\frac{4 x}{27 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{4 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{2 x}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}",1,"((6*x*(-1 + 2*x^3))/((1 + x)^(3/2)*(1 - x + x^2)) + ((2*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(81*Sqrt[1 - x + x^2])","C",1
518,1,23,23,0.039999,"\int \frac{x^2}{(1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[x^2/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","-\frac{2}{9 (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}}","-\frac{2}{9 (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}}",1,"-2/(9*(1 + x)^(3/2)*(1 - x + x^2)^(3/2))","A",1
519,1,409,318,0.7590124,"\int \frac{x}{(1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[x/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","\frac{2 x^2 \left(5 x^3+8\right)}{27 (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}}-\frac{5 (x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{162 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{10 x^2}{27 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{10 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}+\frac{5 \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{9\ 3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}+\frac{2 x^2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}-\frac{10 \left(x^3+1\right)}{27 \sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}",1,"(2*x^2*(8 + 5*x^3))/(27*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)) - (5*(1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(162*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
520,1,178,168,0.4740802,"\int \frac{1}{(1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[1/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","\frac{\frac{6 x \left(7 x^3+10\right)}{(x+1)^{3/2} \left(x^2-x+1\right)}+\frac{7 i (x+1) \sqrt{1+\frac{6 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{6-\frac{36 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{81 \sqrt{x^2-x+1}}","\frac{14 x}{27 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{14 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}+\frac{2 x}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}",1,"((6*x*(10 + 7*x^3))/((1 + x)^(3/2)*(1 - x + x^2)) + ((7*I)*(1 + x)*Sqrt[1 + (6*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[6 - (36*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(81*Sqrt[1 - x + x^2])","C",1
521,1,2539,96,6.0889164,"\int \frac{1}{x (1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","\text{Result too large to show}","\frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left(\sqrt{x^3+1}\right)}{3 \sqrt{x+1} \sqrt{x^2-x+1}}",1,"Sqrt[1 + x]*Sqrt[1 - x + x^2]*(2/(81*(1 + x)^2) + 22/(81*(1 + x)) - (2*(-1 + x))/(27*(1 - x + x^2)^2) - (2*(-21 + 11*x))/(81*(1 - x + x^2))) + 2*(((-I)*(1 + x)*Sqrt[1 - 6/((3 - I*Sqrt[3])*(1 + x))]*Sqrt[1 - 6/((3 + I*Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[-6/(3 - I*Sqrt[3])]/Sqrt[1 + x]], (3 - I*Sqrt[3])/(3 + I*Sqrt[3])])/(Sqrt[6]*Sqrt[-(3 - I*Sqrt[3])^(-1)]*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) + (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]) - (Sqrt[3/2]*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(1 + x)*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x])^2*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(-Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[(Sqrt[(2*(3 - I*Sqrt[3]))/3]*(Sqrt[1/2 + (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))/((Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + 1/Sqrt[1 + x]))]*Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]*((-1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*EllipticF[ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2] - Sqrt[(2*(3 - I*Sqrt[3]))/3]*EllipticPi[((1 + Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]]))/((1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(-Sqrt[1/2 - (I/2)/Sqrt[3]] + Sqrt[1/2 + (I/2)/Sqrt[3]])), ArcSin[Sqrt[((Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] + 6/Sqrt[1 + x]))/((Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])*(Sqrt[6*(3 - I*Sqrt[3])] - 6/Sqrt[1 + x]))]], (Sqrt[3 - I*Sqrt[3]] + Sqrt[3 + I*Sqrt[3]])^2/(Sqrt[3 - I*Sqrt[3]] - Sqrt[3 + I*Sqrt[3]])^2]))/(Sqrt[3 - I*Sqrt[3]]*(-1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(1 - Sqrt[1/2 - (I/2)/Sqrt[3]])*(Sqrt[1/2 - (I/2)/Sqrt[3]] - Sqrt[1/2 + (I/2)/Sqrt[3]])*Sqrt[3 - 3*(1 + x) + (1 + x)^2]))","C",0
522,1,414,349,0.8586972,"\int \frac{1}{x^2 (1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[1/(x^2*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","-\frac{55 x^6+88 x^3+27}{27 x (x+1)^{3/2} \left(x^2-x+1\right)^{3/2}}+\frac{55 (x+1)^{3/2} \left(\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left(x^2-x+1\right)}{(x+1)^2}+\frac{i \sqrt{2} \left(\sqrt{3}+3 i\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left(1-i \sqrt{3}\right) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{x+1}}\right)}{324 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}}","\frac{22}{27 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{55 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{27 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{55 \sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{18\ 3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{55 \left(x^3+1\right)}{27 x \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{55 \left(x^3+1\right)}{27 \sqrt{x+1} \left(x+\sqrt{3}+1\right) \sqrt{x^2-x+1}}+\frac{2}{9 x \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}",1,"-1/27*(27 + 88*x^3 + 55*x^6)/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)) + (55*(1 + x)^(3/2)*((12*Sqrt[(-I)/(3*I + Sqrt[3])]*(1 - x + x^2))/(1 + x)^2 + (3*Sqrt[2]*(1 - I*Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x] + (I*Sqrt[2]*(3*I + Sqrt[3])*Sqrt[(3*I + Sqrt[3] - (6*I)/(1 + x))/(3*I + Sqrt[3])]*Sqrt[(-3*I + Sqrt[3] + (6*I)/(1 + x))/(-3*I + Sqrt[3])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[1 + x]))/(324*Sqrt[(-I)/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2])","C",1
523,1,183,203,0.6484711,"\int \frac{1}{x^3 (1+x)^{5/2} \left(1-x+x^2\right)^{5/2}} \, dx","Integrate[1/(x^3*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]","\frac{-\frac{6 \left(91 x^6+130 x^3+27\right)}{x^2 (x+1)^{3/2}}-\frac{91 i (x+1) \left(x^2-x+1\right) \sqrt{6+\frac{36 i}{\left(\sqrt{3}-3 i\right) (x+1)}} \sqrt{1-\frac{6 i}{\left(\sqrt{3}+3 i\right) (x+1)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{-\frac{i}{\sqrt{3}+3 i}}}}{324 \left(x^2-x+1\right)^{3/2}}","\frac{26}{27 x^2 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{91 \sqrt{2+\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{54 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^2-x+1}}-\frac{91 \left(x^3+1\right)}{54 x^2 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 x^2 \sqrt{x+1} \sqrt{x^2-x+1} \left(x^3+1\right)}",1,"((-6*(27 + 130*x^3 + 91*x^6))/(x^2*(1 + x)^(3/2)) - ((91*I)*(1 + x)*(1 - x + x^2)*Sqrt[6 + (36*I)/((-3*I + Sqrt[3])*(1 + x))]*Sqrt[1 - (6*I)/((3*I + Sqrt[3])*(1 + x))]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(3*I + Sqrt[3])]/Sqrt[1 + x]], (3*I + Sqrt[3])/(3*I - Sqrt[3])])/Sqrt[(-I)/(3*I + Sqrt[3])])/(324*(1 - x + x^2)^(3/2))","C",1
524,1,78,97,0.0519091,"\int \frac{x}{(-1+x)^3 \left(3+5 x+4 x^2\right)^2} \, dx","Integrate[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]","\frac{\frac{184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left(4 x^2+5 x+3\right)+\frac{59248}{x-1}-\frac{25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt{23} \tan ^{-1}\left(\frac{8 x+5}{\sqrt{23}}\right)}{7312896}","\frac{44 x+39}{276 (1-x)^2 \left(4 x^2+5 x+3\right)}-\frac{11 \log \left(4 x^2+5 x+3\right)}{4608}-\frac{97}{4416 (1-x)}-\frac{21}{736 (1-x)^2}+\frac{11 \log (1-x)}{2304}+\frac{6023 \tan ^{-1}\left(\frac{8 x+5}{\sqrt{23}}\right)}{52992 \sqrt{23}}",1,"(-25392/(-1 + x)^2 + 59248/(-1 + x) + (184*(975 + 2204*x))/(3 + 5*x + 4*x^2) + 36138*Sqrt[23]*ArcTan[(5 + 8*x)/Sqrt[23]] + 34914*Log[1 - x] - 17457*Log[3 + 5*x + 4*x^2])/7312896","A",1
525,1,568,490,0.7462609,"\int \frac{x^4 \sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[(x^4*Sqrt[d + e*x])/(a + b*x + c*x^2),x]","-\frac{\sqrt{2} \left(a^2 c^2 \left(e \sqrt{b^2-4 a c}+2 c d\right)+a b c^2 \left(2 d \sqrt{b^2-4 a c}-5 a e\right)-a b^2 c \left(3 e \sqrt{b^2-4 a c}+4 c d\right)+b^4 \left(e \sqrt{b^2-4 a c}+c d\right)+b^3 c \left(5 a e-d \sqrt{b^2-4 a c}\right)+b^5 (-e)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{9/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{2} \left(a^2 c^2 \left(e \sqrt{b^2-4 a c}-2 c d\right)+a b c^2 \left(2 d \sqrt{b^2-4 a c}+5 a e\right)+a b^2 c \left(4 c d-3 e \sqrt{b^2-4 a c}\right)+b^4 \left(e \sqrt{b^2-4 a c}-c d\right)-b^3 c \left(d \sqrt{b^2-4 a c}+5 a e\right)+b^5 e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{9/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \sqrt{d+e x} \left(-7 c^2 e (d+e x) (5 a e-2 b d+3 b e x)+35 b c e^2 (6 a e+b (d+e x))-105 b^3 e^3+c^3 \left(8 d^3-4 d^2 e x+3 d e^2 x^2+15 e^3 x^3\right)\right)}{105 c^4 e^3}","\frac{\sqrt{2} \left(-\frac{-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt{b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{9/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(\frac{-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d}{\sqrt{b^2-4 a c}}-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{9/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b \left(b^2-2 a c\right) \sqrt{d+e x}}{c^4}+\frac{2 (d+e x)^{3/2} \left(c e (b d-a e)+b^2 e^2+c^2 d^2\right)}{3 c^3 e^3}-\frac{2 (d+e x)^{5/2} (b e+2 c d)}{5 c^2 e^3}+\frac{2 (d+e x)^{7/2}}{7 c e^3}",1,"(2*Sqrt[d + e*x]*(-105*b^3*e^3 - 7*c^2*e*(d + e*x)*(-2*b*d + 5*a*e + 3*b*e*x) + c^3*(8*d^3 - 4*d^2*e*x + 3*d*e^2*x^2 + 15*e^3*x^3) + 35*b*c*e^2*(6*a*e + b*(d + e*x))))/(105*c^4*e^3) - (Sqrt[2]*(-(b^5*e) + a*b*c^2*(2*Sqrt[b^2 - 4*a*c]*d - 5*a*e) + b^3*c*(-(Sqrt[b^2 - 4*a*c]*d) + 5*a*e) + b^4*(c*d + Sqrt[b^2 - 4*a*c]*e) + a^2*c^2*(2*c*d + Sqrt[b^2 - 4*a*c]*e) - a*b^2*c*(4*c*d + 3*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(9/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^5*e - b^3*c*(Sqrt[b^2 - 4*a*c]*d + 5*a*e) + a*b*c^2*(2*Sqrt[b^2 - 4*a*c]*d + 5*a*e) + a*b^2*c*(4*c*d - 3*Sqrt[b^2 - 4*a*c]*e) + a^2*c^2*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + b^4*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
526,1,466,326,0.531401,"\int \frac{x^3 \sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[(x^3*Sqrt[d + e*x])/(a + b*x + c*x^2),x]","\frac{2 \sqrt{d+e x} \left(-5 c e (3 a e+b (d+e x))+15 b^2 e^2+c^2 \left(-2 d^2+d e x+3 e^2 x^2\right)\right)}{15 c^3 e^2}+\frac{\sqrt{2} \left(a c^2 \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^2 c \left(4 a e-d \sqrt{b^2-4 a c}\right)-a b c \left(2 e \sqrt{b^2-4 a c}+3 c d\right)+b^3 \left(e \sqrt{b^2-4 a c}+c d\right)+b^4 (-e)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(a c^2 \left(d \sqrt{b^2-4 a c}+2 a e\right)-b^2 c \left(d \sqrt{b^2-4 a c}+4 a e\right)+a b c \left(3 c d-2 e \sqrt{b^2-4 a c}\right)+b^3 \left(e \sqrt{b^2-4 a c}-c d\right)+b^4 e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}","\frac{2 \left(b^2-a c\right) \sqrt{d+e x}}{c^3}+\frac{\left(-\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c}}-\frac{\left(\sqrt{b^2-4 a c} \left(b^2-a c\right)-3 a b c+b^3\right) \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c}}-\frac{2 (d+e x)^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{2 (d+e x)^{5/2}}{5 c e^2}",1,"(2*Sqrt[d + e*x]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x + 3*e^2*x^2) - 5*c*e*(3*a*e + b*(d + e*x))))/(15*c^3*e^2) + (Sqrt[2]*(-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
527,1,375,316,0.4423356,"\int \frac{x^2 \sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[(x^2*Sqrt[d + e*x])/(a + b*x + c*x^2),x]","\frac{\sqrt{2} \left(-b^2 \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(d \sqrt{b^2-4 a c}-3 a e\right)+a c \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{2} \left(b^2 \left(e \sqrt{b^2-4 a c}-c d\right)-b c \left(d \sqrt{b^2-4 a c}+3 a e\right)+a c \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^3 e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \sqrt{d+e x} (c (d+e x)-3 b e)}{3 c^2 e}","\frac{\sqrt{2} \left(-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{5/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{5/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b \sqrt{d+e x}}{c^2}+\frac{2 (d+e x)^{3/2}}{3 c e}",1,"(2*Sqrt[d + e*x]*(-3*b*e + c*(d + e*x)))/(3*c^2*e) + (Sqrt[2]*(b^3*e + b*c*(Sqrt[b^2 - 4*a*c]*d - 3*a*e) - b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) + a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^3*e - b*c*(Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b^2*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
528,1,301,287,0.4059426,"\int \frac{x \sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[(x*Sqrt[d + e*x])/(a + b*x + c*x^2),x]","\frac{\frac{\sqrt{2} \left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}-2 a c e+b^2 e-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+2 \sqrt{c} \sqrt{d+e x}}{c^{3/2}}","\frac{\sqrt{2} \left(-\sqrt{b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \left(\sqrt{b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \sqrt{d+e x}}{c}",1,"(2*Sqrt[c]*Sqrt[d + e*x] + (Sqrt[2]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(-(b*c*d) - c*Sqrt[b^2 - 4*a*c]*d + b^2*e - 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/c^(3/2)","A",1
529,1,175,198,0.3974014,"\int \frac{\sqrt{d+e x}}{a+b x+c x^2} \, dx","Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2),x]","\frac{\sqrt{2} \left(\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)-\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)\right)}{\sqrt{c} \sqrt{b^2-4 a c}}","\frac{\sqrt{2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c}}",1,"(Sqrt[2]*(-(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]]) + Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]]))/(Sqrt[c]*Sqrt[b^2 - 4*a*c])","A",1
530,1,267,275,0.9375203,"\int \frac{\sqrt{d+e x}}{x \left(a+b x+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/(x*(a + b*x + c*x^2)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \sqrt{c} \left(d \sqrt{b^2-4 a c}+2 a e-b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a}","\frac{\sqrt{2} \sqrt{c} \left(d \sqrt{b^2-4 a c}-2 a e+b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \sqrt{c} \left(-d \sqrt{b^2-4 a c}-2 a e+b d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a}",1,"(-2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[2]*Sqrt[c]*(b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a","A",1
531,1,364,368,1.590677,"\int \frac{\sqrt{d+e x}}{x^2 \left(a+b x+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/(x^2*(a + b*x + c*x^2)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(-b d \sqrt{b^2-4 a c}+a e \sqrt{b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{2} \sqrt{c} \left(b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+a b e+2 a c d+b^2 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 (b d-a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}-\frac{a \sqrt{d+e x}}{x}+\frac{a e \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{a^2}","-\frac{\sqrt{2} \sqrt{c} \left(\sqrt{b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \sqrt{c} \left(-b \left(d \sqrt{b^2-4 a c}+a e\right)-a \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 (b d-a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a^2 \sqrt{d}}-\frac{\sqrt{d+e x}}{a x}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a \sqrt{d}}",1,"(-((a*Sqrt[d + e*x])/x) + (a*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (2*(b*d - a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 2*a*c*d - b*Sqrt[b^2 - 4*a*c]*d + a*b*e + a*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a^2","A",1
532,1,516,531,2.3351306,"\int \frac{\sqrt{d+e x}}{x^3 \left(a+b x+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/(x^3*(a + b*x + c*x^2)),x]","-\frac{\frac{3 a^2 e \left(e x \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)-\sqrt{d} \sqrt{d+e x}\right)}{d^{3/2} x}+\frac{2 a^2 \sqrt{d+e x}}{x^2}+\frac{8 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right) \left(-a b e-a c d+b^2 d\right)}{\sqrt{d}}+\frac{4 \sqrt{2} \sqrt{c} \left(b^2 \left(a e-d \sqrt{b^2-4 a c}\right)+a b \left(e \sqrt{b^2-4 a c}+3 c d\right)+a c \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 (-d)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{4 \sqrt{2} \sqrt{c} \left(-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)+a b \left(e \sqrt{b^2-4 a c}-3 c d\right)+a c \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{4 a e (a e-b d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{4 a \sqrt{d+e x} (b d-a e)}{d x}}{4 a^3}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right) \left(-a b e-a c d+b^2 d\right)}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left(b^2 \left(d \sqrt{b^2-4 a c}-a e\right)-a b \left(e \sqrt{b^2-4 a c}+3 c d\right)-a c \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \sqrt{c} \left(-b^2 \left(d \sqrt{b^2-4 a c}+a e\right)-a b \left(3 c d-e \sqrt{b^2-4 a c}\right)+a c \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{e (b d-a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a^2 d^{3/2}}+\frac{\sqrt{d+e x} (b d-a e)}{a^2 d x}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 a d^{3/2}}-\frac{\sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a d x}",1,"-1/4*((2*a^2*Sqrt[d + e*x])/x^2 - (4*a*(b*d - a*e)*Sqrt[d + e*x])/(d*x) - (4*a*e*(-(b*d) + a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2) + (8*(b^2*d - a*c*d - a*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (3*a^2*e*(-(Sqrt[d]*Sqrt[d + e*x]) + e*x*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]))/(d^(3/2)*x) + (4*Sqrt[2]*Sqrt[c]*(-(b^3*d) + a*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(-(Sqrt[b^2 - 4*a*c]*d) + a*e) + a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (4*Sqrt[2]*Sqrt[c]*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a^3","A",1
533,1,808,650,1.1159903,"\int \frac{x^4 (d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(x^4*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]","\frac{2 \sqrt{d+e x} \left((d+e x)^2 \left(8 d^2-20 e x d+35 e^2 x^2\right) c^4-9 e (d+e x)^2 (-2 b d+7 a e+5 b e x) c^3+21 e^2 \left(15 a^2 e^2+10 a b (4 d+e x) e+3 b^2 (d+e x)^2\right) c^2-105 b^2 e^3 (4 b d+9 a e+b e x) c+315 b^4 e^4\right)}{315 c^5 e^3}+\frac{\sqrt{2} \left(-e^2 b^6+e \left(2 c d+\sqrt{b^2-4 a c} e\right) b^5-c \left(c d^2+2 e \left(\sqrt{b^2-4 a c} d-3 a e\right)\right) b^4+c \left(c d \left(\sqrt{b^2-4 a c} d-10 a e\right)-4 a \sqrt{b^2-4 a c} e^2\right) b^3+a c^2 \left(4 c d^2+6 \sqrt{b^2-4 a c} e d-9 a e^2\right) b^2+a c^2 \left(3 a \sqrt{b^2-4 a c} e^2-2 c d \left(\sqrt{b^2-4 a c} d-5 a e\right)\right) b+2 a^2 c^3 \left(e \left(a e-\sqrt{b^2-4 a c} d\right)-c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right)}{c^{11/2} \sqrt{b^2-4 a c} \sqrt{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}}+\frac{\sqrt{2} \left(e^2 b^6+e \left(\sqrt{b^2-4 a c} e-2 c d\right) b^5+c \left(c d^2-2 e \left(\sqrt{b^2-4 a c} d+3 a e\right)\right) b^4+c \left(c d \left(\sqrt{b^2-4 a c} d+10 a e\right)-4 a \sqrt{b^2-4 a c} e^2\right) b^3+a c^2 \left(-4 c d^2+6 \sqrt{b^2-4 a c} e d+9 a e^2\right) b^2+a c^2 \left(3 a \sqrt{b^2-4 a c} e^2-2 c d \left(\sqrt{b^2-4 a c} d+5 a e\right)\right) b-2 a^2 c^3 \left(e \left(\sqrt{b^2-4 a c} d+a e\right)-c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}\right)}{c^{11/2} \sqrt{b^2-4 a c} \sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}}","-\frac{2 \sqrt{d+e x} \left(-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right)}{c^5}+\frac{\sqrt{2} \left(\frac{10 a^2 b c^3 d e-2 a^2 c^3 \left(c d^2-a e^2\right)-b^4 c \left(c d^2-6 a e^2\right)-10 a b^3 c^2 d e+a b^2 c^2 \left(4 c d^2-9 a e^2\right)+b^6 \left(-e^2\right)+2 b^5 c d e}{\sqrt{b^2-4 a c}}+\left(a c e+b^2 (-e)+b c d\right) \left(3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{11/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(\left(a c e+b^2 (-e)+b c d\right) \left(3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right)-\frac{10 a^2 b c^3 d e-2 a^2 c^3 \left(c d^2-a e^2\right)-b^4 c \left(c d^2-6 a e^2\right)-10 a b^3 c^2 d e+a b^2 c^2 \left(4 c d^2-9 a e^2\right)+b^6 \left(-e^2\right)+2 b^5 c d e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{11/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b \left(b^2-2 a c\right) (d+e x)^{3/2}}{3 c^4}+\frac{2 (d+e x)^{5/2} \left(c e (b d-a e)+b^2 e^2+c^2 d^2\right)}{5 c^3 e^3}-\frac{2 (d+e x)^{7/2} (b e+2 c d)}{7 c^2 e^3}+\frac{2 (d+e x)^{9/2}}{9 c e^3}",1,"(2*Sqrt[d + e*x]*(315*b^4*e^4 - 105*b^2*c*e^3*(4*b*d + 9*a*e + b*e*x) - 9*c^3*e*(d + e*x)^2*(-2*b*d + 7*a*e + 5*b*e*x) + c^4*(d + e*x)^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 21*c^2*e^2*(15*a^2*e^2 + 3*b^2*(d + e*x)^2 + 10*a*b*e*(4*d + e*x))))/(315*c^5*e^3) + (Sqrt[2]*(-(b^6*e^2) + b^5*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + a*b^2*c^2*(4*c*d^2 + 6*Sqrt[b^2 - 4*a*c]*d*e - 9*a*e^2) + b^3*c*(-4*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 10*a*e)) + a*b*c^2*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - 2*c*d*(Sqrt[b^2 - 4*a*c]*d - 5*a*e)) - b^4*c*(c*d^2 + 2*e*(Sqrt[b^2 - 4*a*c]*d - 3*a*e)) + 2*a^2*c^3*(-(c*d^2) + e*(-(Sqrt[b^2 - 4*a*c]*d) + a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(11/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(b^6*e^2 + b^5*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + a*b^2*c^2*(-4*c*d^2 + 6*Sqrt[b^2 - 4*a*c]*d*e + 9*a*e^2) - 2*a^2*c^3*(-(c*d^2) + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b^4*c*(c*d^2 - 2*e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + a*b*c^2*(3*a*Sqrt[b^2 - 4*a*c]*e^2 - 2*c*d*(Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + b^3*c*(-4*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 10*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(11/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
534,1,680,581,0.9035142,"\int \frac{x^3 (d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(x^3*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]","-\frac{2 \sqrt{d+e x} \left(7 c^2 e \left(5 a e (4 d+e x)+3 b (d+e x)^2\right)-35 b c e^2 (6 a e+4 b d+b e x)+105 b^3 e^3+3 c^3 (2 d-5 e x) (d+e x)^2\right)}{105 c^4 e^2}-\frac{\sqrt{2} \left(a b c^2 \left(e \left(4 d \sqrt{b^2-4 a c}-5 a e\right)+3 c d^2\right)+a c^2 \left(c d \left(4 a e-d \sqrt{b^2-4 a c}\right)+a e^2 \sqrt{b^2-4 a c}\right)+b^2 c \left(c d \left(d \sqrt{b^2-4 a c}-8 a e\right)-3 a e^2 \sqrt{b^2-4 a c}\right)+b^4 e \left(e \sqrt{b^2-4 a c}+2 c d\right)-b^3 c \left(e \left(2 d \sqrt{b^2-4 a c}-5 a e\right)+c d^2\right)+b^5 \left(-e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{9/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{2} \left(a b c^2 \left(e \left(4 d \sqrt{b^2-4 a c}+5 a e\right)-3 c d^2\right)+a c^2 \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)\right)+b^2 c \left(c d \left(d \sqrt{b^2-4 a c}+8 a e\right)-3 a e^2 \sqrt{b^2-4 a c}\right)+b^4 e \left(e \sqrt{b^2-4 a c}-2 c d\right)+b^3 c \left(c d^2-e \left(2 d \sqrt{b^2-4 a c}+5 a e\right)\right)+b^5 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{9/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}","\frac{\sqrt{2} \left(-\frac{4 a^2 c^3 d e-b^3 c \left(c d^2-5 a e^2\right)-8 a b^2 c^2 d e+a b c^2 \left(3 c d^2-5 a e^2\right)+b^5 \left(-e^2\right)+2 b^4 c d e}{\sqrt{b^2-4 a c}}-b^2 c \left(c d^2-3 a e^2\right)-4 a b c^2 d e+a c^2 \left(c d^2-a e^2\right)+b^4 \left(-e^2\right)+2 b^3 c d e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{9/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(\frac{4 a^2 c^3 d e-b^3 c \left(c d^2-5 a e^2\right)-8 a b^2 c^2 d e+a b c^2 \left(3 c d^2-5 a e^2\right)+b^5 \left(-e^2\right)+2 b^4 c d e}{\sqrt{b^2-4 a c}}-b^2 c \left(c d^2-3 a e^2\right)-4 a b c^2 d e+a c^2 \left(c d^2-a e^2\right)+b^4 \left(-e^2\right)+2 b^3 c d e\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{9/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \left(b^2-a c\right) (d+e x)^{3/2}}{3 c^3}+\frac{2 \sqrt{d+e x} \left(2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right)}{c^4}-\frac{2 (d+e x)^{5/2} (b e+c d)}{5 c^2 e^2}+\frac{2 (d+e x)^{7/2}}{7 c e^2}",1,"(-2*Sqrt[d + e*x]*(105*b^3*e^3 + 3*c^3*(2*d - 5*e*x)*(d + e*x)^2 - 35*b*c*e^2*(4*b*d + 6*a*e + b*e*x) + 7*c^2*e*(3*b*(d + e*x)^2 + 5*a*e*(4*d + e*x))))/(105*c^4*e^2) - (Sqrt[2]*(-(b^5*e^2) + b^4*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b^2*c*(-3*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 8*a*e)) - b^3*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 5*a*e)) + a*b*c^2*(3*c*d^2 + e*(4*Sqrt[b^2 - 4*a*c]*d - 5*a*e)) + a*c^2*(a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(9/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^5*e^2 + b^4*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + a*c^2*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)) + b^3*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + a*b*c^2*(-3*c*d^2 + e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*e)) + b^2*c*(-3*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 8*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(9/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
535,1,538,441,0.6748357,"\int \frac{x^2 (d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(x^2*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]","\frac{2 \sqrt{d+e x} \left(-5 c e (3 a e+4 b d+b e x)+15 b^2 e^2+3 c^2 (d+e x)^2\right)}{15 c^3 e}+\frac{\sqrt{2} \left(2 a c^2 \left(e \left(d \sqrt{b^2-4 a c}-a e\right)+c d^2\right)-b^2 c \left(2 e \left(d \sqrt{b^2-4 a c}-2 a e\right)+c d^2\right)+b c \left(c d \left(d \sqrt{b^2-4 a c}-6 a e\right)-2 a e^2 \sqrt{b^2-4 a c}\right)+b^3 e \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^4 \left(-e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(2 a c^2 \left(e \left(d \sqrt{b^2-4 a c}+a e\right)-c d^2\right)+b^2 c \left(c d^2-2 e \left(d \sqrt{b^2-4 a c}+2 a e\right)\right)+b c \left(c d \left(d \sqrt{b^2-4 a c}+6 a e\right)-2 a e^2 \sqrt{b^2-4 a c}\right)+b^3 e \left(e \sqrt{b^2-4 a c}-2 c d\right)+b^4 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}","-\frac{2 \sqrt{d+e x} \left(a c e+b^2 (-e)+b c d\right)}{c^3}+\frac{\sqrt{2} \left((c d-b e) \left(2 a c e+b^2 (-e)+b c d\right)+\frac{-b^2 c \left(c d^2-4 a e^2\right)-6 a b c^2 d e+2 a c^2 \left(c d^2-a e^2\right)+b^4 \left(-e^2\right)+2 b^3 c d e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{7/2} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left((c d-b e) \left(2 a c e+b^2 (-e)+b c d\right)-\frac{-b^2 c \left(c d^2-4 a e^2\right)-6 a b c^2 d e+2 a c^2 \left(c d^2-a e^2\right)+b^4 \left(-e^2\right)+2 b^3 c d e}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{7/2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 b (d+e x)^{3/2}}{3 c^2}+\frac{2 (d+e x)^{5/2}}{5 c e}",1,"(2*Sqrt[d + e*x]*(15*b^2*e^2 + 3*c^2*(d + e*x)^2 - 5*c*e*(4*b*d + 3*a*e + b*e*x)))/(15*c^3*e) + (Sqrt[2]*(-(b^4*e^2) + b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(-2*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d - 6*a*e)) - b^2*c*(c*d^2 + 2*e*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + 2*a*c^2*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(b^4*e^2 + b^3*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + 2*a*c^2*(-(c*d^2) + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) + b^2*c*(c*d^2 - 2*e*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + b*c*(-2*a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(Sqrt[b^2 - 4*a*c]*d + 6*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])","A",1
536,1,779,453,1.4888798,"\int \frac{x (d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(x*(d + e*x)^(3/2))/(a + b*x + c*x^2),x]","\frac{2 \left(\frac{3 \sqrt{c} d \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{3 \sqrt{c} d \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{3 \left(3 c^2 d e \left(d \sqrt{b^2-4 a c}-2 a e-b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)+b^2 e^3 \left(\sqrt{b^2-4 a c}-b\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{3 \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-3 e \sqrt{d+e x} (b e-2 c d)+c e (d+e x)^{3/2}-3 c d e \sqrt{d+e x}\right)}{3 c^2 e}","\frac{\sqrt{2} \left(b c \left(e \left(2 d \sqrt{b^2-4 a c}-3 a e\right)+c d^2\right)+c \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}-4 a e\right)\right)-b^2 e \left(e \sqrt{b^2-4 a c}+2 c d\right)+b^3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \left(b c \left(c d^2-e \left(2 d \sqrt{b^2-4 a c}+3 a e\right)\right)-c \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)\right)-b^2 e \left(2 c d-e \sqrt{b^2-4 a c}\right)+b^3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \sqrt{d+e x} (c d-b e)}{c^2}+\frac{2 (d+e x)^{3/2}}{3 c}",1,"(2*(-3*c*d*e*Sqrt[d + e*x] - 3*e*(-2*c*d + b*e)*Sqrt[d + e*x] + c*e*(d + e*x)^(3/2) + (3*Sqrt[c]*d*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (3*(2*c^3*d^3 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 3*c^2*d*e*(-(b*d) + Sqrt[b^2 - 4*a*c]*d - 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (3*Sqrt[c]*d*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + (3*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/(3*c^2*e)","A",1
537,1,317,322,0.7046485,"\int \frac{(d+e x)^{3/2}}{a+b x+c x^2} \, dx","Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]","\frac{\frac{\sqrt{2} \left(2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}-b\right)-2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+2 \sqrt{c} e \sqrt{d+e x}}{c^{3/2}}","-\frac{\sqrt{2} \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c^{3/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 e \sqrt{d+e x}}{c}",1,"(2*Sqrt[c]*e*Sqrt[d + e*x] + (Sqrt[2]*(-2*c^2*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/c^(3/2)","A",1
538,1,331,340,1.1598996,"\int \frac{(d+e x)^{3/2}}{x \left(a+b x+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/(x*(a + b*x + c*x^2)),x]","\frac{\frac{\sqrt{2} \left(c d \left(d \sqrt{b^2-4 a c}-4 a e\right)-a e^2 \sqrt{b^2-4 a c}+b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}+\frac{\sqrt{2} \left(c d \left(d \sqrt{b^2-4 a c}+4 a e\right)-a e^2 \sqrt{b^2-4 a c}-b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a}","-\frac{\sqrt{2} \left(-c d \left(d \sqrt{b^2-4 a c}-4 a e\right)+a e^2 \sqrt{b^2-4 a c}-b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \left(-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)+a e^2 \sqrt{b^2-4 a c}+b \left(a e^2+c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a}",1,"(-2*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[2]*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) + b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) - b*(c*d^2 + a*e^2))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a","A",1
539,1,393,403,1.5397218,"\int \frac{(d+e x)^{3/2}}{x^2 \left(a+b x+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/(x^2*(a + b*x + c*x^2)),x]","\frac{\frac{\sqrt{2} \sqrt{c} \left(2 a \left(e \left(d \sqrt{b^2-4 a c}-a e\right)+c d^2\right)+b d \left(2 a e-d \sqrt{b^2-4 a c}\right)-b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{\sqrt{2} \sqrt{c} \left(b d \left(d \sqrt{b^2-4 a c}+2 a e\right)-2 a e \left(d \sqrt{b^2-4 a c}+a e\right)+2 a c d^2-b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+2 \sqrt{d} (b d-2 a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)-\frac{a d \sqrt{d+e x}}{x}+a \sqrt{d} e \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a^2}","-\frac{\sqrt{2} \sqrt{c} \left(-2 a \left(e \left(d \sqrt{b^2-4 a c}-a e\right)+c d^2\right)+b d \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{\sqrt{2} \sqrt{c} \left(-2 a \left(c d^2-e \left(d \sqrt{b^2-4 a c}+a e\right)\right)-b d \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a^2 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 \sqrt{d} (b d-2 a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a^2}-\frac{d \sqrt{d+e x}}{a x}+\frac{\sqrt{d} e \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a}",1,"(-((a*d*Sqrt[d + e*x])/x) + a*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*Sqrt[d]*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (Sqrt[2]*Sqrt[c]*(-(b^2*d^2) + b*d*(-(Sqrt[b^2 - 4*a*c]*d) + 2*a*e) + 2*a*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(-(b^2*d^2) + 2*a*c*d^2 - 2*a*e*(Sqrt[b^2 - 4*a*c]*d + a*e) + b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a^2","A",1
540,1,587,607,2.8570599,"\int \frac{(d+e x)^{3/2}}{x^3 \left(a+b x+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/(x^3*(a + b*x + c*x^2)),x]","\frac{-\frac{2 a^2 d \sqrt{d+e x}}{x^2}+3 a^2 e \left(\frac{\sqrt{d+e x}}{x}-\frac{e \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}\right)-\frac{8 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right) \left(-2 a b d e+a \left(a e^2-c d^2\right)+b^2 d^2\right)}{\sqrt{d}}+\frac{4 \sqrt{2} \sqrt{c} \left(a b \left(e \left(a e-2 d \sqrt{b^2-4 a c}\right)-3 c d^2\right)+a \left(c d \left(4 a e-d \sqrt{b^2-4 a c}\right)+a e^2 \sqrt{b^2-4 a c}\right)+b^2 d \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right)}{\sqrt{b^2-4 a c} \sqrt{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}}-\frac{4 \sqrt{2} \sqrt{c} \left(a b \left(e \left(2 d \sqrt{b^2-4 a c}+a e\right)-3 c d^2\right)+a \left(c d \left(d \sqrt{b^2-4 a c}+4 a e\right)-a e^2 \sqrt{b^2-4 a c}\right)-b^2 d \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{4 a \sqrt{d+e x} (b d-2 a e)}{x}+\frac{4 a e (2 a e-b d) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{4 a^3}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right) \left(-2 a b d e-a \left(c d^2-a e^2\right)+b^2 d^2\right)}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left(-a b \left(e \left(2 d \sqrt{b^2-4 a c}-a e\right)+3 c d^2\right)+a \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}-4 a e\right)\right)+b^2 d \left(d \sqrt{b^2-4 a c}-2 a e\right)+b^3 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{\sqrt{2} \sqrt{c} \left(-a b \left(3 c d^2-e \left(2 d \sqrt{b^2-4 a c}+a e\right)\right)-a \left(a e^2 \sqrt{b^2-4 a c}-c d \left(d \sqrt{b^2-4 a c}+4 a e\right)\right)-b^2 d \left(d \sqrt{b^2-4 a c}+2 a e\right)+b^3 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{\sqrt{d+e x} (b d-2 a e)}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{a^2 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x}}{\sqrt{d}}\right)}{4 a \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x}",1,"((-2*a^2*d*Sqrt[d + e*x])/x^2 + (4*a*(b*d - 2*a*e)*Sqrt[d + e*x])/x + (4*a*e*(-(b*d) + 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] - (8*(b^2*d^2 - 2*a*b*d*e + a*(-(c*d^2) + a*e^2))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + 3*a^2*e*(Sqrt[d + e*x]/x - (e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d]) + (4*Sqrt[2]*Sqrt[c]*(b^3*d^2 + b^2*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*b*(-3*c*d^2 + e*(-2*Sqrt[b^2 - 4*a*c]*d + a*e)) + a*(a*Sqrt[b^2 - 4*a*c]*e^2 + c*d*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (4*Sqrt[2]*Sqrt[c]*(b^3*d^2 - b^2*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d + a*e)) + a*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(4*a^3)","A",1
541,0,0,201,0.319228,"\int \frac{x^m (e+f x)^n}{a+b x+c x^2} \, dx","Integrate[(x^m*(e + f*x)^n)/(a + b*x + c*x^2),x]","\int \frac{x^m (e+f x)^n}{a+b x+c x^2} \, dx","\frac{2 c x^{m+1} (e+f x)^n \left(\frac{f x}{e}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{f x}{e},-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right)}{(m+1) \sqrt{b^2-4 a c} \left(b-\sqrt{b^2-4 a c}\right)}-\frac{2 c x^{m+1} (e+f x)^n \left(\frac{f x}{e}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{f x}{e},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right)}{(m+1) \sqrt{b^2-4 a c} \left(\sqrt{b^2-4 a c}+b\right)}",1,"Integrate[(x^m*(e + f*x)^n)/(a + b*x + c*x^2), x]","F",-1
542,1,261,290,0.8198037,"\int \frac{x^3 (e+f x)^n}{a+b x+c x^2} \, dx","Integrate[(x^3*(e + f*x)^n)/(a + b*x + c*x^2),x]","\frac{(e+f x)^{n+1} \left(\frac{c \left(\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{(n+1) \left(f \left(\sqrt{b^2-4 a c}-b\right)+2 c e\right)}+\frac{c \left(-\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{(n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{b f+c e}{f^2 (n+1)}+\frac{c (e+f x)}{f^2 (n+2)}\right)}{c^2}","\frac{\left(\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{c (n+1) \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{\left(-\frac{b \left(b^2-3 a c\right)}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{c (n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{(b f+c e) (e+f x)^{n+1}}{c^2 f^2 (n+1)}+\frac{(e+f x)^{n+2}}{c f^2 (n+2)}",1,"((e + f*x)^(1 + n)*(-((c*e + b*f)/(f^2*(1 + n))) + (c*(e + f*x))/(f^2*(2 + n)) + (c*(a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)])/((2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)*(1 + n)) + (c*(a - b^2/c - (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/((2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)*(1 + n))))/c^2","A",1
543,1,202,237,0.3730208,"\int \frac{x^2 (e+f x)^n}{a+b x+c x^2} \, dx","Integrate[(x^2*(e + f*x)^n)/(a + b*x + c*x^2),x]","\frac{(e+f x)^{n+1} \left(\frac{\left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{f \left(\sqrt{b^2-4 a c}-b\right)+2 c e}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{2 c e-f \left(\sqrt{b^2-4 a c}+b\right)}+\frac{1}{f}\right)}{c (n+1)}","\frac{\left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{c (n+1) \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{\left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{c (n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}+\frac{(e+f x)^{n+1}}{c f (n+1)}",1,"((e + f*x)^(1 + n)*(f^(-1) + ((b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f) + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)))/(c*(1 + n))","A",1
544,1,183,198,0.2879511,"\int \frac{x (e+f x)^n}{a+b x+c x^2} \, dx","Integrate[(x*(e + f*x)^n)/(a + b*x + c*x^2),x]","\frac{(e+f x)^{n+1} \left(-\frac{\left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{f \left(\sqrt{b^2-4 a c}-b\right)+2 c e}-\frac{\left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{2 c e-f \left(\sqrt{b^2-4 a c}+b\right)}\right)}{n+1}","-\frac{\left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{(n+1) \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}-\frac{\left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{(n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}",1,"((e + f*x)^(1 + n)*(-(((1 - b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)) - ((1 + b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)))/(1 + n)","A",1
545,1,163,191,0.2517923,"\int \frac{(e+f x)^n}{a+b x+c x^2} \, dx","Integrate[(e + f*x)^n/(a + b*x + c*x^2),x]","\frac{2 c (e+f x)^{n+1} \left(\frac{\, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{2 c e-f \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{f \left(\sqrt{b^2-4 a c}-b\right)+2 c e}\right)}{(n+1) \sqrt{b^2-4 a c}}","\frac{2 c (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{(n+1) \sqrt{b^2-4 a c} \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{2 c (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{(n+1) \sqrt{b^2-4 a c} \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}",1,"(2*c*(e + f*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)]/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)) + Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)]/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)))/(Sqrt[b^2 - 4*a*c]*(1 + n))","A",1
546,1,207,242,0.4079566,"\int \frac{(e+f x)^n}{x \left(a+b x+c x^2\right)} \, dx","Integrate[(e + f*x)^n/(x*(a + b*x + c*x^2)),x]","\frac{(e+f x)^{n+1} \left(\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{f \left(\sqrt{b^2-4 a c}-b\right)+2 c e}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{2 c e-f \left(\sqrt{b^2-4 a c}+b\right)}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{e}\right)}{a (n+1)}","\frac{c \left(\frac{b}{\sqrt{b^2-4 a c}}+1\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{a (n+1) \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}+\frac{c \left(1-\frac{b}{\sqrt{b^2-4 a c}}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{a (n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{a e (n+1)}",1,"((e + f*x)^(1 + n)*((c*(1 + b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f) + (c*(1 - b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f) - Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e]/e))/(a*(1 + n))","A",1
547,1,246,296,0.4332574,"\int \frac{(e+f x)^n}{x^2 \left(a+b x+c x^2\right)} \, dx","Integrate[(e + f*x)^n/(x^2*(a + b*x + c*x^2)),x]","\frac{(e+f x)^{n+1} \left(-\frac{c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left(\sqrt{b^2-4 a c}-b\right) f}\right)}{f \left(\sqrt{b^2-4 a c}-b\right)+2 c e}-\frac{c \left(\frac{2 a c-b^2}{\sqrt{b^2-4 a c}}+b\right) \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{2 c e-f \left(\sqrt{b^2-4 a c}+b\right)}+\frac{a f \, _2F_1\left(2,n+1;n+2;\frac{f x}{e}+1\right)}{e^2}+\frac{b \, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{e}\right)}{a^2 (n+1)}","-\frac{c \left(\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b-\sqrt{b^2-4 a c}\right) f}\right)}{a^2 (n+1) \left(2 c e-f \left(b-\sqrt{b^2-4 a c}\right)\right)}-\frac{c \left(b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right) (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left(b+\sqrt{b^2-4 a c}\right) f}\right)}{a^2 (n+1) \left(2 c e-f \left(\sqrt{b^2-4 a c}+b\right)\right)}+\frac{b (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{a^2 e (n+1)}+\frac{f (e+f x)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{f x}{e}+1\right)}{a e^2 (n+1)}",1,"((e + f*x)^(1 + n)*(-((c*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)) - (c*(b + (-b^2 + 2*a*c)/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f) + (b*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e])/e + (a*f*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (f*x)/e])/e^2))/(a^2*(1 + n))","A",1
548,1,134,141,0.0766509,"\int \frac{(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx","Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2),x]","-\frac{8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x \left(240 d^4 g^2+120 d^3 e g (4 f+g x)+70 d^2 e^2 \left(3 f^2+3 f g x+g^2 x^2\right)+10 d e^3 x \left(6 f^2+8 f g x+3 g^2 x^2\right)+e^4 x^2 \left(10 f^2+15 f g x+6 g^2 x^2\right)\right)}{30 e^2}","-\frac{8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{d x^2 \left(4 d^2 g^2+7 d e f g+2 e^2 f^2\right)}{e}-\frac{d^2 x \left(8 d^2 g^2+16 d e f g+7 e^2 f^2\right)}{e^2}-\frac{1}{2} e g x^4 (2 d g+e f)-\frac{1}{3} x^3 (d g+e f) (7 d g+e f)-\frac{1}{5} e^2 g^2 x^5",1,"-1/30*(x*(240*d^4*g^2 + 120*d^3*e*g*(4*f + g*x) + 70*d^2*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) + 10*d*e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2) + e^4*x^2*(10*f^2 + 15*f*g*x + 6*g^2*x^2)))/e^2 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3","A",1
549,1,103,109,0.0538731,"\int \frac{(d+e x)^3 (f+g x)^2}{d^2-e^2 x^2} \, dx","Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2),x]","-\frac{48 d^2 (d g+e f)^2 \log (d-e x)+e x \left(48 d^3 g^2+24 d^2 e g (4 f+g x)+12 d e^2 \left(3 f^2+3 f g x+g^2 x^2\right)+e^3 x \left(6 f^2+8 f g x+3 g^2 x^2\right)\right)}{12 e^3}","-\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x^2 \left(4 d^2 g^2+6 d e f g+e^2 f^2\right)}{2 e}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4",1,"-1/12*(e*x*(48*d^3*g^2 + 24*d^2*e*g*(4*f + g*x) + 12*d*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) + e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2)) + 48*d^2*(e*f + d*g)^2*Log[d - e*x])/e^3","A",1
550,1,73,65,0.0340828,"\int \frac{(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx","Integrate[((d + e*x)^2*(f + g*x)^2)/(d^2 - e^2*x^2),x]","-\frac{e x \left(6 d^2 g^2+3 d e g (4 f+g x)+e^2 \left(3 f^2+3 f g x+g^2 x^2\right)\right)+6 d (d g+e f)^2 \log (d-e x)}{3 e^3}","-\frac{2 d (d g+e f)^2 \log (d-e x)}{e^3}-\frac{2 d g x (d g+e f)}{e^2}-\frac{d (f+g x)^2}{e}-\frac{(f+g x)^3}{3 g}",1,"-1/3*(e*x*(6*d^2*g^2 + 3*d*e*g*(4*f + g*x) + e^2*(3*f^2 + 3*f*g*x + g^2*x^2)) + 6*d*(e*f + d*g)^2*Log[d - e*x])/e^3","A",1
551,1,43,50,0.0196681,"\int \frac{(d+e x) (f+g x)^2}{d^2-e^2 x^2} \, dx","Integrate[((d + e*x)*(f + g*x)^2)/(d^2 - e^2*x^2),x]","-\frac{e g x (2 d g+4 e f+e g x)+2 (d g+e f)^2 \log (d-e x)}{2 e^3}","-\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{g x (d g+e f)}{e^2}-\frac{(f+g x)^2}{2 e}",1,"-1/2*(e*g*x*(4*e*f + 2*d*g + e*g*x) + 2*(e*f + d*g)^2*Log[d - e*x])/e^3","A",1
552,1,55,62,0.0217592,"\int \frac{(f+g x)^2}{d^2-e^2 x^2} \, dx","Integrate[(f + g*x)^2/(d^2 - e^2*x^2),x]","\frac{\left(d^2 g^2+e^2 f^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)-d e g \left(f \log \left(d^2-e^2 x^2\right)+g x\right)}{d e^3}","-\frac{(d g+e f)^2 \log (d-e x)}{2 d e^3}+\frac{(e f-d g)^2 \log (d+e x)}{2 d e^3}-\frac{g^2 x}{e^2}",1,"((e^2*f^2 + d^2*g^2)*ArcTanh[(e*x)/d] - d*e*g*(g*x + f*Log[d^2 - e^2*x^2]))/(d*e^3)","A",1
553,1,82,86,0.0471935,"\int \frac{(f+g x)^2}{(d+e x) \left(d^2-e^2 x^2\right)} \, dx","Integrate[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)),x]","\frac{(e f-d g) ((d+e x) (3 d g+e f) \log (d+e x)+2 d (d g-e f))-(d+e x) (d g+e f)^2 \log (d-e x)}{4 d^2 e^3 (d+e x)}","\frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)}",1,"(-((e*f + d*g)^2*(d + e*x)*Log[d - e*x]) + (e*f - d*g)*(2*d*(-(e*f) + d*g) + (e*f + 3*d*g)*(d + e*x)*Log[d + e*x]))/(4*d^2*e^3*(d + e*x))","A",1
554,1,87,87,0.0745782,"\int \frac{(f+g x)^2}{(d+e x)^2 \left(d^2-e^2 x^2\right)} \, dx","Integrate[(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)),x]","\frac{\frac{2 d (d g-e f) \left(2 d^2 g+d e (2 f+3 g x)+e^2 f x\right)}{(d+e x)^2}+(d g+e f)^2 (-\log (d-e x))+(d g+e f)^2 \log (d+e x)}{8 d^3 e^3}","\frac{(d g+e f)^2 \tanh ^{-1}\left(\frac{e x}{d}\right)}{4 d^3 e^3}-\frac{(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2}",1,"((2*d*(-(e*f) + d*g)*(2*d^2*g + e^2*f*x + d*e*(2*f + 3*g*x)))/(d + e*x)^2 - (e*f + d*g)^2*Log[d - e*x] + (e*f + d*g)^2*Log[d + e*x])/(8*d^3*e^3)","A",1
555,1,122,113,0.0591682,"\int \frac{(f+g x)^2}{(d+e x)^3 \left(d^2-e^2 x^2\right)} \, dx","Integrate[(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)),x]","\frac{-\frac{8 d^3 (e f-d g)^2}{(d+e x)^3}+\frac{6 d^2 \left(3 d^2 g^2-2 d e f g-e^2 f^2\right)}{(d+e x)^2}-\frac{6 d (d g+e f)^2}{d+e x}-3 (d g+e f)^2 \log (d-e x)+3 (d g+e f)^2 \log (d+e x)}{48 d^4 e^3}","\frac{(d g+e f)^2 \tanh ^{-1}\left(\frac{e x}{d}\right)}{8 d^4 e^3}-\frac{(d g+e f)^2}{8 d^3 e^3 (d+e x)}-\frac{(3 d g+e f) (e f-d g)}{8 d^2 e^3 (d+e x)^2}-\frac{(e f-d g)^2}{6 d e^3 (d+e x)^3}",1,"((-8*d^3*(e*f - d*g)^2)/(d + e*x)^3 + (6*d^2*(-(e^2*f^2) - 2*d*e*f*g + 3*d^2*g^2))/(d + e*x)^2 - (6*d*(e*f + d*g)^2)/(d + e*x) - 3*(e*f + d*g)^2*Log[d - e*x] + 3*(e*f + d*g)^2*Log[d + e*x])/(48*d^4*e^3)","A",1
556,1,142,139,0.0899225,"\int \frac{(f+g x)^2}{(d+e x)^4 \left(d^2-e^2 x^2\right)} \, dx","Integrate[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)),x]","-\frac{\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d^2 (d g+e f)^2}{(d+e x)^2}+\frac{8 d^3 \left(-3 d^2 g^2+2 d e f g+e^2 f^2\right)}{(d+e x)^3}+\frac{6 d (d g+e f)^2}{d+e x}+3 (d g+e f)^2 \log (d-e x)-3 (d g+e f)^2 \log (d+e x)}{96 d^5 e^3}","\frac{(d g+e f)^2 \tanh ^{-1}\left(\frac{e x}{d}\right)}{16 d^5 e^3}-\frac{(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac{(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}-\frac{(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{8 d e^3 (d+e x)^4}",1,"-1/96*((12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (8*d^3*(e^2*f^2 + 2*d*e*f*g - 3*d^2*g^2))/(d + e*x)^3 + (6*d^2*(e*f + d*g)^2)/(d + e*x)^2 + (6*d*(e*f + d*g)^2)/(d + e*x) + 3*(e*f + d*g)^2*Log[d - e*x] - 3*(e*f + d*g)^2*Log[d + e*x])/(d^5*e^3)","A",1
557,1,226,218,0.1214282,"\int \frac{(d+e x)^7 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","-\frac{32 d^5 (d g+e f)^2}{e^3 (e x-d)}+\frac{1}{4} e x^4 \left(23 d^2 g^2+14 d e f g+e^2 f^2\right)+\frac{1}{3} d x^3 \left(49 d^2 g^2+46 d e f g+7 e^2 f^2\right)+\frac{d^2 x^2 \left(80 d^2 g^2+98 d e f g+23 e^2 f^2\right)}{2 e}+\frac{16 d^4 \left(9 d^2 g^2+14 d e f g+5 e^2 f^2\right) \log (d-e x)}{e^3}+\frac{d^3 x \left(112 d^2 g^2+160 d e f g+49 e^2 f^2\right)}{e^2}+\frac{1}{5} e^2 g x^5 (7 d g+2 e f)+\frac{1}{6} e^3 g^2 x^6","\frac{32 d^5 (d g+e f)^2}{e^3 (d-e x)}+\frac{16 d^4 (d g+e f) (9 d g+5 e f) \log (d-e x)}{e^3}+\frac{1}{4} e x^4 \left(23 d^2 g^2+14 d e f g+e^2 f^2\right)+\frac{1}{3} d x^3 \left(49 d^2 g^2+46 d e f g+7 e^2 f^2\right)+\frac{d^2 x^2 \left(80 d^2 g^2+98 d e f g+23 e^2 f^2\right)}{2 e}+\frac{d^3 x \left(112 d^2 g^2+160 d e f g+49 e^2 f^2\right)}{e^2}+\frac{1}{5} e^2 g x^5 (7 d g+2 e f)+\frac{1}{6} e^3 g^2 x^6",1,"(d^3*(49*e^2*f^2 + 160*d*e*f*g + 112*d^2*g^2)*x)/e^2 + (d^2*(23*e^2*f^2 + 98*d*e*f*g + 80*d^2*g^2)*x^2)/(2*e) + (d*(7*e^2*f^2 + 46*d*e*f*g + 49*d^2*g^2)*x^3)/3 + (e*(e^2*f^2 + 14*d*e*f*g + 23*d^2*g^2)*x^4)/4 + (e^2*g*(2*e*f + 7*d*g)*x^5)/5 + (e^3*g^2*x^6)/6 - (32*d^5*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (16*d^4*(5*e^2*f^2 + 14*d*e*f*g + 9*d^2*g^2)*Log[d - e*x])/e^3","A",1
558,1,185,177,0.1200451,"\int \frac{(d+e x)^6 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","-\frac{16 d^4 (d g+e f)^2}{e^3 (e x-d)}+\frac{1}{3} x^3 \left(17 d^2 g^2+12 d e f g+e^2 f^2\right)+\frac{d x^2 \left(16 d^2 g^2+17 d e f g+3 e^2 f^2\right)}{e}+\frac{d^2 x \left(48 d^2 g^2+64 d e f g+17 e^2 f^2\right)}{e^2}+\frac{32 d^3 \left(2 d^2 g^2+3 d e f g+e^2 f^2\right) \log (d-e x)}{e^3}+\frac{1}{2} e g x^4 (3 d g+e f)+\frac{1}{5} e^2 g^2 x^5","\frac{16 d^4 (d g+e f)^2}{e^3 (d-e x)}+\frac{32 d^3 (d g+e f) (2 d g+e f) \log (d-e x)}{e^3}+\frac{1}{3} x^3 \left(17 d^2 g^2+12 d e f g+e^2 f^2\right)+\frac{d x^2 \left(16 d^2 g^2+17 d e f g+3 e^2 f^2\right)}{e}+\frac{d^2 x \left(48 d^2 g^2+64 d e f g+17 e^2 f^2\right)}{e^2}+\frac{1}{2} e g x^4 (3 d g+e f)+\frac{1}{5} e^2 g^2 x^5",1,"(d^2*(17*e^2*f^2 + 64*d*e*f*g + 48*d^2*g^2)*x)/e^2 + (d*(3*e^2*f^2 + 17*d*e*f*g + 16*d^2*g^2)*x^2)/e + ((e^2*f^2 + 12*d*e*f*g + 17*d^2*g^2)*x^3)/3 + (e*g*(e*f + 3*d*g)*x^4)/2 + (e^2*g^2*x^5)/5 - (16*d^4*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (32*d^3*(e^2*f^2 + 3*d*e*f*g + 2*d^2*g^2)*Log[d - e*x])/e^3","A",1
559,1,154,146,0.0931432,"\int \frac{(d+e x)^5 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^5*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","-\frac{8 d^3 (d g+e f)^2}{e^3 (e x-d)}+\frac{x^2 \left(12 d^2 g^2+10 d e f g+e^2 f^2\right)}{2 e}+\frac{d x \left(20 d^2 g^2+24 d e f g+5 e^2 f^2\right)}{e^2}+\frac{4 d^2 \left(7 d^2 g^2+10 d e f g+3 e^2 f^2\right) \log (d-e x)}{e^3}+\frac{1}{3} g x^3 (5 d g+2 e f)+\frac{1}{4} e g^2 x^4","\frac{8 d^3 (d g+e f)^2}{e^3 (d-e x)}+\frac{4 d^2 (d g+e f) (7 d g+3 e f) \log (d-e x)}{e^3}+\frac{x^2 \left(12 d^2 g^2+10 d e f g+e^2 f^2\right)}{2 e}+\frac{d x \left(20 d^2 g^2+24 d e f g+5 e^2 f^2\right)}{e^2}+\frac{1}{3} g x^3 (5 d g+2 e f)+\frac{1}{4} e g^2 x^4",1,"(d*(5*e^2*f^2 + 24*d*e*f*g + 20*d^2*g^2)*x)/e^2 + ((e^2*f^2 + 10*d*e*f*g + 12*d^2*g^2)*x^2)/(2*e) + (g*(2*e*f + 5*d*g)*x^3)/3 + (e*g^2*x^4)/4 - (8*d^3*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (4*d^2*(3*e^2*f^2 + 10*d*e*f*g + 7*d^2*g^2)*Log[d - e*x])/e^3","A",1
560,1,115,107,0.0851613,"\int \frac{(d+e x)^4 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","-\frac{4 d^2 (d g+e f)^2}{e^3 (e x-d)}+\frac{x \left(8 d^2 g^2+8 d e f g+e^2 f^2\right)}{e^2}+\frac{4 d \left(3 d^2 g^2+4 d e f g+e^2 f^2\right) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3}","\frac{4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac{x \left(8 d^2 g^2+8 d e f g+e^2 f^2\right)}{e^2}+\frac{4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3}",1,"((e^2*f^2 + 8*d*e*f*g + 8*d^2*g^2)*x)/e^2 + (g*(e*f + 2*d*g)*x^2)/e + (g^2*x^3)/3 - (4*d^2*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (4*d*(e^2*f^2 + 4*d*e*f*g + 3*d^2*g^2)*Log[d - e*x])/e^3","A",1
561,1,83,78,0.0582972,"\int \frac{(d+e x)^3 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","\frac{2 \left(5 d^2 g^2+6 d e f g+e^2 f^2\right) \log (d-e x)+\frac{4 d (d g+e f)^2}{d-e x}+2 e g x (3 d g+2 e f)+e^2 g^2 x^2}{2 e^3}","\frac{2 d (d g+e f)^2}{e^3 (d-e x)}+\frac{(5 d g+e f) (d g+e f) \log (d-e x)}{e^3}+\frac{g x (3 d g+2 e f)}{e^2}+\frac{g^2 x^2}{2 e}",1,"(2*e*g*(2*e*f + 3*d*g)*x + e^2*g^2*x^2 + (4*d*(e*f + d*g)^2)/(d - e*x) + 2*(e^2*f^2 + 6*d*e*f*g + 5*d^2*g^2)*Log[d - e*x])/(2*e^3)","A",1
562,1,46,50,0.0486453,"\int \frac{(d+e x)^2 (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)^2*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","\frac{\frac{(d g+e f)^2}{d-e x}+2 g (d g+e f) \log (d-e x)+e g^2 x}{e^3}","\frac{(d g+e f)^2}{e^3 (d-e x)}+\frac{2 g (d g+e f) \log (d-e x)}{e^3}+\frac{g^2 x}{e^2}",1,"(e*g^2*x + (e*f + d*g)^2/(d - e*x) + 2*g*(e*f + d*g)*Log[d - e*x])/e^3","A",1
563,1,91,86,0.0493488,"\int \frac{(d+e x) (f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[((d + e*x)*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]","\frac{(d-e x) \left(3 d^2 g^2+2 d e f g-e^2 f^2\right) \log (d-e x)+(d-e x) (e f-d g)^2 \log (d+e x)+2 d (d g+e f)^2}{4 d^2 e^3 (d-e x)}","\frac{(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}-\frac{(e f-3 d g) (d g+e f) \log (d-e x)}{4 d^2 e^3}+\frac{(d g+e f)^2}{2 d e^3 (d-e x)}",1,"(2*d*(e*f + d*g)^2 + (-(e^2*f^2) + 2*d*e*f*g + 3*d^2*g^2)*(d - e*x)*Log[d - e*x] + (e*f - d*g)^2*(d - e*x)*Log[d + e*x])/(4*d^2*e^3*(d - e*x))","A",1
564,1,85,74,0.0382126,"\int \frac{(f+g x)^2}{\left(d^2-e^2 x^2\right)^2} \, dx","Integrate[(f + g*x)^2/(d^2 - e^2*x^2)^2,x]","\frac{-2 d^2 f g-d^2 g^2 x-e^2 f^2 x}{2 d^2 e^2 \left(e^2 x^2-d^2\right)}-\frac{\left(d^2 g^2-e^2 f^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)}{2 d^3 e^3}","\frac{(e f-d g) (d g+e f) \tanh ^{-1}\left(\frac{e x}{d}\right)}{2 d^3 e^3}+\frac{(f+g x) \left(d^2 g+e^2 f x\right)}{2 d^2 e^2 \left(d^2-e^2 x^2\right)}",1,"(-2*d^2*f*g - e^2*f^2*x - d^2*g^2*x)/(2*d^2*e^2*(-d^2 + e^2*x^2)) - ((-(e^2*f^2) + d^2*g^2)*ArcTanh[(e*x)/d])/(2*d^3*e^3)","A",1
565,1,139,121,0.1142116,"\int \frac{(f+g x)^2}{(d+e x) \left(d^2-e^2 x^2\right)^2} \, dx","Integrate[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)^2),x]","\frac{\frac{4 d \left(d^2 g^2-e^2 f^2\right)}{d+e x}+\left(d^2 g^2-2 d e f g-3 e^2 f^2\right) \log (d-e x)+\left(-d^2 g^2+2 d e f g+3 e^2 f^2\right) \log (d+e x)-\frac{2 d^2 (e f-d g)^2}{(d+e x)^2}+\frac{2 d (d g+e f)^2}{d-e x}}{16 d^4 e^3}","\frac{(3 e f-d g) (d g+e f) \tanh ^{-1}\left(\frac{e x}{d}\right)}{8 d^4 e^3}+\frac{(d g+e f)^2}{8 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}",1,"((2*d*(e*f + d*g)^2)/(d - e*x) - (2*d^2*(e*f - d*g)^2)/(d + e*x)^2 + (4*d*(-(e^2*f^2) + d^2*g^2))/(d + e*x) + (-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + (3*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(16*d^4*e^3)","A",1
566,1,171,146,0.0949905,"\int \frac{(f+g x)^2}{(d+e x)^2 \left(d^2-e^2 x^2\right)^2} \, dx","Integrate[(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^2),x]","\frac{2 d \left(2 d^5 g^2+2 d^4 e g (f+2 g x)+d^3 e^2 f (g x-4 f)+d^2 e^3 f x (f+6 g x)+3 d e^4 f x^2 (2 f+g x)+3 e^5 f^2 x^3\right)+3 e f (e x-d) (d+e x)^3 (d g+e f) \log (d-e x)+3 e f (d-e x) (d+e x)^3 (d g+e f) \log (d+e x)}{24 d^5 e^3 (d-e x) (d+e x)^3}","\frac{f (d g+e f) \tanh ^{-1}\left(\frac{e x}{d}\right)}{4 d^5 e^2}+\frac{(d g+e f)^2}{16 d^4 e^3 (d-e x)}-\frac{(3 e f-d g) (d g+e f)}{16 d^4 e^3 (d+e x)}-\frac{(e f-d g)^2}{12 d^2 e^3 (d+e x)^3}-\frac{e^2 f^2-d^2 g^2}{8 d^3 e^3 (d+e x)^2}",1,"(2*d*(2*d^5*g^2 + 3*e^5*f^2*x^3 + d^3*e^2*f*(-4*f + g*x) + 3*d*e^4*f*x^2*(2*f + g*x) + 2*d^4*e*g*(f + 2*g*x) + d^2*e^3*f*x*(f + 6*g*x)) + 3*e*f*(e*f + d*g)*(-d + e*x)*(d + e*x)^3*Log[d - e*x] + 3*e*f*(e*f + d*g)*(d - e*x)*(d + e*x)^3*Log[d + e*x])/(24*d^5*e^3*(d - e*x)*(d + e*x)^3)","A",1
567,1,195,178,0.1440433,"\int \frac{(f+g x)^2}{(d+e x)^3 \left(d^2-e^2 x^2\right)^2} \, dx","Integrate[(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)^2),x]","\frac{-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d^2 \left(d^2 g^2-2 d e f g-3 e^2 f^2\right)}{(d+e x)^2}-3 \left(d^2 g^2+6 d e f g+5 e^2 f^2\right) \log (d-e x)+3 \left(d^2 g^2+6 d e f g+5 e^2 f^2\right) \log (d+e x)+\frac{16 d^3 \left(d^2 g^2-e^2 f^2\right)}{(d+e x)^3}+\frac{6 d (d g+e f)^2}{d-e x}-\frac{24 d e f (d g+e f)}{d+e x}}{192 d^6 e^3}","\frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left(\frac{e x}{d}\right)}{32 d^6 e^3}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}",1,"((6*d*(e*f + d*g)^2)/(d - e*x) - (12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (16*d^3*(-(e^2*f^2) + d^2*g^2))/(d + e*x)^3 + (6*d^2*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^2 - (24*d*e*f*(e*f + d*g))/(d + e*x) - 3*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*Log[d + e*x])/(192*d^6*e^3)","A",1
568,1,229,210,0.1821193,"\int \frac{(f+g x)^2}{(d+e x)^4 \left(d^2-e^2 x^2\right)^2} \, dx","Integrate[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2),x]","\frac{-\frac{48 d^5 (e f-d g)^2}{(d+e x)^5}-\frac{15 d \left(d^2 g^2+6 d e f g+5 e^2 f^2\right)}{d+e x}-15 \left(d^2 g^2+4 d e f g+3 e^2 f^2\right) \log (d-e x)+15 \left(d^2 g^2+4 d e f g+3 e^2 f^2\right) \log (d+e x)-\frac{60 d^2 e f (d g+e f)}{(d+e x)^2}+\frac{60 d^4 \left(d^2 g^2-e^2 f^2\right)}{(d+e x)^4}+\frac{20 d^3 \left(d^2 g^2-2 d e f g-3 e^2 f^2\right)}{(d+e x)^3}+\frac{15 d (d g+e f)^2}{d-e x}}{960 d^7 e^3}","\frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left(\frac{e x}{d}\right)}{32 d^7 e^3}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}",1,"((15*d*(e*f + d*g)^2)/(d - e*x) - (48*d^5*(e*f - d*g)^2)/(d + e*x)^5 + (60*d^4*(-(e^2*f^2) + d^2*g^2))/(d + e*x)^4 + (20*d^3*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^3 - (60*d^2*e*f*(e*f + d*g))/(d + e*x)^2 - (15*d*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d + e*x) - 15*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d - e*x] + 15*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d + e*x])/(960*d^7*e^3)","A",1
569,1,193,179,0.0998371,"\int \frac{(d+e x)^7 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{x^2 \left(24 d^2 g^2+14 d e f g+e^2 f^2\right)}{2 e}-\frac{d x \left(56 d^2 g^2+48 d e f g+7 e^2 f^2\right)}{e^2}-\frac{8 d^2 \left(13 d^2 g^2+14 d e f g+3 e^2 f^2\right) \log (d-e x)}{e^3}+\frac{32 d^3 \left(2 d^2 g^2+3 d e f g+e^2 f^2\right)}{e^3 (e x-d)}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{1}{4} e g^2 x^4","\frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac{d x \left(56 d^2 g^2+48 d e f g+7 e^2 f^2\right)}{e^2}-\frac{8 d^2 \left(13 d^2 g^2+14 d e f g+3 e^2 f^2\right) \log (d-e x)}{e^3}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac{1}{4} e g^2 x^4",1,"-((d*(7*e^2*f^2 + 48*d*e*f*g + 56*d^2*g^2)*x)/e^2) - ((e^2*f^2 + 14*d*e*f*g + 24*d^2*g^2)*x^2)/(2*e) - (g*(2*e*f + 7*d*g)*x^3)/3 - (e*g^2*x^4)/4 + (8*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)^2) + (32*d^3*(e^2*f^2 + 3*d*e*f*g + 2*d^2*g^2))/(e^3*(-d + e*x)) - (8*d^2*(3*e^2*f^2 + 14*d*e*f*g + 13*d^2*g^2)*Log[d - e*x])/e^3","A",1
570,1,157,149,0.0833432,"\int \frac{(d+e x)^6 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{x \left(18 d^2 g^2+12 d e f g+e^2 f^2\right)}{e^2}+\frac{4 d^2 \left(7 d^2 g^2+10 d e f g+3 e^2 f^2\right)}{e^3 (e x-d)}-\frac{2 d \left(19 d^2 g^2+18 d e f g+3 e^2 f^2\right) \log (d-e x)}{e^3}-\frac{g x^2 (3 d g+e f)}{e}-\frac{g^2 x^3}{3}","\frac{4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{4 d^2 (d g+e f) (7 d g+3 e f)}{e^3 (d-e x)}-\frac{x \left(18 d^2 g^2+12 d e f g+e^2 f^2\right)}{e^2}-\frac{2 d \left(19 d^2 g^2+18 d e f g+3 e^2 f^2\right) \log (d-e x)}{e^3}-\frac{g x^2 (3 d g+e f)}{e}-\frac{g^2 x^3}{3}",1,"-(((e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/e^2) - (g*(e*f + 3*d*g)*x^2)/e - (g^2*x^3)/3 + (4*d^3*(e*f + d*g)^2)/(e^3*(d - e*x)^2) + (4*d^2*(3*e^2*f^2 + 10*d*e*f*g + 7*d^2*g^2))/(e^3*(-d + e*x)) - (2*d*(3*e^2*f^2 + 18*d*e*f*g + 19*d^2*g^2)*Log[d - e*x])/e^3","A",1
571,1,118,118,0.089735,"\int \frac{(d+e x)^5 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^5*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","-\frac{\frac{8 d \left(3 d^2 g^2+4 d e f g+e^2 f^2\right)}{d-e x}+2 \left(13 d^2 g^2+10 d e f g+e^2 f^2\right) \log (d-e x)-\frac{4 d^2 (d g+e f)^2}{(d-e x)^2}+2 e g x (5 d g+2 e f)+e^2 g^2 x^2}{2 e^3}","\frac{2 d^2 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{\left(13 d^2 g^2+10 d e f g+e^2 f^2\right) \log (d-e x)}{e^3}-\frac{4 d (3 d g+e f) (d g+e f)}{e^3 (d-e x)}-\frac{g x (5 d g+2 e f)}{e^2}-\frac{g^2 x^2}{2 e}",1,"-1/2*(2*e*g*(2*e*f + 5*d*g)*x + e^2*g^2*x^2 - (4*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (8*d*(e^2*f^2 + 4*d*e*f*g + 3*d^2*g^2))/(d - e*x) + 2*(e^2*f^2 + 10*d*e*f*g + 13*d^2*g^2)*Log[d - e*x])/e^3","A",1
572,1,93,81,0.0401804,"\int \frac{(d+e x)^4 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{-4 d^3 g^2+4 d^2 e g (g x-f)+2 d e^2 g x (3 f+g x)-2 g (d-e x)^2 (2 d g+e f) \log (d-e x)+e^3 x \left(f^2-g^2 x^2\right)}{e^3 (d-e x)^2}","-\frac{(d g+e f) (5 d g+e f)}{e^3 (d-e x)}+\frac{d (d g+e f)^2}{e^3 (d-e x)^2}-\frac{2 g (2 d g+e f) \log (d-e x)}{e^3}-\frac{g^2 x}{e^2}",1,"(-4*d^3*g^2 + 4*d^2*e*g*(-f + g*x) + 2*d*e^2*g*x*(3*f + g*x) + e^3*x*(f^2 - g^2*x^2) - 2*g*(e*f + 2*d*g)*(d - e*x)^2*Log[d - e*x])/(e^3*(d - e*x)^2)","A",1
573,1,49,61,0.025837,"\int \frac{(d+e x)^3 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{\frac{(d g+e f) (e (f+4 g x)-3 d g)}{(d-e x)^2}-2 g^2 \log (d-e x)}{2 e^3}","-\frac{2 g (d g+e f)}{e^3 (d-e x)}+\frac{(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac{g^2 \log (d-e x)}{e^3}",1,"(((e*f + d*g)*(-3*d*g + e*(f + 4*g*x)))/(d - e*x)^2 - 2*g^2*Log[d - e*x])/(2*e^3)","A",1
574,1,90,88,0.077444,"\int \frac{(d+e x)^2 (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)^2*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{-\frac{2 d (d g+e f) \left(2 d^2 g-d e (2 f+3 g x)+e^2 f x\right)}{(d-e x)^2}+(e f-d g)^2 (-\log (d-e x))+(e f-d g)^2 \log (d+e x)}{8 d^3 e^3}","\frac{(e f-d g)^2 \tanh ^{-1}\left(\frac{e x}{d}\right)}{4 d^3 e^3}+\frac{(e f-3 d g) (d g+e f)}{4 d^2 e^3 (d-e x)}+\frac{(d g+e f)^2}{4 d e^3 (d-e x)^2}",1,"((-2*d*(e*f + d*g)*(2*d^2*g + e^2*f*x - d*e*(2*f + 3*g*x)))/(d - e*x)^2 - (e*f - d*g)^2*Log[d - e*x] + (e*f - d*g)^2*Log[d + e*x])/(8*d^3*e^3)","A",1
575,1,140,122,0.1023229,"\int \frac{(d+e x) (f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[((d + e*x)*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]","\frac{\frac{4 d e^2 f^2-4 d^3 g^2}{d-e x}+\left(d^2 g^2+2 d e f g-3 e^2 f^2\right) \log (d-e x)+\left(-d^2 g^2-2 d e f g+3 e^2 f^2\right) \log (d+e x)+\frac{2 d^2 (d g+e f)^2}{(d-e x)^2}-\frac{2 d (e f-d g)^2}{d+e x}}{16 d^4 e^3}","\frac{(d g+3 e f) (e f-d g) \tanh ^{-1}\left(\frac{e x}{d}\right)}{8 d^4 e^3}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}",1,"((2*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (4*d*e^2*f^2 - 4*d^3*g^2)/(d - e*x) - (2*d*(e*f - d*g)^2)/(d + e*x) + (-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + (3*e^2*f^2 - 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(16*d^4*e^3)","A",1
576,1,110,127,0.0445012,"\int \frac{(f+g x)^2}{\left(d^2-e^2 x^2\right)^3} \, dx","Integrate[(f + g*x)^2/(d^2 - e^2*x^2)^3,x]","\frac{d^5 e g (4 f+g x)+d^3 e^3 x \left(5 f^2+g^2 x^2\right)+\left(d^2-e^2 x^2\right)^2 \left(3 e^2 f^2-d^2 g^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)-3 d e^5 f^2 x^3}{8 d^5 e^3 \left(d^2-e^2 x^2\right)^2}","\frac{(f+g x) \left(d^2 g+e^2 f x\right)}{4 d^2 e^2 \left(d^2-e^2 x^2\right)^2}+\frac{\left(3 e^2 f^2-d^2 g^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)}{8 d^5 e^3}+\frac{x \left(3 e^2 f^2-d^2 g^2\right)+2 d^2 f g}{8 d^4 e^2 \left(d^2-e^2 x^2\right)}",1,"(-3*d*e^5*f^2*x^3 + d^5*e*g*(4*f + g*x) + d^3*e^3*x*(5*f^2 + g^2*x^2) + (3*e^2*f^2 - d^2*g^2)*(d^2 - e^2*x^2)^2*ArcTanh[(e*x)/d])/(8*d^5*e^3*(d^2 - e^2*x^2)^2)","A",1
577,1,197,188,0.1546496,"\int \frac{(f+g x)^2}{(d+e x) \left(d^2-e^2 x^2\right)^3} \, dx","Integrate[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)^3),x]","\frac{-\frac{4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac{3 d^2 \left(d^2 g^2+2 d e f g-3 e^2 f^2\right)}{(d+e x)^2}+\frac{6 d \left(d^2 g^2-3 e^2 f^2\right)}{d+e x}+3 \left(d^2 g^2-2 d e f g-5 e^2 f^2\right) \log (d-e x)+3 \left(-d^2 g^2+2 d e f g+5 e^2 f^2\right) \log (d+e x)+\frac{3 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{12 d e f (d g+e f)}{d-e x}}{96 d^6 e^3}","\frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}+\frac{\left(-d^2 g^2+2 d e f g+5 e^2 f^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)}{16 d^6 e^3}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}",1,"((3*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (12*d*e*f*(e*f + d*g))/(d - e*x) - (4*d^3*(e*f - d*g)^2)/(d + e*x)^3 + (3*d^2*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2))/(d + e*x)^2 + (6*d*(-3*e^2*f^2 + d^2*g^2))/(d + e*x) + 3*(-5*e^2*f^2 - 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(96*d^6*e^3)","A",1
578,1,244,235,0.174301,"\int \frac{(f+g x)^2}{(d+e x)^2 \left(d^2-e^2 x^2\right)^3} \, dx","Integrate[(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^3),x]","\frac{-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{12 d^2 \left(d^2 g^2-3 e^2 f^2\right)}{(d+e x)^2}+\frac{6 d \left(d^2 g^2+6 d e f g+5 e^2 f^2\right)}{d-e x}+\frac{12 d \left(d^2 g^2-2 d e f g-5 e^2 f^2\right)}{d+e x}+3 \left(d^2 g^2-10 d e f g-15 e^2 f^2\right) \log (d-e x)+3 \left(-d^2 g^2+10 d e f g+15 e^2 f^2\right) \log (d+e x)+\frac{6 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{8 d^3 \left(d^2 g^2+2 d e f g-3 e^2 f^2\right)}{(d+e x)^3}}{384 d^7 e^3}","\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac{(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac{(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}+\frac{\left(-d^2 g^2+10 d e f g+15 e^2 f^2\right) \tanh ^{-1}\left(\frac{e x}{d}\right)}{64 d^7 e^3}-\frac{-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac{3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}",1,"((6*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (6*d*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d - e*x) - (12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (8*d^3*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2))/(d + e*x)^3 + (12*d^2*(-3*e^2*f^2 + d^2*g^2))/(d + e*x)^2 + (12*d*(-5*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x) + 3*(-15*e^2*f^2 - 10*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*Log[d + e*x])/(384*d^7*e^3)","A",1
579,1,193,269,0.9678249,"\int \frac{(d+e x)^3 (f+g x)^5}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((d + e*x)^3*(f + g*x)^5)/(d^2 - e^2*x^2)^(7/2),x]","\frac{\sqrt{d^2-e^2 x^2} \left(\frac{2 (2 e f-23 d g) (d g+e f)^4}{d^2 (d-e x)^2}+\frac{2 (d g+e f)^3 \left(127 d^2 g^2-21 d e f g+2 e^2 f^2\right)}{d^3 (d-e x)}+30 g^4 (3 d g+5 e f)+\frac{6 (d g+e f)^5}{d (d-e x)^3}+15 e g^5 x\right)-15 g^3 \left(13 d^2 g^2+30 d e f g+20 e^2 f^2\right) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{30 e^6}","-\frac{g^3 \left(13 d^2 g^2+30 d e f g+20 e^2 f^2\right) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^6}+\frac{g^4 \sqrt{d^2-e^2 x^2} (3 d g+5 e f)}{e^6}+\frac{(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{15 d^2 e^6 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{(d+e x)^3 (d g+e f)^5}{5 d e^6 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{g^5 x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{(d+e x) (d g+e f)^3 \left(127 d^2 g^2-21 d e f g+2 e^2 f^2\right)}{15 d^3 e^6 \sqrt{d^2-e^2 x^2}}",1,"(Sqrt[d^2 - e^2*x^2]*(30*g^4*(5*e*f + 3*d*g) + 15*e*g^5*x + (6*(e*f + d*g)^5)/(d*(d - e*x)^3) + (2*(2*e*f - 23*d*g)*(e*f + d*g)^4)/(d^2*(d - e*x)^2) + (2*(e*f + d*g)^3*(2*e^2*f^2 - 21*d*e*f*g + 127*d^2*g^2))/(d^3*(d - e*x))) - 15*g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^6)","A",1
580,1,168,215,0.7412235,"\int \frac{(d+e x)^3 (f+g x)^4}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((d + e*x)^3*(f + g*x)^4)/(d^2 - e^2*x^2)^(7/2),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(15 d^3 g^4 (d-e x)^3+2 (d-e x)^2 (d g+e f)^2 \left(36 d^2 g^2-8 d e f g+e^2 f^2\right)+3 d^2 (d g+e f)^4+2 d (d-e x) (e f-9 d g) (d g+e f)^3\right)}{d^3 (d-e x)^3}-15 g^3 (3 d g+4 e f) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{15 e^5}","-\frac{g^3 (3 d g+4 e f) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^5}+\frac{2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{15 d^2 e^5 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{(d+e x)^3 (d g+e f)^4}{5 d e^5 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{e^5}+\frac{2 (d+e x) (d g+e f)^2 \left(36 d^2 g^2-8 d e f g+e^2 f^2\right)}{15 d^3 e^5 \sqrt{d^2-e^2 x^2}}",1,"((Sqrt[d^2 - e^2*x^2]*(3*d^2*(e*f + d*g)^4 + 2*d*(e*f - 9*d*g)*(e*f + d*g)^3*(d - e*x) + 2*(e*f + d*g)^2*(e^2*f^2 - 8*d*e*f*g + 36*d^2*g^2)*(d - e*x)^2 + 15*d^3*g^4*(d - e*x)^3))/(d^3*(d - e*x)^3) - 15*g^3*(4*e*f + 3*d*g)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^5)","A",1
581,1,182,183,0.806618,"\int \frac{(d+e x)^3 (f+g x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(\sqrt{1-\frac{e^2 x^2}{d^2}} (d g+e f) \left(22 d^4 g^2-d^3 e g (16 f+51 g x)+d^2 e^2 \left(7 f^2+33 f g x+32 g^2 x^2\right)-d e^3 f x (6 f+11 g x)+2 e^4 f^2 x^2\right)-15 d^2 g^3 (d-e x)^3 \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{15 d^3 e^4 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{(d+e x)^3 (d g+e f)^3}{5 d e^4 \left(d^2-e^2 x^2\right)^{5/2}}-\frac{g^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^4}+\frac{(d+e x) (d g+e f) \left(32 d^2 g^2-11 d e f g+2 e^2 f^2\right)}{15 d^3 e^4 \sqrt{d^2-e^2 x^2}}",1,"((d + e*x)*((e*f + d*g)*Sqrt[1 - (e^2*x^2)/d^2]*(22*d^4*g^2 + 2*e^4*f^2*x^2 - d*e^3*f*x*(6*f + 11*g*x) - d^3*e*g*(16*f + 51*g*x) + d^2*e^2*(7*f^2 + 33*f*g*x + 32*g^2*x^2)) - 15*d^2*g^3*(d - e*x)^3*ArcSin[(e*x)/d]))/(15*d^3*e^4*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])","A",1
582,1,110,145,0.3928825,"\int \frac{(d+e x)^3 (f+g x)^2}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(2 d^4 g^2-6 d^3 e g (f+g x)+d^2 e^2 \left(7 f^2+18 f g x+7 g^2 x^2\right)-6 d e^3 f x (f+g x)+2 e^4 f^2 x^2\right)}{15 d^3 e^3 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^3 (d g+e f)^2}{5 d e^3 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{(d+e x) \left(7 d^2 g^2-6 d e f g+2 e^2 f^2\right)}{15 d^3 e^3 \sqrt{d^2-e^2 x^2}}",1,"((d + e*x)*(2*d^4*g^2 + 2*e^4*f^2*x^2 - 6*d^3*e*g*(f + g*x) - 6*d*e^3*f*x*(f + g*x) + d^2*e^2*(7*f^2 + 18*f*g*x + 7*g^2*x^2)))/(15*d^3*e^3*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
583,1,83,117,0.2373073,"\int \frac{(d+e x)^3 (f+g x)}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[((d + e*x)^3*(f + g*x))/(d^2 - e^2*x^2)^(7/2),x]","-\frac{(d+e x) \left(3 d^3 g-d^2 e (7 f+9 g x)+3 d e^2 x (2 f+g x)-2 e^3 f x^2\right)}{15 d^3 e^2 (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{(d+e x)^3 (d g+e f)}{5 d e^2 \left(d^2-e^2 x^2\right)^{5/2}}+\frac{2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left(d^2-e^2 x^2\right)^{3/2}}+\frac{x (2 e f-3 d g)}{15 d^3 e \sqrt{d^2-e^2 x^2}}",1,"-1/15*((d + e*x)*(3*d^3*g - 2*e^3*f*x^2 + 3*d*e^2*x*(2*f + g*x) - d^2*e*(7*f + 9*g*x)))/(d^3*e^2*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
584,1,58,103,0.0596357,"\int \frac{(d+e x)^3}{\left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(7/2),x]","\frac{(d+e x) \left(7 d^2-6 d e x+2 e^2 x^2\right)}{15 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}","\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}",1,"((d + e*x)*(7*d^2 - 6*d*e*x + 2*e^2*x^2))/(15*d^3*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])","A",1
585,1,225,242,0.3943146,"\int \frac{(d+e x)^3}{(f+g x) \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/((f + g*x)*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\frac{(d+e x) \left(d^2 g^2-e^2 f^2\right) \left(32 d^4 g^2+3 d^3 e g (8 f-17 g x)+d^2 e^2 \left(7 f^2-27 f g x+22 g^2 x^2\right)+3 d e^3 f x (3 g x-2 f)+2 e^4 f^2 x^2\right)}{d^3 (d-e x)^2 \sqrt{d^2-e^2 x^2}}-15 g^3 \sqrt{e^2 f^2-d^2 g^2} \tan ^{-1}\left(\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right)}{15 (d g-e f) (d g+e f)^4}","\frac{g^3 \tan ^{-1}\left(\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right)}{(d g+e f)^3 \sqrt{e^2 f^2-d^2 g^2}}-\frac{5 d (e f-d g)-e x (11 d g+e f)}{15 d \left(d^2-e^2 x^2\right)^{3/2} (d g+e f)^2}+\frac{4 d (d+e x)}{5 \left(d^2-e^2 x^2\right)^{5/2} (d g+e f)}+\frac{15 d^3 g^2+e x \left(22 d^2 g^2+9 d e f g+2 e^2 f^2\right)}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^3}",1,"(((-(e^2*f^2) + d^2*g^2)*(d + e*x)*(32*d^4*g^2 + 2*e^4*f^2*x^2 + 3*d^3*e*g*(8*f - 17*g*x) + 3*d*e^3*f*x*(-2*f + 3*g*x) + d^2*e^2*(7*f^2 - 27*f*g*x + 22*g^2*x^2)))/(d^3*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]) - 15*g^3*Sqrt[e^2*f^2 - d^2*g^2]*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(15*(-(e*f) + d*g)*(e*f + d*g)^4)","A",1
586,1,341,311,0.6101583,"\int \frac{(d+e x)^3}{(f+g x)^2 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/((f + g*x)^2*(d^2 - e^2*x^2)^(7/2)),x]","\frac{15 e g^3 (4 e f-3 d g) \sqrt{e^2 f^2-d^2 g^2} \tan ^{-1}\left(\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right)+\frac{(d+e x) \left(e^2 f^2-d^2 g^2\right) \left(15 d^6 g^4-9 d^5 e g^3 (8 f+13 g x)+d^4 e^2 g^2 \left(38 f^2+164 f g x+171 g^2 x^2\right)-3 d^3 e^3 g \left(-9 f^3+19 f^2 g x+47 f g^2 x^2+24 g^3 x^3\right)+d^2 e^4 f \left(7 f^3-29 f^2 g x+7 f g^2 x^2+43 g^3 x^3\right)+6 d e^5 f^2 x \left(-f^2+f g x+2 g^2 x^2\right)+2 e^6 f^3 x^2 (f+g x)\right)}{d^3 (d-e x)^2 \sqrt{d^2-e^2 x^2} (f+g x)}}{15 (e f-d g)^2 (d g+e f)^5}","\frac{e g^3 (4 e f-3 d g) \tan ^{-1}\left(\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right)}{(e f-d g) (d g+e f)^4 \sqrt{e^2 f^2-d^2 g^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac{e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left(d^2-e^2 x^2\right)^{3/2} (d g+e f)^3}+\frac{4 d e (d+e x)}{5 \left(d^2-e^2 x^2\right)^{5/2} (d g+e f)^2}+\frac{e \left(45 d^3 g^2+e x \left(57 d^2 g^2+14 d e f g+2 e^2 f^2\right)\right)}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^4}",1,"(((e^2*f^2 - d^2*g^2)*(d + e*x)*(15*d^6*g^4 + 2*e^6*f^3*x^2*(f + g*x) - 9*d^5*e*g^3*(8*f + 13*g*x) + 6*d*e^5*f^2*x*(-f^2 + f*g*x + 2*g^2*x^2) + d^4*e^2*g^2*(38*f^2 + 164*f*g*x + 171*g^2*x^2) - 3*d^3*e^3*g*(-9*f^3 + 19*f^2*g*x + 47*f*g^2*x^2 + 24*g^3*x^3) + d^2*e^4*f*(7*f^3 - 29*f^2*g*x + 7*f*g^2*x^2 + 43*g^3*x^3)))/(d^3*(d - e*x)^2*(f + g*x)*Sqrt[d^2 - e^2*x^2]) + 15*e*g^3*(4*e*f - 3*d*g)*Sqrt[e^2*f^2 - d^2*g^2]*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2])])/(15*(e*f - d*g)^2*(e*f + d*g)^5)","A",1
587,1,387,398,1.1414743,"\int \frac{(d+e x)^3}{(f+g x)^3 \left(d^2-e^2 x^2\right)^{7/2}} \, dx","Integrate[(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)),x]","\frac{\sqrt{d^2-e^2 x^2} \left(\frac{2 e^2 (d g+e f) (17 d g+2 e f)}{d^2 (d-e x)^2}+\frac{2 e^2 \left(107 d^2 g^2+19 d e f g+2 e^2 f^2\right)}{d^3 (d-e x)}+\frac{6 e^2 (d g+e f)^2}{d (d-e x)^3}+\frac{45 e g^4 (3 e f-2 d g)}{(f+g x) (e f-d g)^2}+\frac{15 g^4 (d g+e f)}{(f+g x)^2 (e f-d g)}\right)-\frac{15 i e^2 g^3 \left(13 d^2 g^2-30 d e f g+20 e^2 f^2\right) \log \left(\frac{4 (e f-d g)^2 (d g+e f)^5 \left(\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}+i d^2 g+i e^2 f x\right)}{e^2 g^2 (f+g x) \sqrt{e^2 f^2-d^2 g^2} \left(13 d^2 g^2-30 d e f g+20 e^2 f^2\right)}\right)}{(e f-d g)^2 \sqrt{e^2 f^2-d^2 g^2}}}{30 (d g+e f)^5}","\frac{e^2 g^3 \left(13 d^2 g^2-30 d e f g+20 e^2 f^2\right) \tan ^{-1}\left(\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right)}{2 (e f-d g)^2 (d g+e f)^5 \sqrt{e^2 f^2-d^2 g^2}}+\frac{3 e g^4 \sqrt{d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac{e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left(d^2-e^2 x^2\right)^{3/2} (d g+e f)^4}+\frac{4 d e^2 (d+e x)}{5 \left(d^2-e^2 x^2\right)^{5/2} (d g+e f)^3}+\frac{e^2 \left(90 d^3 g^2+e x \left(107 d^2 g^2+19 d e f g+2 e^2 f^2\right)\right)}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^5}",1,"(Sqrt[d^2 - e^2*x^2]*((6*e^2*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e^2*(e*f + d*g)*(2*e*f + 17*d*g))/(d^2*(d - e*x)^2) + (2*e^2*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2))/(d^3*(d - e*x)) + (15*g^4*(e*f + d*g))/((e*f - d*g)*(f + g*x)^2) + (45*e*g^4*(3*e*f - 2*d*g))/((e*f - d*g)^2*(f + g*x))) - ((15*I)*e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*Log[(4*(e*f - d*g)^2*(e*f + d*g)^5*(I*d^2*g + I*e^2*f*x + Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2]))/(e^2*g^2*Sqrt[e^2*f^2 - d^2*g^2]*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*(f + g*x))])/((e*f - d*g)^2*Sqrt[e^2*f^2 - d^2*g^2]))/(30*(e*f + d*g)^5)","C",1
588,1,91,112,0.0889544,"\int \frac{a+c x^2}{(d+e x)^{3/2} (f+g x)} \, dx","Integrate[(a + c*x^2)/((d + e*x)^(3/2)*(f + g*x)),x]","\frac{2 c (e f-d g) (2 d g+e (f+g x))-2 e^2 \left(a g^2+c f^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{g (d+e x)}{d g-e f}\right)}{e^2 g^2 \sqrt{d+e x} (e f-d g)}","-\frac{2 \left(a e^2+c d^2\right)}{e^2 \sqrt{d+e x} (e f-d g)}-\frac{2 \left(a g^2+c f^2\right) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)}{g^{3/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^2 g}",1,"(2*c*(e*f - d*g)*(2*d*g + e*(f + g*x)) - 2*e^2*(c*f^2 + a*g^2)*Hypergeometric2F1[-1/2, 1, 1/2, (g*(d + e*x))/(-(e*f) + d*g)])/(e^2*g^2*(e*f - d*g)*Sqrt[d + e*x])","C",1
589,1,207,240,0.2448594,"\int \frac{(d+e x)^3 \left(a+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^3*(a + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(495 e (f+g x)^3 \left(a e^2 g^2+c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)-693 (f+g x)^2 (e f-d g) \left(3 a e^2 g^2+c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)-3465 \left(a g^2+c f^2\right) (e f-d g)^3+1155 (f+g x) (e f-d g)^2 \left(3 a e g^2+c f (5 e f-2 d g)\right)-385 c e^2 (f+g x)^4 (5 e f-3 d g)+315 c e^3 (f+g x)^5\right)}{3465 g^6}","\frac{2 e (f+g x)^{7/2} \left(a e^2 g^2+c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)}{7 g^6}-\frac{2 (f+g x)^{5/2} (e f-d g) \left(3 a e^2 g^2+c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)}{5 g^6}-\frac{2 \sqrt{f+g x} \left(a g^2+c f^2\right) (e f-d g)^3}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 \left(3 a e g^2+c f (5 e f-2 d g)\right)}{3 g^6}-\frac{2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6}",1,"(2*Sqrt[f + g*x]*(-3465*(e*f - d*g)^3*(c*f^2 + a*g^2) + 1155*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f + g*x) - 693*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^2 + 495*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^3 - 385*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^4 + 315*c*e^3*(f + g*x)^5))/(3465*g^6)","A",1
590,1,149,175,0.149594,"\int \frac{(d+e x)^2 \left(a+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^2*(a + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(63 (f+g x)^2 \left(a e^2 g^2+c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)+315 \left(a g^2+c f^2\right) (e f-d g)^2-210 (f+g x) (e f-d g) \left(a e g^2+c f (2 e f-d g)\right)-90 c e (f+g x)^3 (2 e f-d g)+35 c e^2 (f+g x)^4\right)}{315 g^5}","\frac{2 (f+g x)^{5/2} \left(a e^2 g^2+c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)}{5 g^5}+\frac{2 \sqrt{f+g x} \left(a g^2+c f^2\right) (e f-d g)^2}{g^5}-\frac{4 (f+g x)^{3/2} (e f-d g) \left(a e g^2+c f (2 e f-d g)\right)}{3 g^5}-\frac{4 c e (f+g x)^{7/2} (2 e f-d g)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5}",1,"(2*Sqrt[f + g*x]*(315*(e*f - d*g)^2*(c*f^2 + a*g^2) - 210*(e*f - d*g)*(a*e*g^2 + c*f*(2*e*f - d*g))*(f + g*x) + 63*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 90*c*e*(2*e*f - d*g)*(f + g*x)^3 + 35*c*e^2*(f + g*x)^4))/(315*g^5)","A",1
591,1,94,113,0.0872857,"\int \frac{(d+e x) \left(a+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)*(a + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(35 a g^2 (3 d g-2 e f+e g x)+7 c d g \left(8 f^2-4 f g x+3 g^2 x^2\right)-3 c e \left(16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right)\right)}{105 g^4}","-\frac{2 \sqrt{f+g x} \left(a g^2+c f^2\right) (e f-d g)}{g^4}+\frac{2 (f+g x)^{3/2} \left(a e g^2+c f (3 e f-2 d g)\right)}{3 g^4}-\frac{2 c (f+g x)^{5/2} (3 e f-d g)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4}",1,"(2*Sqrt[f + g*x]*(35*a*g^2*(-2*e*f + 3*d*g + e*g*x) + 7*c*d*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*c*e*(16*f^3 - 8*f^2*g*x + 6*f*g^2*x^2 - 5*g^3*x^3)))/(105*g^4)","A",1
592,1,44,61,0.0260675,"\int \frac{a+c x^2}{\sqrt{f+g x}} \, dx","Integrate[(a + c*x^2)/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(15 a g^2+c \left(8 f^2-4 f g x+3 g^2 x^2\right)\right)}{15 g^3}","\frac{2 \sqrt{f+g x} \left(a g^2+c f^2\right)}{g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}-\frac{4 c f (f+g x)^{3/2}}{3 g^3}",1,"(2*Sqrt[f + g*x]*(15*a*g^2 + c*(8*f^2 - 4*f*g*x + 3*g^2*x^2)))/(15*g^3)","A",1
593,1,92,104,0.1599679,"\int \frac{a+c x^2}{(d+e x) \sqrt{f+g x}} \, dx","Integrate[(a + c*x^2)/((d + e*x)*Sqrt[f + g*x]),x]","\frac{2 c \sqrt{f+g x} (-3 d g-2 e f+e g x)}{3 e^2 g^2}-\frac{2 \left(a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{5/2} \sqrt{e f-d g}}","-\frac{2 \left(a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{5/2} \sqrt{e f-d g}}-\frac{2 c \sqrt{f+g x} (d g+e f)}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}",1,"(2*c*Sqrt[f + g*x]*(-2*e*f - 3*d*g + e*g*x))/(3*e^2*g^2) - (2*(c*d^2 + a*e^2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*Sqrt[e*f - d*g])","A",1
594,1,171,122,0.2883473,"\int \frac{a+c x^2}{(d+e x)^2 \sqrt{f+g x}} \, dx","Integrate[(a + c*x^2)/((d + e*x)^2*Sqrt[f + g*x]),x]","\frac{\frac{\left(a e^2+c d^2\right) \left(\sqrt{e} \sqrt{f+g x} (d g-e f)+g (d+e x) \sqrt{d g-e f} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{d g-e f}}\right)\right)}{(d+e x) (e f-d g)^2}+\frac{4 c d \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{\sqrt{e f-d g}}+\frac{2 c \sqrt{e} \sqrt{f+g x}}{g}}{e^{5/2}}","-\frac{\sqrt{f+g x} \left(a+\frac{c d^2}{e^2}\right)}{(d+e x) (e f-d g)}+\frac{\left(a e^2 g+c d (4 e f-3 d g)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{5/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{f+g x}}{e^2 g}",1,"((2*c*Sqrt[e]*Sqrt[f + g*x])/g + ((c*d^2 + a*e^2)*(Sqrt[e]*(-(e*f) + d*g)*Sqrt[f + g*x] + g*Sqrt[-(e*f) + d*g]*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[-(e*f) + d*g]]))/((e*f - d*g)^2*(d + e*x)) + (4*c*d*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e*f - d*g])/e^(5/2)","A",1
595,1,207,178,0.8156897,"\int \frac{a+c x^2}{(d+e x)^3 \sqrt{f+g x}} \, dx","Integrate[(a + c*x^2)/((d + e*x)^3*Sqrt[f + g*x]),x]","\frac{2 \left(\frac{\sqrt{e} g^2 \sqrt{f+g x} \left(a e^2+c d^2\right) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{e (f+g x)}{e f-d g}\right)}{(d g-e f)^3}-\frac{c d \left(\sqrt{e} \sqrt{f+g x} (d g-e f)+g (d+e x) \sqrt{d g-e f} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{d g-e f}}\right)\right)}{(d+e x) (e f-d g)^2}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{\sqrt{e f-d g}}\right)}{e^{5/2}}","-\frac{\sqrt{f+g x} \left(a+\frac{c d^2}{e^2}\right)}{2 (d+e x)^2 (e f-d g)}-\frac{\left(3 a e^2 g^2+c \left(3 d^2 g^2-8 d e f g+8 e^2 f^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{4 e^{5/2} (e f-d g)^{5/2}}+\frac{\sqrt{f+g x} \left(3 a e^2 g+c d (8 e f-5 d g)\right)}{4 e^2 (d+e x) (e f-d g)^2}",1,"(2*(-((c*d*(Sqrt[e]*(-(e*f) + d*g)*Sqrt[f + g*x] + g*Sqrt[-(e*f) + d*g]*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[-(e*f) + d*g]]))/((e*f - d*g)^2*(d + e*x))) - (c*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e*f - d*g] + (Sqrt[e]*(c*d^2 + a*e^2)*g^2*Sqrt[f + g*x]*Hypergeometric2F1[1/2, 3, 3/2, (e*(f + g*x))/(e*f - d*g)])/(-(e*f) + d*g)^3))/e^(5/2)","C",1
596,1,207,238,0.2403864,"\int \frac{(d+e x)^3 \left(a+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)^3*(a + c*x^2))/(f + g*x)^(3/2),x]","\frac{2 \left(63 e (f+g x)^3 \left(a e^2 g^2+c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)-105 (f+g x)^2 (e f-d g) \left(3 a e^2 g^2+c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)+315 \left(a g^2+c f^2\right) (e f-d g)^3+315 (f+g x) (e f-d g)^2 \left(3 a e g^2+c f (5 e f-2 d g)\right)-45 c e^2 (f+g x)^4 (5 e f-3 d g)+35 c e^3 (f+g x)^5\right)}{315 g^6 \sqrt{f+g x}}","\frac{2 e (f+g x)^{5/2} \left(a e^2 g^2+c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)}{5 g^6}-\frac{2 (f+g x)^{3/2} (e f-d g) \left(3 a e^2 g^2+c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)}{3 g^6}+\frac{2 \left(a g^2+c f^2\right) (e f-d g)^3}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left(3 a e g^2+c f (5 e f-2 d g)\right)}{g^6}-\frac{2 c e^2 (f+g x)^{7/2} (5 e f-3 d g)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6}",1,"(2*(315*(e*f - d*g)^3*(c*f^2 + a*g^2) + 315*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f + g*x) - 105*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^2 + 63*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^3 - 45*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^4 + 35*c*e^3*(f + g*x)^5))/(315*g^6*Sqrt[f + g*x])","A",1
597,1,149,173,0.1538015,"\int \frac{(d+e x)^2 \left(a+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)^2*(a + c*x^2))/(f + g*x)^(3/2),x]","\frac{2 \left(35 (f+g x)^2 \left(a e^2 g^2+c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)-105 \left(a g^2+c f^2\right) (e f-d g)^2-210 (f+g x) (e f-d g) \left(a e g^2+c f (2 e f-d g)\right)-42 c e (f+g x)^3 (2 e f-d g)+15 c e^2 (f+g x)^4\right)}{105 g^5 \sqrt{f+g x}}","\frac{2 (f+g x)^{3/2} \left(a e^2 g^2+c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)}{3 g^5}-\frac{2 \left(a g^2+c f^2\right) (e f-d g)^2}{g^5 \sqrt{f+g x}}-\frac{4 \sqrt{f+g x} (e f-d g) \left(a e g^2+c f (2 e f-d g)\right)}{g^5}-\frac{4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5}",1,"(2*(-105*(e*f - d*g)^2*(c*f^2 + a*g^2) - 210*(e*f - d*g)*(a*e*g^2 + c*f*(2*e*f - d*g))*(f + g*x) + 35*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 42*c*e*(2*e*f - d*g)*(f + g*x)^3 + 15*c*e^2*(f + g*x)^4))/(105*g^5*Sqrt[f + g*x])","A",1
598,1,92,111,0.0828999,"\int \frac{(d+e x) \left(a+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]","\frac{30 a g^2 (-d g+2 e f+e g x)+10 c d g \left(-8 f^2-4 f g x+g^2 x^2\right)+6 c e \left(16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right)}{15 g^4 \sqrt{f+g x}}","\frac{2 \left(a g^2+c f^2\right) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left(a e g^2+c f (3 e f-2 d g)\right)}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4}",1,"(30*a*g^2*(2*e*f - d*g + e*g*x) + 10*c*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 6*c*e*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3))/(15*g^4*Sqrt[f + g*x])","A",1
599,1,43,59,0.0280299,"\int \frac{a+c x^2}{(f+g x)^{3/2}} \, dx","Integrate[(a + c*x^2)/(f + g*x)^(3/2),x]","\frac{2 \left(c \left(-8 f^2-4 f g x+g^2 x^2\right)-3 a g^2\right)}{3 g^3 \sqrt{f+g x}}","-\frac{2 \left(a g^2+c f^2\right)}{g^3 \sqrt{f+g x}}+\frac{2 c (f+g x)^{3/2}}{3 g^3}-\frac{4 c f \sqrt{f+g x}}{g^3}",1,"(2*(-3*a*g^2 + c*(-8*f^2 - 4*f*g*x + g^2*x^2)))/(3*g^3*Sqrt[f + g*x])","A",1
600,1,90,112,0.0680131,"\int \frac{a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx","Integrate[(a + c*x^2)/((d + e*x)*(f + g*x)^(3/2)),x]","-\frac{2 \left(g^2 \left(a e^2+c d^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)+c (e f-d g) (d g+2 e f+e g x)\right)}{e^2 g^2 \sqrt{f+g x} (d g-e f)}","-\frac{2 \left(a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{3/2} (e f-d g)^{3/2}}+\frac{2 \left(a g^2+c f^2\right)}{g^2 \sqrt{f+g x} (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e g^2}",1,"(-2*(c*(e*f - d*g)*(2*e*f + d*g + e*g*x) + (c*d^2 + a*e^2)*g^2*Hypergeometric2F1[-1/2, 1, 1/2, (e*(f + g*x))/(e*f - d*g)]))/(e^2*g^2*(-(e*f) + d*g)*Sqrt[f + g*x])","C",1
601,1,118,144,0.0824087,"\int \frac{a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx","Integrate[(a + c*x^2)/((d + e*x)^2*(f + g*x)^(3/2)),x]","-\frac{2 \left(g^2 \left(a e^2+c d^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)+2 c d g (e f-d g) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)+c (e f-d g)^2\right)}{e^2 g \sqrt{f+g x} (e f-d g)^2}","-\frac{\sqrt{f+g x} \left(a e^2+c d^2\right)}{e (d+e x) (e f-d g)^2}+\frac{\left(3 a e^2 g+c d (4 e f-d g)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left(a g^2+c f^2\right)}{g \sqrt{f+g x} (e f-d g)^2}",1,"(-2*(c*(e*f - d*g)^2 + 2*c*d*g*(e*f - d*g)*Hypergeometric2F1[-1/2, 1, 1/2, (e*(f + g*x))/(e*f - d*g)] + (c*d^2 + a*e^2)*g^2*Hypergeometric2F1[-1/2, 2, 1/2, (e*(f + g*x))/(e*f - d*g)]))/(e^2*g*(e*f - d*g)^2*Sqrt[f + g*x])","C",1
602,1,140,214,0.0962223,"\int \frac{a+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx","Integrate[(a + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)),x]","\frac{2 \left(g \left(g \left(a e^2+c d^2\right) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)+2 c d (e f-d g) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)\right)+c (e f-d g)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right)\right)}{e^2 \sqrt{f+g x} (e f-d g)^3}","-\frac{\sqrt{f+g x} \left(a e^2+c d^2\right)}{2 e (d+e x)^2 (e f-d g)^2}-\frac{\left(15 a e^2 g^2+c \left(-d^2 g^2+8 d e f g+8 e^2 f^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{4 e^{3/2} (e f-d g)^{7/2}}+\frac{\sqrt{f+g x} \left(7 a e^2 g+c d (8 e f-d g)\right)}{4 e (d+e x) (e f-d g)^3}+\frac{2 \left(a g^2+c f^2\right)}{\sqrt{f+g x} (e f-d g)^3}",1,"(2*(c*(e*f - d*g)^2*Hypergeometric2F1[-1/2, 1, 1/2, (e*(f + g*x))/(e*f - d*g)] + g*(2*c*d*(e*f - d*g)*Hypergeometric2F1[-1/2, 2, 1/2, (e*(f + g*x))/(e*f - d*g)] + (c*d^2 + a*e^2)*g*Hypergeometric2F1[-1/2, 3, 1/2, (e*(f + g*x))/(e*f - d*g)])))/(e^2*(e*f - d*g)^3*Sqrt[f + g*x])","C",1
603,1,155,147,0.5576103,"\int \frac{a+c x^2}{\sqrt{d+e x} \sqrt{f+g x}} \, dx","Integrate[(a + c*x^2)/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]","\frac{\sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \left(8 a e^2 g^2+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right) \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)+c e \sqrt{g} \sqrt{d+e x} (f+g x) (-3 d g-3 e f+2 e g x)}{4 e^3 g^{5/2} \sqrt{f+g x}}","\frac{\left(8 a e^2 g^2+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{4 e^{5/2} g^{5/2}}-\frac{c \sqrt{d+e x} \sqrt{f+g x} (5 d g+3 e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g}",1,"(c*e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(-3*e*f - 3*d*g + 2*e*g*x) + Sqrt[e*f - d*g]*(8*a*e^2*g^2 + c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(4*e^3*g^(5/2)*Sqrt[f + g*x])","A",1
604,1,66,16,0.0941353,"\int \frac{-1+2 x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx","Integrate[(-1 + 2*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]),x]","\frac{\sqrt{x-1} \left(x \sqrt{1-x^2}-2 \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)\right)}{\sqrt{1-x}}+2 \tanh ^{-1}\left(\sqrt{\frac{x-1}{x+1}}\right)","\sqrt{x-1} x \sqrt{x+1}",1,"(Sqrt[-1 + x]*(x*Sqrt[1 - x^2] - 2*ArcSin[Sqrt[1 - x]/Sqrt[2]]))/Sqrt[1 - x] + 2*ArcTanh[Sqrt[(-1 + x)/(1 + x)]]","C",0
605,1,410,411,2.435893,"\int \frac{(d+e x)^{3/2} \sqrt{f+g x}}{a+c x^2} \, dx","Integrate[((d + e*x)^(3/2)*Sqrt[f + g*x])/(a + c*x^2),x]","\frac{-\frac{\left(\sqrt{-a} c d^2+2 a \sqrt{c} d e+(-a)^{3/2} e^2\right) \sqrt{\sqrt{-a} g-\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{a \sqrt{\sqrt{-a} e-\sqrt{c} d}}+\frac{\left(\sqrt{-a} c d^2-2 a \sqrt{c} d e+(-a)^{3/2} e^2\right) \sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{a \sqrt{\sqrt{-a} e+\sqrt{c} d}}+\sqrt{c} e \sqrt{d+e x} \sqrt{f+g x}+\frac{\sqrt{c} \sqrt{e f-d g} (3 d g+e f) \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)}{\sqrt{g} \sqrt{f+g x}}}{c^{3/2}}","\frac{\left(\frac{a \left(a e^2 g-c d (d g+2 e f)\right)}{\sqrt{c}}-\sqrt{-a} \left(c d^2 f-a e (2 d g+e f)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{a c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\left(\sqrt{-a} \left(c d^2 f-a e (2 d g+e f)\right)+\frac{a \left(a e^2 g-c d (d g+2 e f)\right)}{\sqrt{c}}\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{a c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{e \sqrt{d+e x} \sqrt{f+g x}}{c}+\frac{\sqrt{e} (3 d g+e f) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{c \sqrt{g}}",1,"(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[f + g*x] + (Sqrt[c]*Sqrt[e*f - d*g]*(e*f + 3*d*g)*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(Sqrt[g]*Sqrt[f + g*x]) - ((Sqrt[-a]*c*d^2 + 2*a*Sqrt[c]*d*e + (-a)^(3/2)*e^2)*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]) + ((Sqrt[-a]*c*d^2 - 2*a*Sqrt[c]*d*e + (-a)^(3/2)*e^2)*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(a*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/c^(3/2)","A",1
606,1,339,342,1.2725642,"\int \frac{\sqrt{d+e x} \sqrt{f+g x}}{a+c x^2} \, dx","Integrate[(Sqrt[d + e*x]*Sqrt[f + g*x])/(a + c*x^2),x]","\frac{\frac{\frac{\left(\sqrt{-a} \sqrt{c} d-a e\right) \sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}-\frac{\left(\sqrt{-a} \sqrt{c} d+a e\right) \sqrt{\sqrt{-a} g-\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e-\sqrt{c} d}}}{a}+\frac{2 \sqrt{g} \sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)}{\sqrt{f+g x}}}{c}","\frac{\left(-\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left(\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{c}",1,"((2*Sqrt[g]*Sqrt[e*f - d*g]*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/Sqrt[f + g*x] + (-(((Sqrt[-a]*Sqrt[c]*d + a*e)*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]) + ((Sqrt[-a]*Sqrt[c]*d - a*e)*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e])/a)/c","A",1
607,1,229,240,0.3468484,"\int \frac{\sqrt{f+g x}}{\sqrt{d+e x} \left(a+c x^2\right)} \, dx","Integrate[Sqrt[f + g*x]/(Sqrt[d + e*x]*(a + c*x^2)),x]","\frac{\frac{\sqrt{\sqrt{-a} g-\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e-\sqrt{c} d}}-\frac{\sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}}{\sqrt{-a} \sqrt{c}}","\frac{\sqrt{\sqrt{c} f-\sqrt{-a} g} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} e+\sqrt{c} d}}",1,"((Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e] - (Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e])/(Sqrt[-a]*Sqrt[c])","A",1
608,1,265,351,0.6962998,"\int \frac{\sqrt{f+g x}}{(d+e x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^(3/2)*(a + c*x^2)),x]","-\frac{2 e \sqrt{f+g x}}{\sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{a \sqrt{\sqrt{-a} g-\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{(-a)^{3/2} \left(\sqrt{-a} e-\sqrt{c} d\right)^{3/2}}+\frac{a \sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{3/2} \left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2}}","-\frac{2 e \sqrt{f+g x}}{\sqrt{d+e x} \left(a e^2+c d^2\right)}+\frac{\left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(a e^2+c d^2\right) \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left(-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(a e^2+c d^2\right) \sqrt{\sqrt{-a} g+\sqrt{c} f}}",1,"(-2*e*Sqrt[f + g*x])/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (a*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-a)^(3/2)*(-(Sqrt[c]*d) + Sqrt[-a]*e)^(3/2)) + (a*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-a)^(3/2)*(Sqrt[c]*d + Sqrt[-a]*e)^(3/2))","A",1
609,1,353,613,2.8638878,"\int \frac{\sqrt{f+g x}}{(d+e x)^{5/2} \left(a+c x^2\right)} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^(5/2)*(a + c*x^2)),x]","-\frac{2 e \sqrt{f+g x} \left(a e^3 (f+g x)+c d \left(-6 d^2 g+7 d e f-5 d e g x+6 e^2 f x\right)\right)}{3 (d+e x)^{3/2} \left(a e^2+c d^2\right)^2 (e f-d g)}-\frac{\sqrt{c} \sqrt{\sqrt{-a} g-\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\left(\sqrt{-a} e-\sqrt{c} d\right)^{3/2} \left(\sqrt{-a} \sqrt{c} d+a e\right)}-\frac{\sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \left(\sqrt{-a} \sqrt{c} d-a e\right)}","\frac{e \sqrt{f+g x} \left(-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right)}{\sqrt{-a} \sqrt{d+e x} \left(\sqrt{-a} e+\sqrt{c} d\right) \left(a e^2+c d^2\right) (e f-d g)}-\frac{e \sqrt{f+g x} \left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right)}{\sqrt{-a} \sqrt{d+e x} \left(\sqrt{c} d-\sqrt{-a} e\right) \left(a e^2+c d^2\right) (e f-d g)}+\frac{4 e g \sqrt{f+g x}}{3 \sqrt{d+e x} \left(a e^2+c d^2\right) (e f-d g)}-\frac{2 e \sqrt{f+g x}}{3 (d+e x)^{3/2} \left(a e^2+c d^2\right)}+\frac{\sqrt{c} \left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \left(\sqrt{c} d-\sqrt{-a} e\right)^{3/2} \left(a e^2+c d^2\right) \sqrt{\sqrt{c} f-\sqrt{-a} g}}+\frac{\sqrt{c} \left(a \sqrt{c} (e f-d g)+\sqrt{-a} c d f+\sqrt{-a} a e g\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{a \left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \left(a e^2+c d^2\right) \sqrt{\sqrt{-a} g+\sqrt{c} f}}",1,"(-2*e*Sqrt[f + g*x]*(a*e^3*(f + g*x) + c*d*(7*d*e*f - 6*d^2*g + 6*e^2*f*x - 5*d*e*g*x)))/(3*(c*d^2 + a*e^2)^2*(e*f - d*g)*(d + e*x)^(3/2)) - (Sqrt[c]*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-(Sqrt[c]*d) + Sqrt[-a]*e)^(3/2)*(Sqrt[-a]*Sqrt[c]*d + a*e)) - (Sqrt[c]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*(Sqrt[-a]*Sqrt[c]*d - a*e))","A",1
610,1,339,337,1.0905881,"\int \frac{(d+e x)^{3/2}}{\sqrt{f+g x} \left(a+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a + c*x^2)),x]","\frac{\frac{\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(\sqrt{-a} \sqrt{c} d-a e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} g+\sqrt{c} f}}-\frac{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left(\sqrt{-a} \sqrt{c} d+a e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} g-\sqrt{c} f}}}{a}+\frac{2 (e f-d g)^{3/2} \left(\frac{e (f+g x)}{e f-d g}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)}{\sqrt{g} (f+g x)^{3/2}}}{c}","\frac{\left(-2 \sqrt{-a} \sqrt{c} d e-a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left(2 \sqrt{-a} \sqrt{c} d e-a e^2+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{c \sqrt{g}}",1,"((2*(e*f - d*g)^(3/2)*((e*(f + g*x))/(e*f - d*g))^(3/2)*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(Sqrt[g]*(f + g*x)^(3/2)) + (-((Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*(Sqrt[-a]*Sqrt[c]*d + a*e)*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(Sqrt[-a]*Sqrt[c]*d - a*e)*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*f + Sqrt[-a]*g])/a)/c","A",1
611,1,229,240,0.3340984,"\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \left(a+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*(a + c*x^2)),x]","\frac{\frac{\sqrt{\sqrt{-a} e-\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} g-\sqrt{c} f}}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} g+\sqrt{c} f}}}{\sqrt{-a} \sqrt{c}}","\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f}}",1,"((Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g] - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*f + Sqrt[-a]*g])/(Sqrt[-a]*Sqrt[c])","A",1
612,1,225,230,0.1894381,"\int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \left(a+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*(a + c*x^2)),x]","\frac{-\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \sqrt{\sqrt{-a} g-\sqrt{c} f}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}}{\sqrt{-a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}",1,"(-(ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])]/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g])) - ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])]/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]))/Sqrt[-a]","A",1
613,1,287,354,0.6951378,"\int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x} \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + c*x^2)),x]","\frac{\frac{2 \sqrt{-a} e^2 \sqrt{f+g x}}{\sqrt{d+e x} \left(a e^2+c d^2\right) (d g-e f)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\left(\sqrt{-a} e-\sqrt{c} d\right)^{3/2} \sqrt{\sqrt{-a} g-\sqrt{c} f}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \sqrt{\sqrt{-a} g+\sqrt{c} f}}}{\sqrt{-a}}","-\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left(\sqrt{c} d-\sqrt{-a} e\right) (e f-d g)}+\frac{e \sqrt{f+g x}}{\sqrt{-a} \sqrt{d+e x} \left(\sqrt{-a} e+\sqrt{c} d\right) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \left(\sqrt{c} d-\sqrt{-a} e\right)^{3/2} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \sqrt{\sqrt{-a} g+\sqrt{c} f}}",1,"((2*Sqrt[-a]*e^2*Sqrt[f + g*x])/((c*d^2 + a*e^2)*(-(e*f) + d*g)*Sqrt[d + e*x]) + (Sqrt[c]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-(Sqrt[c]*d) + Sqrt[-a]*e)^(3/2)*Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]))/Sqrt[-a]","A",1
614,1,336,625,0.7319693,"\int \frac{(d+e x)^{3/2}}{(f+g x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^(3/2)*(a + c*x^2)),x]","-\left(\left(\frac{d}{\sqrt{-a}}-\frac{e}{\sqrt{c}}\right) \left(\frac{\sqrt{d+e x}}{\sqrt{f+g x} \left(\sqrt{c} f-\sqrt{-a} g\right)}+\frac{\sqrt{\sqrt{-a} e-\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\left(\sqrt{-a} g-\sqrt{c} f\right)^{3/2}}\right)\right)-\left(\frac{a d}{(-a)^{3/2}}-\frac{e}{\sqrt{c}}\right) \left(\frac{\sqrt{d+e x}}{\sqrt{f+g x} \left(\sqrt{-a} g+\sqrt{c} f\right)}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}\right)","\frac{2 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x} \left(a g^2+c f^2\right)}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{\sqrt{g} \left(a g^2+c f^2\right)}-\frac{\sqrt{e} \left(-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{g} \left(a g^2+c f^2\right)}+\frac{\sqrt{e} \left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{g} \left(a g^2+c f^2\right)}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left(-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left(a g^2+c f^2\right)}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left(a g^2+c f^2\right)}",1,"-((d/Sqrt[-a] - e/Sqrt[c])*(Sqrt[d + e*x]/((Sqrt[c]*f - Sqrt[-a]*g)*Sqrt[f + g*x]) + (Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(-(Sqrt[c]*f) + Sqrt[-a]*g)^(3/2))) - ((a*d)/(-a)^(3/2) - e/Sqrt[c])*(Sqrt[d + e*x]/((Sqrt[c]*f + Sqrt[-a]*g)*Sqrt[f + g*x]) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))","A",1
615,1,265,351,0.6412759,"\int \frac{\sqrt{d+e x}}{(f+g x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^(3/2)*(a + c*x^2)),x]","-\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left(a g^2+c f^2\right)}+\frac{a \sqrt{\sqrt{-a} e-\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{(-a)^{3/2} \left(\sqrt{-a} g-\sqrt{c} f\right)^{3/2}}+\frac{a \sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{(-a)^{3/2} \left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}","-\frac{2 g \sqrt{d+e x}}{\sqrt{f+g x} \left(a g^2+c f^2\right)}+\frac{\left(-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left(a g^2+c f^2\right)}-\frac{\left(\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left(a g^2+c f^2\right)}",1,"(-2*g*Sqrt[d + e*x])/((c*f^2 + a*g^2)*Sqrt[f + g*x]) + (a*Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-a)^(3/2)*(-(Sqrt[c]*f) + Sqrt[-a]*g)^(3/2)) + (a*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((-a)^(3/2)*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))","A",1
616,1,287,354,0.777773,"\int \frac{1}{\sqrt{d+e x} (f+g x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*(f + g*x)^(3/2)*(a + c*x^2)),x]","\frac{\frac{2 \sqrt{-a} g^2 \sqrt{d+e x}}{\sqrt{f+g x} \left(a g^2+c f^2\right) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left(\sqrt{-a} g-\sqrt{c} f\right)^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}}{\sqrt{-a}}","\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left(\sqrt{c} f-\sqrt{-a} g\right) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left(\sqrt{-a} g+\sqrt{c} f\right) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left(\sqrt{c} f-\sqrt{-a} g\right)^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}",1,"((2*Sqrt[-a]*g^2*Sqrt[d + e*x])/((e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x]) + (Sqrt[c]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*(-(Sqrt[c]*f) + Sqrt[-a]*g)^(3/2)) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2)))/Sqrt[-a]","A",1
617,1,521,549,2.0651383,"\int \frac{1}{(d+e x)^{3/2} (f+g x)^{3/2} \left(a+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*(f + g*x)^(3/2)*(a + c*x^2)),x]","\frac{\frac{e}{\sqrt{d+e x} \sqrt{f+g x} \left(\sqrt{-a} e-\sqrt{c} d\right)}+\frac{e}{\sqrt{d+e x} \sqrt{f+g x} \left(\sqrt{-a} e+\sqrt{c} d\right)}+\frac{g \sqrt{d+e x} \left(2 \sqrt{-a} e g+\sqrt{c} (d g+e f)\right)}{\sqrt{f+g x} \left(\sqrt{-a} e+\sqrt{c} d\right) \left(\sqrt{-a} g+\sqrt{c} f\right) (e f-d g)}+\frac{\frac{g \sqrt{d+e x} \left(2 \sqrt{-a} e g-\sqrt{c} (d g+e f)\right)}{\sqrt{f+g x} \left(\sqrt{c} f-\sqrt{-a} g\right) (e f-d g)}+\frac{c (e f-d g) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right)}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left(\sqrt{-a} g-\sqrt{c} f\right)^{3/2}}}{\sqrt{c} d-\sqrt{-a} e}+\frac{c (d g-e f) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}}{\sqrt{-a} (e f-d g)}","-\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left(\sqrt{c} d-\sqrt{-a} e\right) (e f-d g)}+\frac{e}{\sqrt{-a} \sqrt{d+e x} \sqrt{f+g x} \left(\sqrt{-a} e+\sqrt{c} d\right) (e f-d g)}+\frac{g \sqrt{d+e x} \left(2 \sqrt{-a} e g-\sqrt{c} (d g+e f)\right)}{\sqrt{-a} \sqrt{f+g x} \left(\sqrt{c} d-\sqrt{-a} e\right) \left(\sqrt{c} f-\sqrt{-a} g\right) (e f-d g)^2}+\frac{g \sqrt{d+e x} \left(2 \sqrt{-a} e g+\sqrt{c} (d g+e f)\right)}{\sqrt{-a} \sqrt{f+g x} \left(\sqrt{-a} e+\sqrt{c} d\right) \left(\sqrt{-a} g+\sqrt{c} f\right) (e f-d g)^2}+\frac{c \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right)}{\sqrt{-a} \left(\sqrt{c} d-\sqrt{-a} e\right)^{3/2} \left(\sqrt{c} f-\sqrt{-a} g\right)^{3/2}}-\frac{c \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right)}{\sqrt{-a} \left(\sqrt{-a} e+\sqrt{c} d\right)^{3/2} \left(\sqrt{-a} g+\sqrt{c} f\right)^{3/2}}",1,"(e/((-(Sqrt[c]*d) + Sqrt[-a]*e)*Sqrt[d + e*x]*Sqrt[f + g*x]) + e/((Sqrt[c]*d + Sqrt[-a]*e)*Sqrt[d + e*x]*Sqrt[f + g*x]) + (g*(2*Sqrt[-a]*e*g + Sqrt[c]*(e*f + d*g))*Sqrt[d + e*x])/((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + ((g*(2*Sqrt[-a]*e*g - Sqrt[c]*(e*f + d*g))*Sqrt[d + e*x])/((Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + (c*(e*f - d*g)*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*(-(Sqrt[c]*f) + Sqrt[-a]*g)^(3/2)))/(Sqrt[c]*d - Sqrt[-a]*e) + (c*(-(e*f) + d*g)*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/((Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2)))/(Sqrt[-a]*(e*f - d*g))","A",1
618,1,63,65,0.0611741,"\int \frac{\sqrt{x}}{\sqrt{1+x} \left(1+x^2\right)} \, dx","Integrate[Sqrt[x]/(Sqrt[1 + x]*(1 + x^2)),x]","\frac{1}{2} \left(-(-1+i)^{3/2} \tan ^{-1}\left(\sqrt{-1+i} \sqrt{\frac{x}{x+1}}\right)-(1+i)^{3/2} \tanh ^{-1}\left(\sqrt{1+i} \sqrt{\frac{x}{x+1}}\right)\right)","-\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{1-i} \sqrt{x}}{\sqrt{x+1}}\right)-\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{1+i} \sqrt{x}}{\sqrt{x+1}}\right)",1,"(-((-1 + I)^(3/2)*ArcTan[Sqrt[-1 + I]*Sqrt[x/(1 + x)]]) - (1 + I)^(3/2)*ArcTanh[Sqrt[1 + I]*Sqrt[x/(1 + x)]])/2","A",1
619,1,91,80,0.13616,"\int \frac{(f+g x)^2 \sqrt{1-x^2}}{(1-x)^4} \, dx","Integrate[((f + g*x)^2*Sqrt[1 - x^2])/(1 - x)^4,x]","\frac{\sqrt{1-x^2} \left((x+1)^{3/2} \left(f^2 (x-4)+f g (2-8 x)+g^2 (x-4)\right)-20 \sqrt{2} g^2 (x-1) \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1-x}{2}\right)\right)}{15 (x-1)^3 \sqrt{x+1}}","\frac{(x+1)^4 (f+g)^2}{5 \left(1-x^2\right)^{5/2}}+\frac{(x+1)^3 (f-9 g) (f+g)}{15 \left(1-x^2\right)^{3/2}}+\frac{2 g^2 (x+1)}{\sqrt{1-x^2}}-g^2 \sin ^{-1}(x)",1,"(Sqrt[1 - x^2]*((f*g*(2 - 8*x) + f^2*(-4 + x) + g^2*(-4 + x))*(1 + x)^(3/2) - 20*Sqrt[2]*g^2*(-1 + x)*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - x)/2]))/(15*(-1 + x)^3*Sqrt[1 + x])","C",1
620,1,148,107,0.2961953,"\int \frac{\left(1-a^2 x^2\right)^{3/2}}{(1-a x)^2 (c+d x)} \, dx","Integrate[(1 - a^2*x^2)^(3/2)/((1 - a*x)^2*(c + d*x)),x]","-\frac{\frac{i (d-a c)^2 \log \left(\frac{2 d^3 \left(\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}+i a^2 c x+i d\right)}{(d-a c)^2 \sqrt{a^2 c^2-d^2} (c+d x)}\right)}{\sqrt{a^2 c^2-d^2}}+d \sqrt{1-a^2 x^2}+(a c-2 d) \sin ^{-1}(a x)}{d^2}","\frac{(a c-d)^2 \tan ^{-1}\left(\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right)}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2}",1,"-((d*Sqrt[1 - a^2*x^2] + (a*c - 2*d)*ArcSin[a*x] + (I*(-(a*c) + d)^2*Log[(2*d^3*(I*d + I*a^2*c*x + Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2]))/((-(a*c) + d)^2*Sqrt[a^2*c^2 - d^2]*(c + d*x))])/Sqrt[a^2*c^2 - d^2])/d^2)","C",1
621,1,120,107,0.1079955,"\int \frac{(1+a x)^2}{(c+d x) \sqrt{1-a^2 x^2}} \, dx","Integrate[(1 + a*x)^2/((c + d*x)*Sqrt[1 - a^2*x^2]),x]","\frac{(a c-d) \sqrt{a^2 c^2-d^2} \tan ^{-1}\left(\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right)}{d^2 (a c+d)}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-d) \sin ^{-1}(a x)}{d^2}+\frac{\sin ^{-1}(a x)}{d}","\frac{(a c-d)^2 \tan ^{-1}\left(\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right)}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2}",1,"-(Sqrt[1 - a^2*x^2]/d) - ((a*c - d)*ArcSin[a*x])/d^2 + ArcSin[a*x]/d + ((a*c - d)*Sqrt[a^2*c^2 - d^2]*ArcTan[(d + a^2*c*x)/(Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2])])/(d^2*(a*c + d))","A",1
622,1,1034,851,10.4708877,"\int (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]","\frac{2 \sqrt{f+g x} \left(\frac{2 \left(-3 a^2 e^2 (26 e f+231 d g) g^4+9 a c \left(6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right) g^2+c^2 f^2 \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right)\right) \left(c x^2+a\right) g^2}{f+g x}-\left(c x^2+a\right) \left(150 a^2 e^3 g^4-2 a c e \left(\left(-23 f^2+16 g x f+45 g^2 x^2\right) e^2+33 d g (4 f+7 g x) e+495 d^2 g^2\right) g^2+c^2 \left(\left(64 f^4-48 g x f^3+40 g^2 x^2 f^2-35 g^3 x^3 f-315 g^4 x^4\right) e^3-33 d g \left(8 f^3-6 g x f^2+5 g^2 x^2 f+35 g^3 x^3\right) e^2-99 d^2 g^2 \left(-4 f^2+3 g x f+15 g^2 x^2\right) e-231 d^3 g^3 (f+3 g x)\right)\right) g^2+\frac{2 \sqrt{a} \left(\sqrt{c} f+i \sqrt{a} g\right) \left(75 a^2 e^3 g^4-3 i a^{3/2} \sqrt{c} e^2 (e f+231 d g) g^3-3 a c e \left(2 e^2 f^2-33 d e g f+165 d^2 g^2\right) g^2+3 i \sqrt{a} c^{3/2} \left(16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f+231 d^3 g^3\right) g+c^2 f \left(-64 e^3 f^3+264 d e^2 g f^2-396 d^2 e g^2 f+231 d^3 g^3\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} \sqrt{f+g x} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) g}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{2 \sqrt{c} \left(\sqrt{a} g-i \sqrt{c} f\right) \left(-3 a^2 e^2 (26 e f+231 d g) g^4+9 a c \left(6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right) g^2+c^2 f^2 \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} \sqrt{f+g x} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{3465 c^2 g^6 \sqrt{c x^2+a}}","\frac{2 \sqrt{f+g x} \sqrt{c x^2+a} (d+e x)^4}{11 e}+\frac{4 \sqrt{-a} \left(3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left(6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right) g^2-c^2 f^2 \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right)\right) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3465 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{4 \sqrt{-a} \left(c f^2+a g^2\right) \left(75 a^2 e^3 g^4-3 a c e \left(2 e^2 f^2-33 d e g f+165 d^2 g^2\right) g^2-c^2 f \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{c x^2+a}}{99 g^4}+\frac{2 e \left(18 a e^2 g^2-c \left(29 e^2 f^2-96 d e g f+81 d^2 g^2\right)\right) (f+g x)^{5/2} \sqrt{c x^2+a}}{693 c g^4}-\frac{2 \left(2 a e^2 g^2 (74 e f-231 d g)-c \left(233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right)\right) (f+g x)^{3/2} \sqrt{c x^2+a}}{3465 c g^4}-\frac{2 \left(150 a^2 e^4 g^4-6 a c e^2 \left(2 e^2 f^2-33 d e g f+165 d^2 g^2\right) g^2+c^2 \left(187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right)\right) \sqrt{f+g x} \sqrt{c x^2+a}}{3465 c^2 e g^4}",1,"(2*Sqrt[f + g*x]*((2*g^2*(-3*a^2*e^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*(a + c*x^2))/(f + g*x) - g^2*(a + c*x^2)*(150*a^2*e^3*g^4 - 2*a*c*e*g^2*(495*d^2*g^2 + 33*d*e*g*(4*f + 7*g*x) + e^2*(-23*f^2 + 16*f*g*x + 45*g^2*x^2)) + c^2*(-231*d^3*g^3*(f + 3*g*x) - 99*d^2*e*g^2*(-4*f^2 + 3*f*g*x + 15*g^2*x^2) - 33*d*e^2*g*(8*f^3 - 6*f^2*g*x + 5*f*g^2*x^2 + 35*g^3*x^3) + e^3*(64*f^4 - 48*f^3*g*x + 40*f^2*g^2*x^2 - 35*f*g^3*x^3 - 315*g^4*x^4))) + (2*Sqrt[c]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-3*a^2*e^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (2*Sqrt[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*(75*a^2*e^3*g^4 - (3*I)*a^(3/2)*Sqrt[c]*e^2*g^3*(e*f + 231*d*g) - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 264*d*e^2*f^2*g - 396*d^2*e*f*g^2 + 231*d^3*g^3) + (3*I)*Sqrt[a]*c^(3/2)*g*(16*e^3*f^3 - 66*d*e^2*f^2*g + 99*d^2*e*f*g^2 + 231*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(3465*c^2*g^6*Sqrt[a + c*x^2])","C",1
623,1,809,635,7.2606348,"\int (d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]","\frac{\sqrt{f+g x} \left(\frac{2 \left(c x^2+a\right) \left(2 a e (4 e f+30 d g+7 e g x) g^2+c \left(\left(8 f^3-6 g x f^2+5 g^2 x^2 f+35 g^3 x^3\right) e^2+6 d g \left(-4 f^2+3 g x f+15 g^2 x^2\right) e+21 d^2 g^2 (f+3 g x)\right)\right)}{c g^3}-\frac{4 \left(\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(21 a^2 e^2 g^4-3 a c \left(-3 e^2 f^2+16 d e g f+21 d^2 g^2\right) g^2+c^2 f^2 \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right)\right) \left(c x^2+a\right) g^2+\sqrt{a} \sqrt{c} \left(\sqrt{c} f+i \sqrt{a} g\right) \left(21 i a^{3/2} e^2 g^3-3 a \sqrt{c} e (e f-10 d g) g^2-3 i \sqrt{a} c \left(-2 e^2 f^2+6 d e g f+21 d^2 g^2\right) g+c^{3/2} f \left(-8 e^2 f^2+24 d e g f-21 d^2 g^2\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) g-i \sqrt{c} \left(\sqrt{c} f+i \sqrt{a} g\right) \left(21 a^2 e^2 g^4-3 a c \left(-3 e^2 f^2+16 d e g f+21 d^2 g^2\right) g^2+c^2 f^2 \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c^2 g^5 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)}\right)}{315 \sqrt{c x^2+a}}","\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(21 a^2 e^2 g^4+3 a c g^2 \left(-21 d^2 g^2-16 d e f g+3 e^2 f^2\right)+c^2 f^2 \left(21 d^2 g^2-24 d e f g+8 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(3 a e g^2 (e f-10 d g)+c f \left(21 d^2 g^2-24 d e f g+8 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} (f+g x)^{3/2} \left(7 a e^2 g^2-c \left(21 d^2 g^2-24 d e f g+8 e^2 f^2\right)\right)}{315 c g^3}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left(6 a e^2 g^2 (e f-10 d g)-c \left(-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right)\right)}{315 c e g^3}+\frac{2 e \sqrt{a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 e}",1,"(Sqrt[f + g*x]*((2*(a + c*x^2)*(2*a*e*g^2*(4*e*f + 30*d*g + 7*e*g*x) + c*(21*d^2*g^2*(f + 3*g*x) + 6*d*e*g*(-4*f^2 + 3*f*g*x + 15*g^2*x^2) + e^2*(8*f^3 - 6*f^2*g*x + 5*f*g^2*x^2 + 35*g^3*x^3))))/(c*g^3) - (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21*d^2*g^2))*(a + c*x^2) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((21*I)*a^(3/2)*e^2*g^3 - 3*a*Sqrt[c]*e*g^2*(e*f - 10*d*g) + c^(3/2)*f*(-8*e^2*f^2 + 24*d*e*f*g - 21*d^2*g^2) - (3*I)*Sqrt[a]*c*g*(-2*e^2*f^2 + 6*d*e*f*g + 21*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c^2*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(315*Sqrt[a + c*x^2])","C",1
624,1,610,434,4.5779328,"\int (d+e x) \sqrt{f+g x} \sqrt{a+c x^2} \, dx","Integrate[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2],x]","\frac{\sqrt{f+g x} \left(\frac{2 \left(a+c x^2\right) \left(10 a e g^2+7 c d g (f+3 g x)+c e \left(-4 f^2+3 f g x+15 g^2 x^2\right)\right)}{c g^2}+\frac{4 \left(g^2 \left(a+c x^2\right) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right)+i \sqrt{c} (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(c f^2 (7 d g-4 e f)-a g^2 (21 d g+8 e f)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\sqrt{a} g (f+g x)^{3/2} \left(-\sqrt{a} g+i \sqrt{c} f\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(3 \sqrt{a} \sqrt{c} g (7 d g+e f)+5 i a e g^2+i c f (4 e f-7 d g)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c g^4 (f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{105 \sqrt{a+c x^2}}","\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(5 a e g^2+c f (4 e f-7 d g)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(a g^2 (21 d g+8 e f)+c f^2 (4 e f-7 d g)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 \sqrt{c} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left(5 a e g^2-3 c g x (7 d g+e f)+c f (4 e f-7 d g)\right)}{105 c g^2}+\frac{2 e \left(a+c x^2\right)^{3/2} \sqrt{f+g x}}{7 c}",1,"(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e*g^2 + 7*c*d*g*(f + 3*g*x) + c*e*(-4*f^2 + 3*f*g*x + 15*g^2*x^2)))/(c*g^2) + (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*(a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(c*f^2*(-4*e*f + 7*d*g) - a*g^2*(8*e*f + 21*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*g*(I*Sqrt[c]*f - Sqrt[a]*g)*((5*I)*a*e*g^2 + I*c*f*(4*e*f - 7*d*g) + 3*Sqrt[a]*Sqrt[c]*g*(e*f + 7*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2])","C",1
625,1,536,362,2.911913,"\int \sqrt{f+g x} \sqrt{a+c x^2} \, dx","Integrate[Sqrt[f + g*x]*Sqrt[a + c*x^2],x]","\frac{\sqrt{f+g x} \left(\frac{2 \left(a+c x^2\right) (f+3 g x)}{g}-\frac{4 \left(\sqrt{c} (f+g x)^{3/2} \left(-3 a^{3/2} g^3+\sqrt{a} c f^2 g+3 i a \sqrt{c} f g^2-i c^{3/2} f^3\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(-3 a^2 g^2+a c \left(f^2-3 g^2 x^2\right)+c^2 f^2 x^2\right)-\sqrt{a} \sqrt{c} g (f+g x)^{3/2} \left(4 i \sqrt{a} \sqrt{c} f g-3 a g^2+c f^2\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c g^3 (f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{15 \sqrt{a+c x^2}}","-\frac{4 \sqrt{-a} f \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(c f^2-3 a g^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} (f+g x)^{3/2}}{5 g}-\frac{4 f \sqrt{a+c x^2} \sqrt{f+g x}}{15 g}",1,"(Sqrt[f + g*x]*((2*(f + 3*g*x)*(a + c*x^2))/g - (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-3*a^2*g^2 + c^2*f^2*x^2 + a*c*(f^2 - 3*g^2*x^2)) + Sqrt[c]*((-I)*c^(3/2)*f^3 + Sqrt[a]*c*f^2*g + (3*I)*a*Sqrt[c]*f*g^2 - 3*a^(3/2)*g^3)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - Sqrt[a]*Sqrt[c]*g*(c*f^2 + (4*I)*Sqrt[a]*Sqrt[c]*f*g - 3*a*g^2)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(15*Sqrt[a + c*x^2])","C",1
626,1,1216,683,8.9756583,"\int \frac{\sqrt{f+g x} \sqrt{a+c x^2}}{d+e x} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x),x]","\frac{\left(\frac{2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3}{(f+g x)^2}-\frac{4 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{f+g x}-\frac{6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{(f+g x)^2}+2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+\frac{12 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{f+g x}+\frac{2 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{(f+g x)^2}+\frac{2 \sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (e f-3 d g) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}+\frac{2 e \left(3 \sqrt{c} d-i \sqrt{a} e\right) g \left(\sqrt{a} g-i \sqrt{c} f\right) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}+\frac{6 i c d^2 g^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}+\frac{6 i a e^2 g^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}-6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\frac{6 a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{(f+g x)^2}\right) (f+g x)^{3/2}}{3 e^3 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{c (f+g x)^2 \left(\frac{f}{f+g x}-1\right)^2}{g^2}+a}}+\frac{2 \sqrt{c x^2+a} \sqrt{f+g x}}{3 e}","-\frac{2 \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}+\frac{2 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} (e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 e^2 g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (e f-3 d g) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 e^2 g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(2 a e^2 g-3 c d (e f-d g)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 e}",1,"(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*e) + ((f + g*x)^(3/2)*(2*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 6*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (2*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (6*c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (2*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (6*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (4*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (12*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (2*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (2*e*(3*Sqrt[c]*d - I*Sqrt[a]*e)*g*((-I)*Sqrt[c]*f + Sqrt[a]*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*a*e^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(3*e^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f + g*x)^2*(-1 + f/(f + g*x))^2)/g^2])","C",1
627,1,1331,650,7.0273931,"\int \frac{\sqrt{f+g x} \sqrt{a+c x^2}}{(d+e x)^2} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^2,x]","\frac{\sqrt{f+g x} \left(-\frac{\left(c x^2+a\right) e^2}{d+e x}-\frac{-3 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+6 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-3 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f+2 i c d e g \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f-3 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+3 \sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (d g-e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e \left(\sqrt{c} f+i \sqrt{a} g\right) \left(\sqrt{a} e g-i \sqrt{c} (2 e f-3 d g)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-3 i c d^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i a e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+3 a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) (f+g x)}\right)}{e^3 \sqrt{c x^2+a}}","-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g-c d (2 e f-3 d g)\right) \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{e (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{3 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(Sqrt[f + g*x]*(-((e^2*(a + c*x^2))/(d + e*x)) - (-3*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 3*c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 3*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 6*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 6*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 3*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 3*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(Sqrt[c]*f + I*Sqrt[a]*g)*(Sqrt[a]*e*g - I*Sqrt[c]*(2*e*f - 3*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (2*I)*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (3*I)*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*a*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*(f + g*x))))/(e^3*Sqrt[a + c*x^2])","C",1
628,1,2703,1205,11.2286858,"\int \frac{\sqrt{f+g x} \sqrt{a+c x^2}}{(d+e x)^3} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^3,x]","\text{Result too large to show}","\frac{\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right) \left(a e^2 g-c d (2 e f-3 d g)\right)^2}{4 e^3 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\sqrt{-a} \sqrt{c} \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(a e^2 g-c d (2 e f-3 d g)\right)}{4 e^2 \left(c d^2+a e^2\right) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(a e^2 g-c d (2 e f-3 d g)\right)}{4 e^2 \left(c d^2+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(a e^2 g-c d (2 e f-3 d g)\right)}{4 e^3 \left(c d^2+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\sqrt{f+g x} \sqrt{c x^2+a} \left(a e^2 g-c d (2 e f-3 d g)\right)}{4 e \left(c d^2+a e^2\right) (e f-d g) (d+e x)}-\frac{3 \sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{2 e^3 \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{c (e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^3 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\sqrt{f+g x} \sqrt{c x^2+a}}{2 e (d+e x)^2}",1,"Sqrt[f + g*x]*Sqrt[a + c*x^2]*(-1/2*1/(e*(d + e*x)^2) + (2*c*d*e*f - 3*c*d^2*g - a*e^2*g)/(4*e*(c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x))) + ((f + g*x)^(3/2)*(-2*c^2*d*e^3*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 5*c^2*d^2*e^2*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*c*e^4*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*c^2*d^3*e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*c*d*e^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - (2*c^2*d*e^3*f^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (5*c^2*d^2*e^2*f^3*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (a*c*e^4*f^3*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*c^2*d^3*e*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*a*c*d*e^3*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (5*a*c*d^2*e^2*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (a^2*e^4*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*a*c*d^3*e*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (a^2*d*e^3*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (4*c^2*d*e^3*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) - (10*c^2*d^2*e^2*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) - (2*a*c*e^4*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (6*c^2*d^3*e*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (2*a*c*d*e^3*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - d*g)*(a*e^2*g + c*d*(-2*e*f + 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (e*(Sqrt[c]*d - I*Sqrt[a]*e)*g*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(a*e^2*g) - (2*I)*Sqrt[a]*Sqrt[c]*e*(e*f - d*g) + c*d*(-4*e*f + 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((4*I)*a*c*e^4*f^2*g*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - ((4*I)*c^2*d^3*e*f*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - ((12*I)*a*c*d*e^3*f*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c^2*d^4*g^3*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*a*c*d^2*e^2*g^3*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - (I*a^2*e^4*g^3*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(4*e^3*(c*d^2 + a*e^2)*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^2*Sqrt[a + (c*(f + g*x)^2*(-1 + f/(f + g*x))^2)/g^2])","C",1
629,1,864,666,7.8807639,"\int \frac{(d+e x)^3 \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^3*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(-c \left(c x^2+a\right) \left(c \left(\left(64 f^3-48 g x f^2+40 g^2 x^2 f-35 g^3 x^3\right) e^3-27 d g \left(8 f^2-6 g x f+5 g^2 x^2\right) e^2+63 d^2 g^2 (4 f-3 g x) e-105 d^3 g^3\right)-2 a e^2 g^2 (-11 e f+45 d g+7 e g x)\right) g^2-\frac{2 \left(\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(21 a^2 e^3 g^4-3 a c e \left(10 e^2 f^2-39 d e g f+63 d^2 g^2\right) g^2+c^2 f \left(-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right)\right) \left(c x^2+a\right) g^2+\sqrt{a} \sqrt{c} \left(\sqrt{c} f+i \sqrt{a} g\right) \left(21 i a^{3/2} e^3 g^3-9 a \sqrt{c} e^2 (2 e f-5 d g) g^2-3 i \sqrt{a} c e \left(16 e^2 f^2-54 d e g f+63 d^2 g^2\right) g+c^{3/2} \left(64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) g-\sqrt{c} \left(i \sqrt{c} f-\sqrt{a} g\right) \left(21 a^2 e^3 g^4-3 a c e \left(10 e^2 f^2-39 d e g f+63 d^2 g^2\right) g^2+c^2 f \left(-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)}\right)}{315 c^2 g^6 \sqrt{c x^2+a}}","\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(21 a^2 e^3 g^4-3 a c e g^2 \left(63 d^2 g^2-39 d e f g+10 e^2 f^2\right)-c^2 f \left(-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(9 a e^2 g^2 (2 e f-5 d g)-c \left(-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left(7 a e^2 g^2+c \left(42 d^2 g^2-111 d e f g+64 e^2 f^2\right)\right)}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left(9 a e^2 g^2 (2 e f-5 d g)+c \left(-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right)\right)}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g}",1,"(2*Sqrt[f + g*x]*(-(c*g^2*(a + c*x^2)*(-2*a*e^2*g^2*(-11*e*f + 45*d*g + 7*e*g*x) + c*(-105*d^3*g^3 + 63*d^2*e*g^2*(4*f - 3*g*x) - 27*d*e^2*g*(8*f^2 - 6*f*g*x + 5*g^2*x^2) + e^3*(64*f^3 - 48*f^2*g*x + 40*f*g^2*x^2 - 35*g^3*x^3)))) - (2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*(a + c*x^2) - Sqrt[c]*(I*Sqrt[c]*f - Sqrt[a]*g)*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((21*I)*a^(3/2)*e^3*g^3 - 9*a*Sqrt[c]*e^2*g^2*(2*e*f - 5*d*g) - (3*I)*Sqrt[a]*c*e*g*(16*e^2*f^2 - 54*d*e*f*g + 63*d^2*g^2) + c^(3/2)*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(315*c^2*g^6*Sqrt[a + c*x^2])","C",1
630,1,712,508,5.2601798,"\int \frac{(d+e x)^2 \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^2*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(g^2 \left(a+c x^2\right) \left(10 a e^2 g^2+c \left(35 d^2 g^2+14 d e g (3 g x-4 f)+3 e^2 \left(8 f^2-6 f g x+5 g^2 x^2\right)\right)\right)-\frac{2 \left(g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(a^2 e g^2 (13 e f-42 d g)+a c \left(35 d^2 f g^2-14 d e g \left(4 f^2+3 g^2 x^2\right)+e^2 \left(24 f^3+13 f g^2 x^2\right)\right)+c^2 f x^2 \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right)+\sqrt{a} g (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(6 i \sqrt{a} \sqrt{c} e g (3 e f-7 d g)+5 a e^2 g^2+c \left(-35 d^2 g^2+56 d e f g-24 e^2 f^2\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i \sqrt{c} (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(a e g^2 (13 e f-42 d g)+c f \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{(f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{105 c g^5 \sqrt{a+c x^2}}","\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(5 a e^2 g^2-c \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(a e g^2 (13 e f-42 d g)+c f \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 \sqrt{c} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left(5 a e^2 g^2+c \left(10 d^2 g^2-34 d e f g+21 e^2 f^2\right)\right)}{105 c g^3}-\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{35 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 g}",1,"(2*Sqrt[f + g*x]*(g^2*(a + c*x^2)*(10*a*e^2*g^2 + c*(35*d^2*g^2 + 14*d*e*g*(-4*f + 3*g*x) + 3*e^2*(8*f^2 - 6*f*g*x + 5*g^2*x^2))) - (2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a^2*e*g^2*(13*e*f - 42*d*g) + c^2*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2)*x^2 + a*c*(35*d^2*f*g^2 - 14*d*e*g*(4*f^2 + 3*g^2*x^2) + e^2*(24*f^3 + 13*f*g^2*x^2))) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*(5*a*e^2*g^2 + (6*I)*Sqrt[a]*Sqrt[c]*e*g*(3*e*f - 7*d*g) + c*(-24*e^2*f^2 + 56*d*e*f*g - 35*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*c*g^5*Sqrt[a + c*x^2])","C",1
631,1,545,364,4.8341173,"\int \frac{(d+e x) \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]","\frac{\sqrt{f+g x} \left(\frac{2 \left(a+c x^2\right) (5 d g-4 e f+3 e g x)}{g^2}+\frac{4 \left(g^2 \left(a+c x^2\right) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left(3 a e g^2+c f (4 e f-5 d g)\right)-\sqrt{c} (f+g x)^{3/2} \left(-\sqrt{a} g+i \sqrt{c} f\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(3 a e g^2+c f (4 e f-5 d g)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\sqrt{a} \sqrt{c} g (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(\sqrt{c} (5 d g-4 e f)+3 i \sqrt{a} e g\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c g^4 (f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{15 \sqrt{a+c x^2}}","\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) (4 e f-5 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 \sqrt{c} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} (-5 d g+4 e f-3 e g x)}{15 g^2}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(3 a e g^2+c f (4 e f-5 d g)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 \sqrt{c} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(Sqrt[f + g*x]*((2*(-4*e*f + 5*d*g + 3*e*g*x)*(a + c*x^2))/g^2 + (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(3*a*e*g^2 + c*f*(4*e*f - 5*d*g))*(a + c*x^2) - Sqrt[c]*(I*Sqrt[c]*f - Sqrt[a]*g)*(3*a*e*g^2 + c*f*(4*e*f - 5*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((3*I)*Sqrt[a]*e*g + Sqrt[c]*(-4*e*f + 5*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(15*Sqrt[a + c*x^2])","C",1
632,1,456,322,1.9860461,"\int \frac{\sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[Sqrt[a + c*x^2]/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(g^2 \left(a+c x^2\right)-\frac{2 \left(f g^2 \left(a+c x^2\right) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\sqrt{a} g (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\sqrt{c} f (f+g x)^{3/2} \left(\sqrt{a} g-i \sqrt{c} f\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{(f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right)}{3 g^3 \sqrt{a+c x^2}}","-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 g}",1,"(2*Sqrt[f + g*x]*(g^2*(a + c*x^2) - (2*(f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a + c*x^2) + Sqrt[c]*f*((-I)*Sqrt[c]*f + Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - Sqrt[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(3*g^3*Sqrt[a + c*x^2])","C",1
633,1,1096,473,4.3107113,"\int \frac{\sqrt{a+c x^2}}{(d+e x) \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + c*x^2]/((d + e*x)*Sqrt[f + g*x]),x]","-\frac{2 \left(-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-2 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f-a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+\sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (d g-e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e \left(i \sqrt{c} d+\sqrt{a} e\right) g \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i c d^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i a e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}\right)}{e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}","-\frac{2 \sqrt{\frac{c x^2}{a}+1} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}+\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (d g+e f) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e^2 g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(-2*(-(c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(I*Sqrt[c]*d + Sqrt[a]*e)*g*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*a*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(e^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
634,1,1336,694,6.40916,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^2 \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + c*x^2]/((d + e*x)^2*Sqrt[f + g*x]),x]","\frac{\sqrt{f+g x} \left(\frac{c x^2+a}{d+e x}-\frac{c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3-2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2-c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2+c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f+2 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f-2 i c d e g \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f+a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f-c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+i \sqrt{c} e \left(\sqrt{c} f+i \sqrt{a} g\right) (d g-e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e \left(\sqrt{c} f+i \sqrt{a} g\right) \left(\sqrt{a} e g+i \sqrt{c} (2 e f-d g)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+i c d^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i a e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{e^2 g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) (f+g x)}\right)}{(d g-e f) \sqrt{c x^2+a}}","-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} (e f-d g)}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g+c d (2 e f-d g)\right) \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) (e f-d g)}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{(d+e x) (e f-d g)}+\frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e \sqrt{a+c x^2} \sqrt{f+g x} (e f-d g)}-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e \sqrt{a+c x^2} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(Sqrt[f + g*x]*((a + c*x^2)/(d + e*x) - (c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 2*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + I*Sqrt[c]*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(-(e*f) + d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(Sqrt[c]*f + I*Sqrt[a]*g)*(Sqrt[a]*e*g + I*Sqrt[c]*(2*e*f - d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (2*I)*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + I*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*a*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*(f + g*x))))/((-(e*f) + d*g)*Sqrt[a + c*x^2])","C",1
635,1,2197,1241,10.6367648,"\int \frac{\sqrt{a+c x^2}}{(d+e x)^3 \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + c*x^2]/((d + e*x)^3*Sqrt[f + g*x]),x]","\text{Result too large to show}","\frac{\sqrt{-a} \sqrt{c} \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(3 a g e^2+c d (2 e f+d g)\right)}{4 e \left(c d^2+a e^2\right) (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(3 a g e^2+c d (2 e f+d g)\right)}{4 e \left(c d^2+a e^2\right) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) \left(3 a g e^2+c d (2 e f+d g)\right)}{4 e^2 \left(c d^2+a e^2\right) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right) \left(3 a g e^2+c d (2 e f+d g)\right)}{4 e^2 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{\sqrt{f+g x} \sqrt{c x^2+a} \left(3 a g e^2+c d (2 e f+d g)\right)}{4 \left(c d^2+a e^2\right) (e f-d g)^2 (d+e x)}+\frac{\sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{2 e^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{c (e f+d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^2 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{\sqrt{f+g x} \sqrt{c x^2+a}}{2 (e f-d g) (d+e x)^2}",1,"(c^2*d^2*f^3 - 3*a*c*e^2*f^3 - (2*c^2*d*e*f^4)/g + (c^2*d^3*f^2*g)/e + a*c*d*e*f^2*g + a*c*d^2*f*g^2 - 3*a^2*e^2*f*g^2 + (a*c*d^3*g^3)/e + 3*a^2*d*e*g^3 - 2*c^2*d^2*f^2*(f + g*x) + 6*a*c*e^2*f^2*(f + g*x) + (4*c^2*d*e*f^3*(f + g*x))/g - (2*c^2*d^3*f*g*(f + g*x))/e - 6*a*c*d*e*f*g*(f + g*x) + c^2*d^2*f*(f + g*x)^2 - 3*a*c*e^2*f*(f + g*x)^2 - (2*c^2*d*e*f^2*(f + g*x)^2)/g + (c^2*d^3*g*(f + g*x)^2)/e + 3*a*c*d*e*g*(f + g*x)^2 - ((e*f - d*g)*(f + g*x)*(a + c*x^2)*(a*e^2*(2*e*f - 5*d*g - 3*e*g*x) - c*d*(3*d^2*g + 2*e^2*f*x + d*e*g*x)))/(d + e*x)^2 + (Sqrt[c]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*(3*a*e^2*g + c*d*(2*e*f + d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((I*Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g)*(-3*a*e^2*g - (6*I)*Sqrt[a]*Sqrt[c]*e*(e*f - d*g) + c*d*(-4*e*f + d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((4*I)*a*c*e^2*f^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((4*I)*c^2*d^3*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) - ((4*I)*a*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((6*I)*a*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - (I*c^2*d^4*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(e^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((3*I)*a^2*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(4*(c*d^2 + a*e^2)*(e*f - d*g)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
636,1,747,531,6.1601612,"\int \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx","Integrate[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]","\frac{2 \sqrt{f+g x} \left(\frac{g \sqrt{f+g x} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(25 a^{3/2} e^3 g^2+\sqrt{a} c e \left(-105 d^2 g^2+42 d e f g-8 e^2 f^2\right)+3 i a \sqrt{c} e^2 g (2 e f-63 d g)+105 i c^{3/2} d^3 g^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{g^2 \left(-a^2 e^2 g^2 (189 d g+19 e f)+a c \left(105 d^3 g^3+105 d^2 e f g^2-21 d e^2 g \left(2 f^2+9 g^2 x^2\right)+e^3 \left(8 f^3-19 f g^2 x^2\right)\right)+c^2 x^2 \left(105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right)\right)}{f+g x}-\left(g^2 \left(a+c x^2\right) \left(25 a e^3 g^2+c e \left(-105 d^2 g^2-21 d e g (f+3 g x)+e^2 \left(4 f^2-3 f g x-15 g^2 x^2\right)\right)\right)\right)+i c \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(c \left(105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right)-a e^2 g^2 (189 d g+19 e f)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{105 c^2 g^4 \sqrt{a+c x^2}}","-\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(25 a e^2 g^2-c \left(105 d^2 g^2-42 d e f g+8 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 c^{5/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(a e^2 g^2 (189 d g+19 e f)-c \left(105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{105 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} \left(25 a e^2 g^2+c \left(-90 d^2 g^2+12 d e f g+7 e^2 f^2\right)\right)}{105 c^2 g^2}+\frac{2 e^2 \sqrt{a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac{2 e \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 c}",1,"(2*Sqrt[f + g*x]*(-(g^2*(a + c*x^2)*(25*a*e^3*g^2 + c*e*(-105*d^2*g^2 - 21*d*e*g*(f + 3*g*x) + e^2*(4*f^2 - 3*f*g*x - 15*g^2*x^2)))) + (g^2*(-(a^2*e^2*g^2*(19*e*f + 189*d*g)) + c^2*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3)*x^2 + a*c*(105*d^2*e*f*g^2 + 105*d^3*g^3 - 21*d*e^2*g*(2*f^2 + 9*g^2*x^2) + e^3*(8*f^3 - 19*f*g^2*x^2))))/(f + g*x) + I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-(a*e^2*g^2*(19*e*f + 189*d*g)) + c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (g*(Sqrt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(3/2)*d^3*g^2 + 25*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2 + 42*d*e*f*g - 105*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(105*c^2*g^4*Sqrt[a + c*x^2])","C",1
637,1,596,410,4.2967994,"\int \frac{(d+e x)^2 \sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx","Integrate[((d + e*x)^2*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]","\frac{2 \sqrt{f+g x} \left(\frac{g^2 \left(-9 a^2 e^2 g^2+a c \left(15 d^2 g^2+10 d e f g-\left(e^2 \left(2 f^2+9 g^2 x^2\right)\right)\right)+c^2 x^2 \left(15 d^2 g^2+10 d e f g-2 e^2 f^2\right)\right)}{f+g x}-i c \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(9 a e^2 g^2+c \left(-15 d^2 g^2-10 d e f g+2 e^2 f^2\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\frac{\sqrt{c} g \sqrt{f+g x} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(2 \sqrt{a} \sqrt{c} e (e f-5 d g)-9 i a e^2 g+15 i c d^2 g\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+c e g^2 \left(a+c x^2\right) (10 d g+e (f+3 g x))\right)}{15 c^2 g^3 \sqrt{a+c x^2}}","\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(9 a e^2 g^2+c \left(-15 d^2 g^2-10 d e f g+2 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) (e f-5 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x} (7 d g+e f)}{15 c g}+\frac{2 e \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c}",1,"(2*Sqrt[f + g*x]*(c*e*g^2*(a + c*x^2)*(10*d*g + e*(f + 3*g*x)) + (g^2*(-9*a^2*e^2*g^2 + c^2*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*g^2)*x^2 + a*c*(10*d*e*f*g + 15*d^2*g^2 - e^2*(2*f^2 + 9*g^2*x^2))))/(f + g*x) - I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(9*a*e^2*g^2 + c*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((15*I)*c*d^2*g - (9*I)*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 5*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(15*c^2*g^3*Sqrt[a + c*x^2])","C",1
638,1,464,331,3.3479151,"\int \frac{(d+e x) \sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx","Integrate[((d + e*x)*Sqrt[f + g*x])/Sqrt[a + c*x^2],x]","\frac{2 \sqrt{f+g x} \left(\frac{i c \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (3 d g+e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{g^2}+\frac{\left(a+c x^2\right) (3 d g+e f)}{f+g x}+\frac{i \sqrt{f+g x} \left(3 \sqrt{c} d+i \sqrt{a} e\right) \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+e \left(a+c x^2\right)\right)}{3 c \sqrt{a+c x^2}}","\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \left(a g^2+c f^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 c^{3/2} g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (3 d g+e f) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 \sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 e \sqrt{a+c x^2} \sqrt{f+g x}}{3 c}",1,"(2*Sqrt[f + g*x]*(e*(a + c*x^2) + ((e*f + 3*d*g)*(a + c*x^2))/(f + g*x) + (I*c*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f + 3*d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/g^2 + (I*(3*Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])))/(3*c*Sqrt[a + c*x^2])","C",1
639,1,294,136,0.4703291,"\int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/Sqrt[a + c*x^2],x]","\frac{2 i \sqrt{f+g x} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(\sqrt{a}+i \sqrt{c} x\right)}{\sqrt{a} g-i \sqrt{c} f}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{g \left(\sqrt{c} x+i \sqrt{a}\right)}}}","-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"((2*I)*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*Sqrt[f + g*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[c]*g*Sqrt[(Sqrt[c]*(f + g*x))/(g*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt[a + c*x^2])","C",1
640,1,300,319,1.1815325,"\int \frac{\sqrt{f+g x}}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)*Sqrt[a + c*x^2]),x]","-\frac{2 i \sqrt{f+g x} \sqrt{\frac{g \left(\sqrt{a}+i \sqrt{c} x\right)}{\sqrt{a} g-i \sqrt{c} f}} \left(F\left(i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-\Pi \left(\frac{e \left(f-\frac{i \sqrt{a} g}{\sqrt{c}}\right)}{e f-d g};i \sinh ^{-1}\left(\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{g \left(\sqrt{c} x+i \sqrt{a}\right)}}}","-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{f+g x}}",1,"((-2*I)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*Sqrt[f + g*x]*(EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - EllipticPi[(e*(f - (I*Sqrt[a]*g)/Sqrt[c]))/(e*f - d*g), I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(e*Sqrt[(Sqrt[c]*(f + g*x))/(g*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt[a + c*x^2])","C",1
641,1,1330,698,6.0089064,"\int \frac{\sqrt{f+g x}}{(d+e x)^2 \sqrt{a+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^2*Sqrt[a + c*x^2]),x]","\frac{\sqrt{f+g x} \left(-\frac{\left(c x^2+a\right) e^2}{d+e x}-\frac{-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-2 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f-2 i c d e g \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f-a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+\sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (d g-e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e \left(i \sqrt{c} d+\sqrt{a} e\right) g \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+i c d^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i a e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) (f+g x)}\right)}{\left(a e^3+c d^2 e\right) \sqrt{c x^2+a}}","-\frac{e \sqrt{a+c x^2} \sqrt{f+g x}}{(d+e x) \left(a e^2+c d^2\right)}+\frac{\sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(a e^2+c d^2\right)}-\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{e \sqrt{a+c x^2} \sqrt{f+g x} \left(a e^2+c d^2\right)}-\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \left(a e^2+c d^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g+c d (2 e f-d g)\right) \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(a e^2+c d^2\right)}",1,"(Sqrt[f + g*x]*(-((e^2*(a + c*x^2))/(d + e*x)) - (-(c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(I*Sqrt[c]*d + Sqrt[a]*e)*g*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (2*I)*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + I*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*a*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*(f + g*x))))/((c*d^2*e + a*e^3)*Sqrt[a + c*x^2])","C",1
642,1,2450,1246,10.966926,"\int \frac{\sqrt{f+g x}}{(d+e x)^3 \sqrt{a+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^3*Sqrt[a + c*x^2]),x]","\text{Result too large to show}","-\frac{\left(a g e^2+c d (6 e f-5 d g)\right) \sqrt{f+g x} \sqrt{c x^2+a} e}{4 \left(c d^2+a e^2\right)^2 (e f-d g) (d+e x)}-\frac{\sqrt{f+g x} \sqrt{c x^2+a} e}{2 \left(c d^2+a e^2\right) (d+e x)^2}-\frac{\sqrt{-a} \sqrt{c} \left(a g e^2+c d (6 e f-5 d g)\right) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{4 \left(c d^2+a e^2\right)^2 (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{\sqrt{-a} \sqrt{c} f \left(a g e^2+c d (6 e f-5 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{4 \left(c d^2+a e^2\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{\sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{2 \left(c d^2+a e^2\right) \sqrt{f+g x} \sqrt{c x^2+a} e}-\frac{\sqrt{-a} \sqrt{c} d g \left(a g e^2+c d (6 e f-5 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{4 \left(c d^2+a e^2\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a} e}+\frac{c (e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{\left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right) \sqrt{f+g x} \sqrt{c x^2+a} e}+\frac{\left(a g e^2+c d (6 e f-5 d g)\right) \left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{4 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a} e}",1,"(-11*c^2*d^2*e^2*f^3 + a*c*e^4*f^3 + (6*c^2*d*e^3*f^4)/g + 5*c^2*d^3*e*f^2*g + 5*a*c*d*e^3*f^2*g - 11*a*c*d^2*e^2*f*g^2 + a^2*e^4*f*g^2 + 5*a*c*d^3*e*g^3 - a^2*d*e^3*g^3 + 22*c^2*d^2*e^2*f^2*(f + g*x) - 2*a*c*e^4*f^2*(f + g*x) - (12*c^2*d*e^3*f^3*(f + g*x))/g - 10*c^2*d^3*e*f*g*(f + g*x) + 2*a*c*d*e^3*f*g*(f + g*x) - 11*c^2*d^2*e^2*f*(f + g*x)^2 + a*c*e^4*f*(f + g*x)^2 + (6*c^2*d*e^3*f^2*(f + g*x)^2)/g + 5*c^2*d^3*e*g*(f + g*x)^2 - a*c*d*e^3*g*(f + g*x)^2 - (e^2*(e*f - d*g)*(f + g*x)*(a + c*x^2)*(2*(c*d^2 + a*e^2)*(e*f - d*g) + (a*e^2*g + c*d*(6*e*f - 5*d*g))*(d + e*x)))/(d + e*x)^2 + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - d*g)*(a*e^2*g + c*d*(6*e*f - 5*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + (e*(I*Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g)*(a*e^2*g + (2*I)*Sqrt[a]*Sqrt[c]*e*(e*f - d*g) + c*d*(-4*e*f + 5*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((8*I)*c^2*d^2*e^2*f^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - ((4*I)*a*c*e^4*f^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - ((12*I)*c^2*d^3*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((12*I)*a*c*d*e^3*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((3*I)*c^2*d^4*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - ((10*I)*a*c*d^2*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - (I*a^2*e^4*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(4*e*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
643,1,1440,600,9.5762279,"\int \frac{(f+g x)^{5/2}}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[(f + g*x)^(5/2)/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{2 \sqrt{f+g x} \sqrt{c x^2+a} g^2}{3 c e}+\frac{2 (f+g x)^{3/2} \left(\frac{7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3}{(f+g x)^2}+\frac{3 i c e^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f^2}{\sqrt{f+g x}}-\frac{14 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{f+g x}-\frac{3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2}{(f+g x)^2}-\frac{6 i c d e g \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f}{\sqrt{f+g x}}+7 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+\frac{6 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{f+g x}+\frac{7 a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f}{(f+g x)^2}+\frac{\sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (7 e f-3 d g) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}+\frac{i e \left(\sqrt{c} f+i \sqrt{a} g\right) \left(i \sqrt{a} e g+\sqrt{c} (6 e f-3 d g)\right) \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}+\frac{3 i c d^2 g^2 \sqrt{-\frac{f}{f+g x}-\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \sqrt{-\frac{f}{f+g x}+\frac{i \sqrt{a} g}{\sqrt{c} (f+g x)}+1} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{f+g x}}-3 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}-\frac{3 a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{(f+g x)^2}\right)}{3 c e^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{c (f+g x)^2 \left(\frac{f}{f+g x}-1\right)^2}{g^2}+a}}","\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g^2+c \left(-3 d^2 g^2+6 d e f g-2 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right)|\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right)}{3 c^{3/2} e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 (e f-d g)^2 \sqrt{\frac{g \left(\sqrt{-a}-\sqrt{c} x\right)}{\sqrt{-a} g+\sqrt{c} f}} \sqrt{-\frac{g \left(\sqrt{-a}+\sqrt{c} x\right)}{\sqrt{c} f-\sqrt{-a} g}} \Pi \left(\frac{e \left(f+\frac{\sqrt{-a} g}{\sqrt{c}}\right)}{e f-d g};\sin ^{-1}\left(\sqrt{\frac{c}{c f+\sqrt{-a} \sqrt{c} g}} \sqrt{f+g x}\right)|\frac{\sqrt{c} f+\sqrt{-a} g}{\sqrt{c} f-\sqrt{-a} g}\right)}{e^3 \sqrt{a+c x^2} \sqrt{\frac{c}{\sqrt{-a} \sqrt{c} g+c f}}}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (7 e f-3 d g) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{a \sqrt{c} x}{(-a)^{3/2}}+1}}{\sqrt{2}}\right)|\frac{2 a g}{a g-\sqrt{-a} \sqrt{c} f}\right)}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 g^2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 c e}",1,"(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*c*e) + (2*(f + g*x)^(3/2)*(7*c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (7*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (7*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (3*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (14*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (6*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + (Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(7*e*f - 3*d*g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + (I*e*(Sqrt[c]*f + I*Sqrt[a]*g)*(I*Sqrt[a]*e*g + Sqrt[c]*(6*e*f - 3*d*g))*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*e^2*f^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] - ((6*I)*c*d*e*f*g*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((3*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x]))/(3*c*e^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + (c*(f + g*x)^2*(-1 + f/(f + g*x))^2)/g^2])","C",1
644,1,927,469,1.078729,"\int \frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+c x^2}} \, dx","Integrate[(f + g*x)^(3/2)/((d + e*x)*Sqrt[a + c*x^2]),x]","\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}} \left(-\frac{\sqrt{a} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right) f^2}{i \sqrt{c} d+\sqrt{a} e}+\frac{2 i \sqrt{a} g \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right) f}{\sqrt{c} e}+\frac{2 \sqrt{a} d g \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right) f}{\sqrt{a} e^2+i \sqrt{c} d e}-\frac{i \sqrt{a} d g^2 \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right)}{\sqrt{c} e^2}+\frac{g \sqrt{\frac{g \left(i \sqrt{c} x+\sqrt{a}\right)}{\sqrt{a} g-i \sqrt{c} f}} \left(\sqrt{c} x+i \sqrt{a}\right) \left(\left(\sqrt{c} f+i \sqrt{a} g\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-i \sqrt{a} g F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c e \sqrt{\frac{g \left(\sqrt{a}-i \sqrt{c} x\right)}{i \sqrt{c} f+\sqrt{a} g}}}-\frac{\sqrt{a} d^2 g^2 \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 \sqrt{a} e}{i \sqrt{c} d+\sqrt{a} e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{c} x}{\sqrt{a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{a} g}{i \sqrt{c} f+\sqrt{a} g}\right)}{e^2 \left(i \sqrt{c} d+\sqrt{a} e\right)}\right)}{\sqrt{f+g x} \sqrt{c x^2+a}}","-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{e^2 \sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]*(((2*I)*Sqrt[a]*f*g*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e) - (I*Sqrt[a]*d*g^2*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e^2) + (g*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*(I*Sqrt[a] + Sqrt[c]*x)*((Sqrt[c]*f + I*Sqrt[a]*g)*EllipticE[ArcSin[Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - I*Sqrt[a]*g*EllipticF[ArcSin[Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c*e*Sqrt[(g*(Sqrt[a] - I*Sqrt[c]*x))/(I*Sqrt[c]*f + Sqrt[a]*g)]) - (Sqrt[a]*f^2*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(I*Sqrt[c]*d + Sqrt[a]*e) + (2*Sqrt[a]*d*f*g*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(I*Sqrt[c]*d*e + Sqrt[a]*e^2) - (Sqrt[a]*d^2*g^2*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(e^2*(I*Sqrt[c]*d + Sqrt[a]*e))))/(Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
645,1,625,457,4.40401,"\int \frac{(d+e x)^3}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^3/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\frac{2 \sqrt{f+g x} \left(\frac{\sqrt{c} g \sqrt{f+g x} \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(9 a^{3/2} e^3 g^2+\sqrt{a} c e \left(-45 d^2 g^2+30 d e f g-8 e^2 f^2\right)-i a \sqrt{c} e^2 g (15 d g+2 e f)+15 i c^{3/2} d^3 g^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{e g^2 \left(-9 a^2 e^2 g^2+a c \left(45 d^2 g^2-30 d e f g+e^2 \left(8 f^2-9 g^2 x^2\right)\right)+c^2 x^2 \left(45 d^2 g^2-30 d e f g+8 e^2 f^2\right)\right)}{f+g x}+i c e \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(c \left(45 d^2 g^2-30 d e f g+8 e^2 f^2\right)-9 a e^2 g^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+c e^2 g^2 \left(a+c x^2\right) (15 d g-4 e f+3 e g x)\right)}{15 c^2 g^4 \sqrt{a+c x^2}}","\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left(9 a e^2 g^2-c \left(45 d^2 g^2-30 d e f g+8 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g^2 (7 e f-15 d g)-c \left(-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{15 c^{3/2} g^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{8 e^2 \sqrt{a+c x^2} \sqrt{f+g x} (e f-3 d g)}{15 c g^2}+\frac{2 e^2 \sqrt{a+c x^2} (d+e x) \sqrt{f+g x}}{5 c g}",1,"(2*Sqrt[f + g*x]*(c*e^2*g^2*(-4*e*f + 15*d*g + 3*e*g*x)*(a + c*x^2) + (e*g^2*(-9*a^2*e^2*g^2 + c^2*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2)*x^2 + a*c*(-30*d*e*f*g + 45*d^2*g^2 + e^2*(8*f^2 - 9*g^2*x^2))))/(f + g*x) + I*c*e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-9*a*e^2*g^2 + c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*g*((15*I)*c^(3/2)*d^3*g^2 + 9*a^(3/2)*e^3*g^2 - I*a*Sqrt[c]*e^2*g*(2*e*f + 15*d*g) + Sqrt[a]*c*e*(-8*e^2*f^2 + 30*d*e*f*g - 45*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(15*c^2*g^4*Sqrt[a + c*x^2])","C",1
646,1,473,356,3.5428439,"\int \frac{(d+e x)^2}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)^2/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\frac{2 \sqrt{f+g x} \left(\frac{g \sqrt{f+g x} \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(2 \sqrt{a} \sqrt{c} e (e f-3 d g)-i a e^2 g+3 i c d^2 g\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}-\frac{2 e g^2 \left(a+c x^2\right) (e f-3 d g)}{f+g x}-2 i c e \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-3 d g) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e^2 g^2 \left(a+c x^2\right)\right)}{3 c g^3 \sqrt{a+c x^2}}","-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(g^2 \left(3 c d^2-a e^2\right)+2 c e f (e f-3 d g)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 c^{3/2} g^2 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} (e f-3 d g) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 \sqrt{c} g^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{2 e^2 \sqrt{a+c x^2} \sqrt{f+g x}}{3 c g}",1,"(2*Sqrt[f + g*x]*(e^2*g^2*(a + c*x^2) - (2*e*g^2*(e*f - 3*d*g)*(a + c*x^2))/(f + g*x) - (2*I)*c*e*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - 3*d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (g*((3*I)*c*d^2*g - I*a*e^2*g + 2*Sqrt[a]*Sqrt[c]*e*(e*f - 3*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]))/(3*c*g^3*Sqrt[a + c*x^2])","C",1
647,1,439,288,1.6759832,"\int \frac{d+e x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","-\frac{2 \left(\sqrt{c} g (f+g x)^{3/2} \left(\sqrt{a} e-i \sqrt{c} d\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-e g^2 \left(a+c x^2\right) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}+i \sqrt{c} e (f+g x)^{3/2} \left(\sqrt{c} f+i \sqrt{a} g\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{c g^2 \sqrt{a+c x^2} \sqrt{f+g x} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}","\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{2 \sqrt{-a} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} g \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}",1,"(-2*(-(e*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a + c*x^2)) + I*Sqrt[c]*e*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
648,1,186,136,0.2143885,"\int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\frac{2 i (f+g x) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)}{g \sqrt{a+c x^2} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}","-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{c} \sqrt{a+c x^2} \sqrt{f+g x}}",1,"((2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*Sqrt[a + c*x^2])","C",1
649,1,311,167,0.9122017,"\int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","-\frac{2 i (f+g x) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-\Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{\sqrt{a+c x^2} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)}","-\frac{2 \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right)}",1,"((-2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*Sqrt[a + c*x^2])","C",1
650,1,1349,746,6.7433228,"\int \frac{1}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\frac{\sqrt{f+g x} \left(\frac{2 \left(-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+2 c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2-c e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f-2 c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f-2 i c d e g \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f-a e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+c d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+\sqrt{c} e \left(\sqrt{a} g-i \sqrt{c} f\right) (d g-e f) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\left(\sqrt{c} d-i \sqrt{a} e\right) g \left(\sqrt{a} e g+i \sqrt{c} (e f-2 d g)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+3 i c d^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+i a e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+a d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}\right)}{g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (d g-e f) (f+g x)}-\frac{2 e^2 \left(c x^2+a\right)}{d+e x}\right)}{2 \left(c d^2+a e^2\right) (e f-d g) \sqrt{c x^2+a}}","-\frac{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}{(d+e x) \left(a e^2+c d^2\right) (e f-d g)}+\frac{\sqrt{-a} \sqrt{c} e f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(a e^2+c d^2\right) (e f-d g)}-\frac{\sqrt{-a} \sqrt{c} d g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(a e^2+c d^2\right) (e f-d g)}-\frac{\sqrt{-a} \sqrt{c} e \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \left(a e^2+c d^2\right) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left(a e^2 g-c d (2 e f-3 d g)\right) \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(a e^2+c d^2\right) (e f-d g)}",1,"(Sqrt[f + g*x]*((-2*e^2*(a + c*x^2))/(d + e*x) + (2*(-(c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - c*e^2*f*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*d - I*Sqrt[a]*e)*g*(Sqrt[a]*e*g + I*Sqrt[c]*(e*f - 2*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (2*I)*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (3*I)*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + I*a*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-(e*f) + d*g)*(f + g*x))))/(2*(c*d^2 + a*e^2)*(e*f - d*g)*Sqrt[a + c*x^2])","C",1
651,1,2491,1257,11.7618591,"\int \frac{1}{(d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\text{Result too large to show}","\frac{3 \left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{f+g x} \sqrt{c x^2+a} e^2}{4 \left(c d^2+a e^2\right)^2 (e f-d g)^2 (d+e x)}-\frac{\sqrt{f+g x} \sqrt{c x^2+a} e^2}{2 \left(c d^2+a e^2\right) (e f-d g) (d+e x)^2}+\frac{3 \sqrt{-a} \sqrt{c} \left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) e}{4 \left(c d^2+a e^2\right)^2 (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{3 \sqrt{-a} \sqrt{c} f \left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) e}{4 \left(c d^2+a e^2\right)^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{3 \sqrt{-a} \sqrt{c} d g \left(a e^2 g-c d (2 e f-3 d g)\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{4 \left(c d^2+a e^2\right)^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{\sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{2 \left(c d^2+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{3 \left(a e^2 g-c d (2 e f-3 d g)\right)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{4 \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right)^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{c (e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{\left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) \left(c d^2+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}",1,"(-15*c^2*d^2*e^2*f^3 - 3*a*c*e^4*f^3 + (6*c^2*d*e^3*f^4)/g + 9*c^2*d^3*e*f^2*g + 9*a*c*d*e^3*f^2*g - 15*a*c*d^2*e^2*f*g^2 - 3*a^2*e^4*f*g^2 + 9*a*c*d^3*e*g^3 + 3*a^2*d*e^3*g^3 + 30*c^2*d^2*e^2*f^2*(f + g*x) + 6*a*c*e^4*f^2*(f + g*x) - (12*c^2*d*e^3*f^3*(f + g*x))/g - 18*c^2*d^3*e*f*g*(f + g*x) - 6*a*c*d*e^3*f*g*(f + g*x) - 15*c^2*d^2*e^2*f*(f + g*x)^2 - 3*a*c*e^4*f*(f + g*x)^2 + (6*c^2*d*e^3*f^2*(f + g*x)^2)/g + 9*c^2*d^3*e*g*(f + g*x)^2 + 3*a*c*d*e^3*g*(f + g*x)^2 - (e^2*(e*f - d*g)*(f + g*x)*(a + c*x^2)*(2*(c*d^2 + a*e^2)*(e*f - d*g) - 3*(a*e^2*g + c*d*(-2*e*f + 3*d*g))*(d + e*x)))/(d + e*x)^2 + (3*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-(e*f) + d*g)*(a*e^2*g + c*d*(-2*e*f + 3*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/(g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) + ((Sqrt[c]*d - I*Sqrt[a]*e)*(3*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(e*f - 2*d*g) - Sqrt[a]*c*e*(2*e^2*f^2 - 6*d*e*f*g + d^2*g^2) - I*c^(3/2)*d*(4*e^2*f^2 - 9*d*e*f*g + 8*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((8*I)*c^2*d^2*e^2*f^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - ((4*I)*a*c*e^4*f^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - ((20*I)*c^2*d^3*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((4*I)*a*c*d*e^3*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((15*I)*c^2*d^4*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((6*I)*a*c*d^2*e^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + ((3*I)*a^2*e^4*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
652,1,468,387,3.2264595,"\int \frac{1}{(d+e x) (f+g x)^{3/2} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^(3/2)*Sqrt[a + c*x^2]),x]","\frac{2 i (f+g x) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left(\left(\sqrt{c} (d g-2 e f)+i \sqrt{a} e g\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\sqrt{c} (e f-d g) E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+e \left(\sqrt{c} f-i \sqrt{a} g\right) \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)\right)}{\sqrt{a+c x^2} \left(\sqrt{c} f-i \sqrt{a} g\right) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)^2}","\frac{2 g^2 \sqrt{a+c x^2}}{\sqrt{f+g x} \left(a g^2+c f^2\right) (e f-d g)}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{\sqrt{a+c x^2} \left(a g^2+c f^2\right) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 e \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) (e f-d g)}",1,"((2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(Sqrt[c]*(e*f - d*g)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (I*Sqrt[a]*e*g + Sqrt[c]*(-2*e*f + d*g))*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(Sqrt[c]*f - I*Sqrt[a]*g)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/((Sqrt[c]*f - I*Sqrt[a]*g)*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^2*Sqrt[a + c*x^2])","C",1
653,1,1917,818,8.9727706,"\int \frac{1}{(d+e x) (f+g x)^{5/2} \sqrt{a+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^(5/2)*Sqrt[a + c*x^2]),x]","\frac{2 \left(g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g) \left(c x^2+a\right) \left(a (4 e f-d g+3 e g x) g^2+c f (e f (8 f+7 g x)-d g (5 f+4 g x))\right)-(f+g x) \left(7 c^2 e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^5-14 c^2 e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^4-3 i c^2 e^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f^4-11 c^2 d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^4+7 c^2 e^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f^3+22 c^2 d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^3+4 c^2 d^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3+10 a c e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^3-11 c^2 d e g \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f^2-8 c^2 d^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2-6 a c e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f^2-6 i a c e^2 g^2 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right) f^2-14 a c d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f^2+4 c^2 d^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f+3 a c e^2 g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2 f+6 a c d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x) f+4 a c d^2 g^4 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f+3 a^2 e^2 g^4 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} f-3 a c d e g^3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)^2+\sqrt{c} \left(\sqrt{a} g-i \sqrt{c} f\right) (e f-d g) \left(3 a e g^2+c f (7 e f-4 d g)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)+\left(\sqrt{c} f+i \sqrt{a} g\right) \left(3 a^{3/2} e^2 g^3+3 i a \sqrt{c} e (2 e f-d g) g^2+\sqrt{a} c \left(2 e^2 f^2+2 d e g f-d^2 g^2\right) g+3 i c^{3/2} f \left(3 e^2 f^2-3 d e g f+d^2 g^2\right)\right) \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-3 i a^2 e^2 g^4 \sqrt{\frac{g \left(x+\frac{i \sqrt{a}}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i \sqrt{a} g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right)-3 a^2 d e g^5 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}\right)\right)}{3 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)^3 \left(c f^2+a g^2\right)^2 (f+g x)^{3/2} \sqrt{c x^2+a}}","-\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left(\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right) e^2}{\left(\frac{\sqrt{c} d}{\sqrt{-a}}+e\right) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right) e}{(e f-d g)^2 \left(c f^2+a g^2\right) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{2 g^2 \sqrt{c x^2+a} e}{(e f-d g)^2 \left(c f^2+a g^2\right) \sqrt{f+g x}}+\frac{8 \sqrt{-a} c^{3/2} f g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 (e f-d g) \left(c f^2+a g^2\right)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right)|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right)}{3 (e f-d g) \left(c f^2+a g^2\right) \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{8 c f g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left(c f^2+a g^2\right)^2 \sqrt{f+g x}}+\frac{2 g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left(c f^2+a g^2\right) (f+g x)^{3/2}}",1,"(2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*(a + c*x^2)*(a*g^2*(4*e*f - d*g + 3*e*g*x) + c*f*(-(d*g*(5*f + 4*g*x)) + e*f*(8*f + 7*g*x))) - (f + g*x)*(7*c^2*e^2*f^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 11*c^2*d*e*f^4*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 4*c^2*d^2*f^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 10*a*c*e^2*f^3*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 14*a*c*d*e*f^2*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 4*a*c*d^2*f*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + 3*a^2*e^2*f*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 3*a^2*d*e*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - 14*c^2*e^2*f^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 22*c^2*d*e*f^3*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 8*c^2*d^2*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 6*a*c*e^2*f^2*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 6*a*c*d*e*f*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) + 7*c^2*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - 11*c^2*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 4*c^2*d^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + 3*a*c*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 - 3*a*c*d*e*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + Sqrt[c]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - d*g)*(3*a*e*g^2 + c*f*(7*e*f - 4*d*g))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (Sqrt[c]*f + I*Sqrt[a]*g)*(3*a^(3/2)*e^2*g^3 + (3*I)*a*Sqrt[c]*e*g^2*(2*e*f - d*g) + Sqrt[a]*c*g*(2*e^2*f^2 + 2*d*e*f*g - d^2*g^2) + (3*I)*c^(3/2)*f*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (3*I)*c^2*e^2*f^4*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (6*I)*a*c*e^2*f^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - (3*I)*a^2*e^2*g^4*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])))/(3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^3*(c*f^2 + a*g^2)^2*(f + g*x)^(3/2)*Sqrt[a + c*x^2])","C",1
654,1,261,110,0.9450509,"\int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{1+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 + c*x^2]),x]","-\frac{2 i (f+g x) \sqrt{\frac{g \left(x+\frac{i}{\sqrt{c}}\right)}{f+g x}} \sqrt{-\frac{-g x+\frac{i g}{\sqrt{c}}}{f+g x}} \left(F\left(i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right)-\Pi \left(\frac{\sqrt{c} (e f-d g)}{e \left(\sqrt{c} f+i g\right)};i \sinh ^{-1}\left(\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right)|\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right)\right)}{\sqrt{c x^2+1} \sqrt{-f-\frac{i g}{\sqrt{c}}} (e f-d g)}","-\frac{2 \sqrt{\frac{\sqrt{-c} (f+g x)}{\sqrt{-c} f+g}} \Pi \left(\frac{2 e}{\sqrt{-c} d+e};\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c} x}}{\sqrt{2}}\right)|\frac{2 g}{\sqrt{-c} f+g}\right)}{\left(\sqrt{-c} d+e\right) \sqrt{f+g x}}",1,"((-2*I)*Sqrt[(g*(I/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(EllipticF[I*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*f + I*g)] - EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*g)), I*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*f + I*g)]))/(Sqrt[-f - (I*g)/Sqrt[c]]*(e*f - d*g)*Sqrt[1 + c*x^2])","C",1
655,1,344,454,1.4036762,"\int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a+c x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]","\frac{\sqrt{2} \left(\sqrt{c} x+i \sqrt{a}\right) \sqrt{d+e x} \sqrt{\frac{\frac{i \sqrt{c} d x}{\sqrt{a}}-\frac{i \sqrt{a} e}{\sqrt{c}}+d+e x}{d+e x}} \sqrt{\frac{(f+g x) \left(\sqrt{a} e+i \sqrt{c} d\right)}{(d+e x) \left(\sqrt{a} g+i \sqrt{c} f\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{(e f-d g) \left(\sqrt{c} x+i \sqrt{a}\right)}{\left(\sqrt{c} f-i \sqrt{a} g\right) (d+e x)}}\right)|-\frac{\frac{i \sqrt{c} d f}{\sqrt{a}}-e f+d g+\frac{i \sqrt{a} e g}{\sqrt{c}}}{2 e f-2 d g}\right)}{\sqrt{a+c x^2} \sqrt{f+g x} \left(\sqrt{c} d-i \sqrt{a} e\right) \sqrt{\frac{\left(\sqrt{c} x+i \sqrt{a}\right) (e f-d g)}{(d+e x) \left(\sqrt{c} f-i \sqrt{a} g\right)}}}","-\frac{(d+e x) \sqrt[4]{a g^2+c f^2} \sqrt{\frac{\left(a+c x^2\right) (e f-d g)^2}{(d+e x)^2 \left(a g^2+c f^2\right)}} \left(\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right) \sqrt{\frac{\frac{(f+g x)^2 \left(a e^2+c d^2\right)}{(d+e x)^2 \left(a g^2+c f^2\right)}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left(a g^2+c f^2\right)}+1}{\left(\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c d^2+a e^2} \sqrt{f+g x}}{\sqrt[4]{c f^2+a g^2} \sqrt{d+e x}}\right)|\frac{1}{2} \left(\frac{c d f+a e g}{\sqrt{c d^2+a e^2} \sqrt{c f^2+a g^2}}+1\right)\right)}{\sqrt{a+c x^2} \sqrt[4]{a e^2+c d^2} (e f-d g) \sqrt{\frac{(f+g x)^2 \left(a e^2+c d^2\right)}{(d+e x)^2 \left(a g^2+c f^2\right)}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left(a g^2+c f^2\right)}+1}}",1,"(Sqrt[2]*(I*Sqrt[a] + Sqrt[c]*x)*Sqrt[d + e*x]*Sqrt[(d - (I*Sqrt[a]*e)/Sqrt[c] + (I*Sqrt[c]*d*x)/Sqrt[a] + e*x)/(d + e*x)]*Sqrt[((I*Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))/((I*Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]*EllipticF[ArcSin[Sqrt[((e*f - d*g)*(I*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], -(((I*Sqrt[c]*d*f)/Sqrt[a] - e*f + d*g + (I*Sqrt[a]*e*g)/Sqrt[c])/(2*e*f - 2*d*g))])/((Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[((e*f - d*g)*(I*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x]*Sqrt[a + c*x^2])","C",1
656,1,107,52,0.2362048,"\int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \sqrt{-1+2 x^2}} \, dx","Integrate[1/(Sqrt[-1 + x]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2]),x]","-\frac{2 (x-1)^{3/2} \sqrt{\frac{x+1}{1-x}} \sqrt{\frac{1-2 x^2}{(x-1)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{2}+2+\frac{1}{x-1}}}{2^{3/4}}\right)|4 \left(-4+3 \sqrt{2}\right)\right)}{\sqrt{3+2 \sqrt{2}} \sqrt{x+1} \sqrt{2 x^2-1}}","\frac{\sqrt{1-2 x^2} \sqrt{1-x^2} F\left(\left.\sin ^{-1}(x)\right|2\right)}{\sqrt{x-1} \sqrt{x+1} \sqrt{2 x^2-1}}",1,"(-2*(-1 + x)^(3/2)*Sqrt[(1 + x)/(1 - x)]*Sqrt[(1 - 2*x^2)/(-1 + x)^2]*EllipticF[ArcSin[Sqrt[2 + Sqrt[2] + (-1 + x)^(-1)]/2^(3/4)], 4*(-4 + 3*Sqrt[2])])/(Sqrt[3 + 2*Sqrt[2]]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2])","B",0
657,1,136,269,0.1267684,"\int \frac{\sqrt{d+e x} (f+g x)^3}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x)^3)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (7 f+g x)-2 a c^2 d^2 e g \left(35 f^2+14 f g x+3 g^2 x^2\right)+c^3 d^3 \left(35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right)\right)}{35 c^4 d^4 \sqrt{d+e x}}","-\frac{16 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2 \left(2 a e^2 g-c d (3 e f-d g)\right)}{35 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{35 c^3 d^3 e}+\frac{12 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{35 c^2 d^2 \sqrt{d+e x}}+\frac{2 (f+g x)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{7 c d \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(7*f + g*x) - 2*a*c^2*d^2*e*g*(35*f^2 + 14*f*g*x + 3*g^2*x^2) + c^3*d^3*(35*f^3 + 35*f^2*g*x + 21*f*g^2*x^2 + 5*g^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])","A",1
658,1,89,200,0.0760128,"\int \frac{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(8 a^2 e^2 g^2-4 a c d e g (5 f+g x)+c^2 d^2 \left(15 f^2+10 f g x+3 g^2 x^2\right)\right)}{15 c^3 d^3 \sqrt{d+e x}}","-\frac{8 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(2 a e^2 g-c d (3 e f-d g)\right)}{15 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{15 c^2 d^2 e}+\frac{2 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 c d \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(5*f + g*x) + c^2*d^2*(15*f^2 + 10*f*g*x + 3*g^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])","A",1
659,1,53,125,0.0465844,"\int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} (c d (3 f+g x)-2 a e g)}{3 c^2 d^2 \sqrt{d+e x}}","\frac{2 g \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c d e}-\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(2 a e^2 g-c d (3 e f-d g)\right)}{3 c^2 d^2 e \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e*g + c*d*(3*f + g*x)))/(3*c^2*d^2*Sqrt[d + e*x])","A",1
660,1,35,46,0.0169554,"\int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)}}{c d \sqrt{d+e x}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)])/(c*d*Sqrt[d + e*x])","A",1
661,1,93,80,0.0415008,"\int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{g} \sqrt{(d+e x) (a e+c d x)} \sqrt{c d f-a e g}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{\sqrt{g} \sqrt{c d f-a e g}}",1,"(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[g]*Sqrt[c*d*f - a*e*g]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
662,1,136,140,0.1062807,"\int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{d+e x} \left(\sqrt{g} (a e+c d x) \sqrt{c d f-a e g}+c d (f+g x) \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)\right)}{\sqrt{g} (f+g x) \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^{3/2}}","\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)}+\frac{c d \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{\sqrt{g} (c d f-a e g)^{3/2}}",1,"(Sqrt[d + e*x]*(Sqrt[g]*Sqrt[c*d*f - a*e*g]*(a*e + c*d*x) + c*d*Sqrt[a*e + c*d*x]*(f + g*x)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(Sqrt[g]*(c*d*f - a*e*g)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x))","A",1
663,1,77,213,0.0450198,"\int \frac{\sqrt{d+e x}}{(f+g x)^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 c^2 d^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{\sqrt{d+e x} (c d f-a e g)^3}","\frac{3 c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 \sqrt{g} (c d f-a e g)^{5/2}}+\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}",1,"(2*c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 3, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^3*Sqrt[d + e*x])","C",1
664,1,77,280,0.0442917,"\int \frac{\sqrt{d+e x}}{(f+g x)^4 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 c^3 d^3 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{\sqrt{d+e x} (c d f-a e g)^4}","\frac{5 c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{8 \sqrt{g} (c d f-a e g)^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}",1,"(2*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 4, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^4*Sqrt[d + e*x])","C",1
665,1,134,257,0.0992194,"\int \frac{(d+e x)^{3/2} (f+g x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x} \left(16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (g x-5 f)-2 a c^2 d^2 e g \left(-15 f^2+10 f g x+g^2 x^2\right)+c^3 d^3 \left(-5 f^3+15 f^2 g x+5 f g^2 x^2+g^3 x^3\right)\right)}{5 c^4 d^4 \sqrt{(d+e x) (a e+c d x)}}","-\frac{16 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(2 a e^2 g-c d (3 e f-d g)\right)}{5 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g^2 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[d + e*x]*(16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(-5*f + g*x) - 2*a*c^2*d^2*e*g*(-15*f^2 + 10*f*g*x + g^2*x^2) + c^3*d^3*(-5*f^3 + 15*f^2*g*x + 5*f*g^2*x^2 + g^3*x^3)))/(5*c^4*d^4*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
666,1,88,181,0.0676477,"\int \frac{(d+e x)^{3/2} (f+g x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x} \left(-8 a^2 e^2 g^2-4 a c d e g (g x-3 f)+c^2 d^2 \left(-3 f^2+6 f g x+g^2 x^2\right)\right)}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}}","-\frac{8 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(2 a e^2 g-c d (3 e f-d g)\right)}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[d + e*x]*(-8*a^2*e^2*g^2 - 4*a*c*d*e*g*(-3*f + g*x) + c^2*d^2*(-3*f^2 + 6*f*g*x + g^2*x^2)))/(3*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
667,1,51,150,0.0404644,"\int \frac{(d+e x)^{3/2} (f+g x)}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x} (2 a e g+c d (g x-f))}{c^2 d^2 \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(2 a e^2 g-c d (d g+e f)\right)}{c^2 d^2 \sqrt{d+e x} \left(c d^2-a e^2\right)}-\frac{2 (d+e x)^{3/2} (c d f-a e g)}{c d \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[d + e*x]*(2*a*e*g + c*d*(-f + g*x)))/(c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
668,1,35,46,0.0124232,"\int \frac{(d+e x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x}}{c d \sqrt{(d+e x) (a e+c d x)}}","-\frac{2 \sqrt{d+e x}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x])/(c*d*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
669,1,71,133,0.0275519,"\int \frac{(d+e x)^{3/2}}{(f+g x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)}","-\frac{2 \sqrt{d+e x}}{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}-\frac{2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{(c d f-a e g)^{3/2}}",1,"(-2*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 1, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
670,1,73,202,0.0327497,"\int \frac{(d+e x)^{3/2}}{(f+g x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c d \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^2}","-\frac{3 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}-\frac{3 c d \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{(c d f-a e g)^{5/2}}",1,"(-2*c*d*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 2, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
671,1,77,274,0.0377947,"\int \frac{(d+e x)^{3/2}}{(f+g x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 c^2 d^2 \sqrt{d+e x} \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^3}","-\frac{15 c^2 d^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 (c d f-a e g)^{7/2}}-\frac{15 c d g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{5 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(-2*c^2*d^2*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 3, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])","C",1
672,1,131,239,0.1096668,"\int \frac{(d+e x)^{5/2} (f+g x)^3}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (d+e x)^{3/2} \left(-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-6 a c^2 d^2 e g \left(f^2-6 f g x+g^2 x^2\right)+c^3 d^3 \left(-f^3-9 f^2 g x+9 f g^2 x^2+g^3 x^3\right)\right)}{3 c^4 d^4 ((d+e x) (a e+c d x))^{3/2}}","-\frac{16 g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(2 a e^2 g-c d (3 e f-d g)\right)}{3 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g^3 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^3 d^3 e}-\frac{4 g \sqrt{d+e x} (f+g x)^2}{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^3}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(d + e*x)^(3/2)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(f - g*x) - 6*a*c^2*d^2*e*g*(f^2 - 6*f*g*x + g^2*x^2) + c^3*d^3*(-f^3 - 9*f^2*g*x + 9*f*g^2*x^2 + g^3*x^3)))/(3*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
673,1,87,211,0.070023,"\int \frac{(d+e x)^{5/2} (f+g x)^2}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x)^2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","\frac{2 (d+e x)^{3/2} \left(8 a^2 e^2 g^2-4 a c d e g (f-3 g x)-c^2 d^2 \left(f^2+6 f g x-3 g^2 x^2\right)\right)}{3 c^3 d^3 ((d+e x) (a e+c d x))^{3/2}}","-\frac{8 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(2 a e^2 g-c d (d g+e f)\right)}{3 c^3 d^3 \sqrt{d+e x} \left(c d^2-a e^2\right)}-\frac{8 g (d+e x)^{3/2} (c d f-a e g)}{3 c^2 d^2 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(2*(d + e*x)^(3/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(f - 3*g*x) - c^2*d^2*(f^2 + 6*f*g*x - 3*g^2*x^2)))/(3*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
674,1,52,154,0.0476714,"\int \frac{(d+e x)^{5/2} (f+g x)}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} (2 a e g+c d (f+3 g x))}{3 c^2 d^2 ((d+e x) (a e+c d x))^{3/2}}","\frac{2 \sqrt{d+e x} \left(2 a e^2 g+c d (e f-3 d g)\right)}{3 c^2 d^2 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^{5/2} (c d f-a e g)}{3 c d \left(c d^2-a e^2\right) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2)*(2*a*e*g + c*d*(f + 3*g*x)))/(3*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
675,1,37,48,0.0292557,"\int \frac{(d+e x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2}}{3 c d ((d+e x) (a e+c d x))^{3/2}}","-\frac{2 (d+e x)^{3/2}}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2))/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
676,1,73,188,0.035671,"\int \frac{(d+e x)^{5/2}}{(f+g x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 ((d+e x) (a e+c d x))^{3/2} (a e g-c d f)}","\frac{2 g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{(c d f-a e g)^{5/2}}+\frac{2 g \sqrt{d+e x}}{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(2*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 1, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(-(c*d*f) + a*e*g)*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
677,1,75,268,0.0452366,"\int \frac{(d+e x)^{5/2}}{(f+g x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c d (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^2}","\frac{5 c d g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{(c d f-a e g)^{7/2}}+\frac{5 g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{10 g \sqrt{d+e x}}{3 (f+g x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(-2*c*d*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 2, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(c*d*f - a*e*g)^2*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
678,1,79,342,0.0511158,"\int \frac{(d+e x)^{5/2}}{(f+g x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","-\frac{2 c^2 d^2 (d+e x)^{3/2} \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^3}","\frac{35 c^2 d^2 g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 (c d f-a e g)^{9/2}}+\frac{35 c d g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^4}+\frac{35 g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{6 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^3}+\frac{14 g \sqrt{d+e x}}{3 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(-2*c^2*d^2*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 3, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(c*d*f - a*e*g)^3*((a*e + c*d*x)*(d + e*x))^(3/2))","C",1
679,1,195,336,0.177055,"\int \frac{(f+g x)^4 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (11 f+3 g x)+48 a^2 c^2 d^2 e^2 g^2 \left(33 f^2+22 f g x+5 g^2 x^2\right)-8 a c^3 d^3 e g \left(231 f^3+297 f^2 g x+165 f g^2 x^2+35 g^3 x^3\right)+c^4 d^4 \left(1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right)\right)}{3465 c^5 d^5 (d+e x)^{3/2}}","-\frac{128 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)^3 \left(2 a e^2 g-c d (5 e f-3 d g)\right)}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac{128 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt{d+e x}}+\frac{32 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{11 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(11*f + 3*g*x) + 48*a^2*c^2*d^2*e^2*g^2*(33*f^2 + 22*f*g*x + 5*g^2*x^2) - 8*a*c^3*d^3*e*g*(231*f^3 + 297*f^2*g*x + 165*f*g^2*x^2 + 35*g^3*x^3) + c^4*d^4*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4)))/(3465*c^5*d^5*(d + e*x)^(3/2))","A",1
680,1,136,269,0.1172124,"\int \frac{(f+g x)^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (3 f+g x)-6 a c^2 d^2 e g \left(21 f^2+18 f g x+5 g^2 x^2\right)+c^3 d^3 \left(105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right)\right)}{315 c^4 d^4 (d+e x)^{3/2}}","-\frac{16 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)^2 \left(2 a e^2 g-c d (5 e f-3 d g)\right)}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac{16 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt{d+e x}}+\frac{4 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{9 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(3*f + g*x) - 6*a*c^2*d^2*e*g*(21*f^2 + 18*f*g*x + 5*g^2*x^2) + c^3*d^3*(105*f^3 + 189*f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2))","A",1
681,1,90,200,0.078537,"\int \frac{(f+g x)^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(8 a^2 e^2 g^2-4 a c d e g (7 f+3 g x)+c^2 d^2 \left(35 f^2+42 f g x+15 g^2 x^2\right)\right)}{105 c^3 d^3 (d+e x)^{3/2}}","-\frac{8 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g) \left(2 a e^2 g-c d (5 e f-3 d g)\right)}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{8 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt{d+e x}}+\frac{2 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{7 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(7*f + 3*g*x) + c^2*d^2*(35*f^2 + 42*f*g*x + 15*g^2*x^2)))/(105*c^3*d^3*(d + e*x)^(3/2))","A",1
682,1,54,125,0.0493701,"\int \frac{(f+g x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+3 g x)-2 a e g)}{15 c^2 d^2 (d+e x)^{3/2}}","\frac{2 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 c d e \sqrt{d+e x}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} \left(2 a e^2 g-c d (5 e f-3 d g)\right)}{15 c^2 d^2 e (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*a*e*g + c*d*(5*f + 3*g*x)))/(15*c^2*d^2*(d + e*x)^(3/2))","A",1
683,1,37,48,0.024102,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/Sqrt[d + e*x],x]","\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 c d (d+e x)^{3/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 c d (d+e x)^{3/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*c*d*(d + e*x)^(3/2))","A",1
684,1,101,124,0.1306027,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\sqrt{g}-\frac{\sqrt{c d f-a e g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{a e+c d x}}\right)}{g^{3/2} \sqrt{d+e x}}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g \sqrt{d+e x}}-\frac{2 \sqrt{c d f-a e g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{3/2}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[g] - (Sqrt[c*d*f - a*e*g]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/Sqrt[a*e + c*d*x]))/(g^(3/2)*Sqrt[d + e*x])","A",1
685,1,110,132,0.1916336,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^2} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^2),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{c d \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{a e+c d x} \sqrt{c d f-a e g}}-\frac{\sqrt{g}}{f+g x}\right)}{g^{3/2} \sqrt{d+e x}}","\frac{c d \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{3/2} \sqrt{c d f-a e g}}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x)}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[g]/(f + g*x)) + (c*d*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x])))/(g^(3/2)*Sqrt[d + e*x])","A",1
686,1,79,207,0.0400342,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^3} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^3),x]","\frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 (d+e x)^{3/2} (c d f-a e g)^3}","\frac{c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2}",1,"(2*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(c*d*f - a*e*g)^3*(d + e*x)^(3/2))","C",1
687,1,79,277,0.0433992,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^4} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^4),x]","\frac{2 c^3 d^3 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(\frac{3}{2},4;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 (d+e x)^{3/2} (c d f-a e g)^4}","\frac{c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{8 g^{3/2} (c d f-a e g)^{5/2}}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3}",1,"(2*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[3/2, 4, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(c*d*f - a*e*g)^4*(d + e*x)^(3/2))","C",1
688,1,79,347,0.0426804,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^5} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^5),x]","\frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left(\frac{3}{2},5;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 (d+e x)^{3/2} (c d f-a e g)^5}","\frac{5 c^4 d^4 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac{5 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{96 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{24 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x)^4}",1,"(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2)*Hypergeometric2F1[3/2, 5, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*(c*d*f - a*e*g)^5*(d + e*x)^(3/2))","C",1
689,1,195,336,0.2311829,"\int \frac{(f+g x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (13 f+5 g x)+16 a^2 c^2 d^2 e^2 g^2 \left(143 f^2+130 f g x+35 g^2 x^2\right)-8 a c^3 d^3 e g \left(429 f^3+715 f^2 g x+455 f g^2 x^2+105 g^3 x^3\right)+c^4 d^4 \left(3003 f^4+8580 f^3 g x+10010 f^2 g^2 x^2+5460 f g^3 x^3+1155 g^4 x^4\right)\right)}{15015 c^5 d^5 (d+e x)^{5/2}}","-\frac{128 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)^3 \left(2 a e^2 g-c d (7 e f-5 d g)\right)}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac{128 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac{32 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{16 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{13 c d (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(13*f + 5*g*x) + 16*a^2*c^2*d^2*e^2*g^2*(143*f^2 + 130*f*g*x + 35*g^2*x^2) - 8*a*c^3*d^3*e*g*(429*f^3 + 715*f^2*g*x + 455*f*g^2*x^2 + 105*g^3*x^3) + c^4*d^4*(3003*f^4 + 8580*f^3*g*x + 10010*f^2*g^2*x^2 + 5460*f*g^3*x^3 + 1155*g^4*x^4)))/(15015*c^5*d^5*(d + e*x)^(5/2))","A",1
690,1,137,269,0.1658623,"\int \frac{(f+g x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (11 f+5 g x)-2 a c^2 d^2 e g \left(99 f^2+110 f g x+35 g^2 x^2\right)+c^3 d^3 \left(231 f^3+495 f^2 g x+385 f g^2 x^2+105 g^3 x^3\right)\right)}{1155 c^4 d^4 (d+e x)^{5/2}}","-\frac{16 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)^2 \left(2 a e^2 g-c d (7 e f-5 d g)\right)}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac{16 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac{4 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{11 c d (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(11*f + 5*g*x) - 2*a*c^2*d^2*e*g*(99*f^2 + 110*f*g*x + 35*g^2*x^2) + c^3*d^3*(231*f^3 + 495*f^2*g*x + 385*f*g^2*x^2 + 105*g^3*x^3)))/(1155*c^4*d^4*(d + e*x)^(5/2))","A",1
691,1,90,200,0.1187932,"\int \frac{(f+g x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(8 a^2 e^2 g^2-4 a c d e g (9 f+5 g x)+c^2 d^2 \left(63 f^2+90 f g x+35 g^2 x^2\right)\right)}{315 c^3 d^3 (d+e x)^{5/2}}","-\frac{8 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g) \left(2 a e^2 g-c d (7 e f-5 d g)\right)}{315 c^3 d^3 e (d+e x)^{5/2}}+\frac{8 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)}{63 c^2 d^2 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{9 c d (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(9*f + 5*g*x) + c^2*d^2*(63*f^2 + 90*f*g*x + 35*g^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))","A",1
692,1,54,125,0.0743613,"\int \frac{(f+g x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+5 g x)-2 a e g)}{35 c^2 d^2 (d+e x)^{5/2}}","\frac{2 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} \left(2 a e^2 g-c d (7 e f-5 d g)\right)}{35 c^2 d^2 e (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-2*a*e*g + c*d*(7*f + 5*g*x)))/(35*c^2*d^2*(d + e*x)^(5/2))","A",1
693,1,37,48,0.0324417,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 c d (d+e x)^{5/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 c d (d+e x)^{5/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*c*d*(d + e*x)^(5/2))","A",1
694,1,132,179,0.2641019,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)),x]","\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \left(\sqrt{g} \sqrt{a e+c d x} (4 a e g+c d (g x-3 f))+3 (c d f-a e g)^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)\right)}{3 g^{5/2} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 (c d f-a e g)^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{5/2}}-\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{g^2 \sqrt{d+e x}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g (d+e x)^{3/2}}",1,"(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(4*a*e*g + c*d*(-3*f + g*x)) + 3*(c*d*f - a*e*g)^(3/2)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(3*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
695,1,75,178,0.0513932,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^2),x]","\frac{2 c d ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 (d+e x)^{5/2} (c d f-a e g)^2}","-\frac{3 c d \sqrt{c d f-a e g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{5/2}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{g (d+e x)^{3/2} (f+g x)}+\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g^2 \sqrt{d+e x}}",1,"(2*c*d*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*(c*d*f - a*e*g)^2*(d + e*x)^(5/2))","C",1
696,1,135,195,0.32323,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^3),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{3 c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{a e+c d x} \sqrt{c d f-a e g}}-\frac{\sqrt{g} (2 a e g+c d (3 f+5 g x))}{(f+g x)^2}\right)}{4 g^{5/2} \sqrt{d+e x}}","\frac{3 c^2 d^2 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 g^{5/2} \sqrt{c d f-a e g}}-\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g^2 \sqrt{d+e x} (f+g x)}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 g (d+e x)^{3/2} (f+g x)^2}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[g]*(2*a*e*g + c*d*(3*f + 5*g*x)))/(f + g*x)^2) + (3*c^2*d^2*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x])))/(4*g^(5/2)*Sqrt[d + e*x])","A",1
697,1,79,265,0.0624593,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^4),x]","\frac{2 c^3 d^3 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(\frac{5}{2},4;\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 (d+e x)^{5/2} (c d f-a e g)^4}","\frac{c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{8 g^{5/2} (c d f-a e g)^{3/2}}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 g^2 \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g^2 \sqrt{d+e x} (f+g x)^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g (d+e x)^{3/2} (f+g x)^3}",1,"(2*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*(c*d*f - a*e*g)^4*(d + e*x)^(5/2))","C",1
698,1,79,335,0.0607073,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^5),x]","\frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(\frac{5}{2},5;\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 (d+e x)^{5/2} (c d f-a e g)^5}","\frac{3 c^4 d^4 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{64 g^{5/2} (c d f-a e g)^{5/2}}+\frac{3 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 g^2 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32 g^2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 g^2 \sqrt{d+e x} (f+g x)^3}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}",1,"(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*(c*d*f - a*e*g)^5*(d + e*x)^(5/2))","C",1
699,1,79,405,0.0644347,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^6),x]","\frac{2 c^5 d^5 ((d+e x) (a e+c d x))^{5/2} \, _2F_1\left(\frac{5}{2},6;\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 (d+e x)^{5/2} (c d f-a e g)^6}","\frac{3 c^5 d^5 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{128 g^{5/2} (c d f-a e g)^{7/2}}+\frac{3 c^4 d^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 g^2 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 g^2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{80 g^2 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{3 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{40 g^2 \sqrt{d+e x} (f+g x)^4}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}",1,"(2*c^5*d^5*((a*e + c*d*x)*(d + e*x))^(5/2)*Hypergeometric2F1[5/2, 6, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*(c*d*f - a*e*g)^6*(d + e*x)^(5/2))","C",1
700,1,205,336,0.208749,"\int \frac{(f+g x)^4 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (15 f+7 g x)+48 a^2 c^2 d^2 e^2 g^2 \left(65 f^2+70 f g x+21 g^2 x^2\right)-8 a c^3 d^3 e g \left(715 f^3+1365 f^2 g x+945 f g^2 x^2+231 g^3 x^3\right)+c^4 d^4 \left(6435 f^4+20020 f^3 g x+24570 f^2 g^2 x^2+13860 f g^3 x^3+3003 g^4 x^4\right)\right)}{45045 c^5 d^5 \sqrt{d+e x}}","-\frac{128 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)^3 \left(2 a e^2 g-c d (9 e f-7 d g)\right)}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac{128 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac{32 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{15 c d (d+e x)^{7/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(15*f + 7*g*x) + 48*a^2*c^2*d^2*e^2*g^2*(65*f^2 + 70*f*g*x + 21*g^2*x^2) - 8*a*c^3*d^3*e*g*(715*f^3 + 1365*f^2*g*x + 945*f*g^2*x^2 + 231*g^3*x^3) + c^4*d^4*(6435*f^4 + 20020*f^3*g*x + 24570*f^2*g^2*x^2 + 13860*f*g^3*x^3 + 3003*g^4*x^4)))/(45045*c^5*d^5*Sqrt[d + e*x])","A",1
701,1,147,269,0.1592756,"\int \frac{(f+g x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (13 f+7 g x)-2 a c^2 d^2 e g \left(143 f^2+182 f g x+63 g^2 x^2\right)+c^3 d^3 \left(429 f^3+1001 f^2 g x+819 f g^2 x^2+231 g^3 x^3\right)\right)}{3003 c^4 d^4 \sqrt{d+e x}}","-\frac{16 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)^2 \left(2 a e^2 g-c d (9 e f-7 d g)\right)}{3003 c^4 d^4 e (d+e x)^{7/2}}+\frac{16 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)^2}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac{12 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{13 c d (d+e x)^{7/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(13*f + 7*g*x) - 2*a*c^2*d^2*e*g*(143*f^2 + 182*f*g*x + 63*g^2*x^2) + c^3*d^3*(429*f^3 + 1001*f^2*g*x + 819*f*g^2*x^2 + 231*g^3*x^3)))/(3003*c^4*d^4*Sqrt[d + e*x])","A",1
702,1,100,200,0.1124498,"\int \frac{(f+g x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(8 a^2 e^2 g^2-4 a c d e g (11 f+7 g x)+c^2 d^2 \left(99 f^2+154 f g x+63 g^2 x^2\right)\right)}{693 c^3 d^3 \sqrt{d+e x}}","-\frac{8 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g) \left(2 a e^2 g-c d (9 e f-7 d g)\right)}{693 c^3 d^3 e (d+e x)^{7/2}}+\frac{8 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} (c d f-a e g)}{99 c^2 d^2 e (d+e x)^{5/2}}+\frac{2 (f+g x)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{11 c d (d+e x)^{7/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(11*f + 7*g*x) + c^2*d^2*(99*f^2 + 154*f*g*x + 63*g^2*x^2)))/(693*c^3*d^3*Sqrt[d + e*x])","A",1
703,1,64,125,0.0787029,"\int \frac{(f+g x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} (c d (9 f+7 g x)-2 a e g)}{63 c^2 d^2 \sqrt{d+e x}}","\frac{2 g \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{9 c d e (d+e x)^{5/2}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2} \left(2 a e^2 g-c d (9 e f-7 d g)\right)}{63 c^2 d^2 e (d+e x)^{7/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e*g + c*d*(9*f + 7*g*x)))/(63*c^2*d^2*Sqrt[d + e*x])","A",1
704,1,37,48,0.0395746,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2),x]","\frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 c d (d+e x)^{7/2}}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{7 c d (d+e x)^{7/2}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*c*d*(d + e*x)^(7/2))","A",1
705,1,145,236,0.3626157,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)),x]","\frac{((d+e x) (a e+c d x))^{5/2} \left(-\frac{10 (c d f-a e g)^{5/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{g^{5/2} (a e+c d x)^{5/2}}+\frac{10 (a e g-c d f) (4 a e g+c d (g x-3 f))}{3 g^2 (a e+c d x)^2}+2\right)}{5 g (d+e x)^{5/2}}","-\frac{2 (c d f-a e g)^{5/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{7/2}}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{g^3 \sqrt{d+e x}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{3 g^2 (d+e x)^{3/2}}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 g (d+e x)^{5/2}}",1,"(((a*e + c*d*x)*(d + e*x))^(5/2)*(2 + (10*(-(c*d*f) + a*e*g)*(4*a*e*g + c*d*(-3*f + g*x)))/(3*g^2*(a*e + c*d*x)^2) - (10*(c*d*f - a*e*g)^(5/2)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(g^(5/2)*(a*e + c*d*x)^(5/2))))/(5*g*(d + e*x)^(5/2))","A",1
706,1,75,235,0.0677796,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^2),x]","\frac{2 c d ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 (d+e x)^{7/2} (c d f-a e g)^2}","\frac{5 c d (c d f-a e g)^{3/2} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{7/2}}-\frac{5 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt{d+e x}}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g^2 (d+e x)^{3/2}}",1,"(2*c*d*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*(c*d*f - a*e*g)^2*(d + e*x)^(7/2))","C",1
707,1,79,246,0.0721953,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^3),x]","\frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 (d+e x)^{7/2} (c d f-a e g)^3}","-\frac{15 c^2 d^2 \sqrt{c d f-a e g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 g^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g^3 \sqrt{d+e x}}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}",1,"(2*c^2*d^2*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*(c*d*f - a*e*g)^3*(d + e*x)^(7/2))","C",1
708,1,171,253,0.370687,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^4} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^4),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{15 c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{a e+c d x} \sqrt{c d f-a e g}}-\frac{\sqrt{g} \left(8 a^2 e^2 g^2+2 a c d e g (5 f+13 g x)+c^2 d^2 \left(15 f^2+40 f g x+33 g^2 x^2\right)\right)}{(f+g x)^3}\right)}{24 g^{7/2} \sqrt{d+e x}}","\frac{5 c^3 d^3 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{8 g^{7/2} \sqrt{c d f-a e g}}-\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{8 g^3 \sqrt{d+e x} (f+g x)}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[g]*(8*a^2*e^2*g^2 + 2*a*c*d*e*g*(5*f + 13*g*x) + c^2*d^2*(15*f^2 + 40*f*g*x + 33*g^2*x^2)))/(f + g*x)^3) + (15*c^3*d^3*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x])))/(24*g^(7/2)*Sqrt[d + e*x])","A",1
709,1,79,323,0.0752783,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^5),x]","\frac{2 c^4 d^4 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(\frac{7}{2},5;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 (d+e x)^{7/2} (c d f-a e g)^5}","\frac{5 c^4 d^4 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{64 g^{7/2} (c d f-a e g)^{3/2}}+\frac{5 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{5 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32 g^3 \sqrt{d+e x} (f+g x)^2}-\frac{5 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}",1,"(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*(c*d*f - a*e*g)^5*(d + e*x)^(7/2))","C",1
710,1,79,393,0.0801812,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^6),x]","\frac{2 c^5 d^5 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(\frac{7}{2},6;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 (d+e x)^{7/2} (c d f-a e g)^6}","\frac{3 c^5 d^5 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{128 g^{7/2} (c d f-a e g)^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{128 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{64 g^3 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{16 g^3 \sqrt{d+e x} (f+g x)^3}-\frac{c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}",1,"(2*c^5*d^5*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*(c*d*f - a*e*g)^6*(d + e*x)^(7/2))","C",1
711,1,79,463,0.0824825,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^7),x]","\frac{2 c^6 d^6 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left(\frac{7}{2},7;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 (d+e x)^{7/2} (c d f-a e g)^7}","\frac{5 c^6 d^6 \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{512 g^{7/2} (c d f-a e g)^{7/2}}+\frac{5 c^5 d^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{512 g^3 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{5 c^4 d^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{768 g^3 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac{c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{192 g^3 \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}-\frac{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{32 g^3 \sqrt{d+e x} (f+g x)^4}-\frac{c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac{\left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}",1,"(2*c^6*d^6*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 7, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*(c*d*f - a*e*g)^7*(d + e*x)^(7/2))","C",1
712,1,269,313,0.6061695,"\int \frac{\sqrt{d+e x} (f+g x)^{5/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x)^(5/2))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a e+c d x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \left(15 a^2 e^2 g^2-10 a c d e g (4 f+g x)+c^2 d^2 \left(33 f^2+26 f g x+8 g^2 x^2\right)\right)+15 \sqrt{c d} (c d f-a e g)^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{24 c^{7/2} d^{7/2} \sqrt{g} \sqrt{(d+e x) (a e+c d x)} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}}","\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{8 c^{7/2} d^{7/2} \sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{8 c^3 d^3 \sqrt{d+e x}}+\frac{5 (f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{12 c^2 d^2 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c d \sqrt{d+e x}}",1,"(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Sqrt[f + g*x]*(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*(15*a^2*e^2*g^2 - 10*a*c*d*e*g*(4*f + g*x) + c^2*d^2*(33*f^2 + 26*f*g*x + 8*g^2*x^2)) + 15*Sqrt[c*d]*(c*d*f - a*e*g)^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(24*c^(7/2)*d^(7/2)*Sqrt[g]*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)])","A",1
713,1,234,244,0.4730953,"\int \frac{\sqrt{d+e x} (f+g x)^{3/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x)^(3/2))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a e+c d x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} (c d (5 f+2 g x)-3 a e g)+3 \sqrt{c d} (c d f-a e g)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{4 c^{5/2} d^{5/2} \sqrt{g} \sqrt{(d+e x) (a e+c d x)} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}}","\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{4 c^{5/2} d^{5/2} \sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{4 c^2 d^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 c d \sqrt{d+e x}}",1,"(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Sqrt[f + g*x]*(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*(-3*a*e*g + c*d*(5*f + 2*g*x)) + 3*Sqrt[c*d]*(c*d*f - a*e*g)^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(4*c^(5/2)*d^(5/2)*Sqrt[g]*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)])","A",1
714,1,213,169,0.191958,"\int \frac{\sqrt{d+e x} \sqrt{f+g x}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*Sqrt[f + g*x])/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a e+c d x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}+\sqrt{c d} \sqrt{c d f-a e g} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{c^{3/2} d^{3/2} \sqrt{g} \sqrt{(d+e x) (a e+c d x)} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}}","\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{c^{3/2} d^{3/2} \sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d \sqrt{d+e x}}",1,"(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Sqrt[f + g*x]*(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)] + Sqrt[c*d]*Sqrt[c*d*f - a*e*g]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(c^(3/2)*d^(3/2)*Sqrt[g]*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)])","A",1
715,1,160,105,0.1120341,"\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{c d} \sqrt{d+e x} \sqrt{a e+c d x} \sqrt{c d f-a e g} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{c^{3/2} d^{3/2} \sqrt{g} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[c*d]*Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/(c^(3/2)*d^(3/2)*Sqrt[g]*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
716,1,50,61,0.0307751,"\int \frac{\sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)}","\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)])/((c*d*f - a*e*g)*Sqrt[d + e*x]*Sqrt[f + g*x])","A",1
717,1,69,129,0.0548106,"\int \frac{\sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} (c d (3 f+2 g x)-a e g)}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}","\frac{4 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(a*e*g) + c*d*(3*f + 2*g*x)))/(3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2))","A",1
718,1,105,198,0.0856971,"\int \frac{\sqrt{d+e x}}{(f+g x)^{7/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(3 a^2 e^2 g^2-2 a c d e g (5 f+2 g x)+c^2 d^2 \left(15 f^2+20 f g x+8 g^2 x^2\right)\right)}{15 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^3}","\frac{16 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{15 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^2*g^2 - 2*a*c*d*e*g*(5*f + 2*g*x) + c^2*d^2*(15*f^2 + 20*f*g*x + 8*g^2*x^2)))/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(5/2))","A",1
719,1,152,267,0.1201175,"\int \frac{\sqrt{d+e x}}{(f+g x)^{9/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[Sqrt[d + e*x]/((f + g*x)^(9/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-5 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (7 f+2 g x)-a c^2 d^2 e g \left(35 f^2+28 f g x+8 g^2 x^2\right)+c^3 d^3 \left(35 f^3+70 f^2 g x+56 f g^2 x^2+16 g^3 x^3\right)\right)}{35 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)^4}","\frac{32 c^3 d^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{12 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{7 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-5*a^3*e^3*g^3 + 3*a^2*c*d*e^2*g^2*(7*f + 2*g*x) - a*c^2*d^2*e*g*(35*f^2 + 28*f*g*x + 8*g^2*x^2) + c^3*d^3*(35*f^3 + 70*f^2*g*x + 56*f*g^2*x^2 + 16*g^3*x^3)))/(35*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f + g*x)^(7/2))","A",1
720,1,100,301,0.1175788,"\int \frac{(d+e x)^{3/2} (f+g x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^(5/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} (f+g x)^{5/2} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{c d \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{5/2}}","\frac{15 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{4 c^{7/2} d^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{15 g \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{4 c^3 d^3 \sqrt{d+e x}}+\frac{5 g (f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 c^2 d^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^{5/2}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x]*(f + g*x)^(5/2)*Hypergeometric2F1[-5/2, -1/2, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*Sqrt[(a*e + c*d*x)*(d + e*x)]*((c*d*(f + g*x))/(c*d*f - a*e*g))^(5/2))","C",1
721,1,100,227,0.0839271,"\int \frac{(d+e x)^{3/2} (f+g x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^(3/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} (f+g x)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{c d \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{3/2}}","\frac{3 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{c^{5/2} d^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(-2*Sqrt[d + e*x]*(f + g*x)^(3/2)*Hypergeometric2F1[-3/2, -1/2, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*Sqrt[(a*e + c*d*x)*(d + e*x)]*((c*d*(f + g*x))/(c*d*f - a*e*g))^(3/2))","C",1
722,1,176,161,0.3689008,"\int \frac{(d+e x)^{3/2} \sqrt{f+g x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","\frac{2 \sqrt{d+e x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x} \sqrt{c d f-a e g} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)-(c d)^{3/2} (f+g x)\right)}{(c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","\frac{2 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{c^{3/2} d^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(2*Sqrt[d + e*x]*(-((c*d)^(3/2)*(f + g*x)) + Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[c*d*f - a*e*g]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/((c*d)^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
723,1,50,61,0.0269301,"\int \frac{(d+e x)^{3/2}}{\sqrt{f+g x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)}","-\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(-2*Sqrt[d + e*x]*Sqrt[f + g*x])/((c*d*f - a*e*g)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
724,1,64,124,0.0492145,"\int \frac{(d+e x)^{3/2}}{(f+g x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \sqrt{d+e x} (a e g+c d (f+2 g x))}{\sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^2}","-\frac{4 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(-2*Sqrt[d + e*x]*(a*e*g + c*d*(f + 2*g*x)))/((c*d*f - a*e*g)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
725,1,105,192,0.067982,"\int \frac{(d+e x)^{3/2}}{(f+g x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \sqrt{d+e x} \left(-a^2 e^2 g^2+2 a c d e g (3 f+2 g x)+c^2 d^2 \left(3 f^2+12 f g x+8 g^2 x^2\right)\right)}{3 (f+g x)^{3/2} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^3}","-\frac{16 c d g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}-\frac{8 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(-2*Sqrt[d + e*x]*(-(a^2*e^2*g^2) + 2*a*c*d*e*g*(3*f + 2*g*x) + c^2*d^2*(3*f^2 + 12*f*g*x + 8*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^(3/2))","A",1
726,1,150,262,0.0944768,"\int \frac{(d+e x)^{3/2}}{(f+g x)^{7/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]","-\frac{2 \sqrt{d+e x} \left(a^3 e^3 g^3-a^2 c d e^2 g^2 (5 f+2 g x)+a c^2 d^2 e g \left(15 f^2+20 f g x+8 g^2 x^2\right)+c^3 d^3 \left(5 f^3+30 f^2 g x+40 f g^2 x^2+16 g^3 x^3\right)\right)}{5 (f+g x)^{5/2} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^4}","-\frac{32 c^2 d^2 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}-\frac{16 c d g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac{12 g \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(-2*Sqrt[d + e*x]*(a^3*e^3*g^3 - a^2*c*d*e^2*g^2*(5*f + 2*g*x) + a*c^2*d^2*e*g*(15*f^2 + 20*f*g*x + 8*g^2*x^2) + c^3*d^3*(5*f^3 + 30*f^2*g*x + 40*f*g^2*x^2 + 16*g^3*x^3)))/(5*(c*d*f - a*e*g)^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^(5/2))","A",1
727,1,102,289,0.1400919,"\int \frac{(d+e x)^{5/2} (f+g x)^{5/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x)^(5/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} (f+g x)^{5/2} \, _2F_1\left(-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 c d ((d+e x) (a e+c d x))^{3/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{5/2}}","\frac{5 g^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{c^{7/2} d^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{5 g^2 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c^3 d^3 \sqrt{d+e x}}-\frac{10 g \sqrt{d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2)*(f + g*x)^(5/2)*Hypergeometric2F1[-5/2, -3/2, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^(5/2))","C",1
728,1,102,219,0.105218,"\int \frac{(d+e x)^{5/2} (f+g x)^{3/2}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x)^(3/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} (f+g x)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 c d ((d+e x) (a e+c d x))^{3/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{3/2}}","\frac{2 g^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{c^{5/2} d^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 g \sqrt{d+e x} \sqrt{f+g x}}{c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}",1,"(-2*(d + e*x)^(3/2)*(f + g*x)^(3/2)*Hypergeometric2F1[-3/2, -3/2, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^(3/2))","C",1
729,1,52,63,0.0305775,"\int \frac{(d+e x)^{5/2} \sqrt{f+g x}}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)}","-\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(-2*(d + e*x)^(3/2)*(f + g*x)^(3/2))/(3*(c*d*f - a*e*g)*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
730,1,68,128,0.0553788,"\int \frac{(d+e x)^{5/2}}{\sqrt{f+g x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 (d+e x)^{3/2} \sqrt{f+g x} (3 a e g-c d (f-2 g x))}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^2}","\frac{4 g \sqrt{d+e x} \sqrt{f+g x}}{3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2} \sqrt{f+g x}}{3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(2*(d + e*x)^(3/2)*Sqrt[f + g*x]*(3*a*e*g - c*d*(f - 2*g*x)))/(3*(c*d*f - a*e*g)^2*((a*e + c*d*x)*(d + e*x))^(3/2))","A",1
731,1,103,194,0.065619,"\int \frac{(d+e x)^{5/2}}{(f+g x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 (d+e x)^{3/2} \left(3 a^2 e^2 g^2+6 a c d e g (f+2 g x)+c^2 d^2 \left(-f^2+4 f g x+8 g^2 x^2\right)\right)}{3 \sqrt{f+g x} ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^3}","\frac{16 g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 g \sqrt{d+e x}}{3 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 \sqrt{f+g x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(2*(d + e*x)^(3/2)*(3*a^2*e^2*g^2 + 6*a*c*d*e*g*(f + 2*g*x) + c^2*d^2*(-f^2 + 4*f*g*x + 8*g^2*x^2)))/(3*(c*d*f - a*e*g)^3*((a*e + c*d*x)*(d + e*x))^(3/2)*Sqrt[f + g*x])","A",1
732,1,152,260,0.0969737,"\int \frac{(d+e x)^{5/2}}{(f+g x)^{5/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]","\frac{2 (d+e x)^{3/2} \left(-a^3 e^3 g^3+3 a^2 c d e^2 g^2 (3 f+2 g x)+3 a c^2 d^2 e g \left(3 f^2+12 f g x+8 g^2 x^2\right)+c^3 d^3 \left(-f^3+6 f^2 g x+24 f g^2 x^2+16 g^3 x^3\right)\right)}{3 (f+g x)^{3/2} ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^4}","\frac{32 c d g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 g^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{4 g \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(2*(d + e*x)^(3/2)*(-(a^3*e^3*g^3) + 3*a^2*c*d*e^2*g^2*(3*f + 2*g*x) + 3*a*c^2*d^2*e*g*(3*f^2 + 12*f*g*x + 8*g^2*x^2) + c^3*d^3*(-f^3 + 6*f^2*g*x + 24*f*g^2*x^2 + 16*g^3*x^3)))/(3*(c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*x))^(3/2)*(f + g*x)^(3/2))","A",1
733,1,300,385,1.1557045,"\int \frac{(f+g x)^{5/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{\sqrt{c d} \sqrt{d+e x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left(15 a^3 e^3 g^3-5 a^2 c d e^2 g^2 (11 f+2 g x)+a c^2 d^2 e g \left(73 f^2+36 f g x+8 g^2 x^2\right)+c^3 d^3 \left(15 f^3+118 f^2 g x+136 f g^2 x^2+48 g^3 x^3\right)\right)-15 \sqrt{a e+c d x} (c d f-a e g)^{9/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{192 c^{9/2} d^{9/2} g^{3/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","-\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{4 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(\frac{a e}{c d}-\frac{f}{g}\right)}{24 \sqrt{d+e x}}",1,"(Sqrt[c*d]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(15*a^3*e^3*g^3 - 5*a^2*c*d*e^2*g^2*(11*f + 2*g*x) + a*c^2*d^2*e*g*(73*f^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(15*f^3 + 118*f^2*g*x + 136*f*g^2*x^2 + 48*g^3*x^3)) - 15*(c*d*f - a*e*g)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(192*c^(9/2)*d^(9/2)*g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
734,1,255,313,0.8441725,"\int \frac{(f+g x)^{3/2} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{\sqrt{c d} \sqrt{d+e x} \left(-\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left(3 a^2 e^2 g^2-2 a c d e g (4 f+g x)-c^2 d^2 \left(3 f^2+14 f g x+8 g^2 x^2\right)\right)-3 \sqrt{a e+c d x} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{24 c^{7/2} d^{7/2} g^{3/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{8 c^{5/2} d^{5/2} g^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{8 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 g \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(\frac{a e}{c d}-\frac{f}{g}\right)}{12 \sqrt{d+e x}}",1,"(Sqrt[c*d]*Sqrt[d + e*x]*(-(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(3*a^2*e^2*g^2 - 2*a*c*d*e*g*(4*f + g*x) - c^2*d^2*(3*f^2 + 14*f*g*x + 8*g^2*x^2))) - 3*(c*d*f - a*e*g)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(24*c^(7/2)*d^(7/2)*g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
735,1,215,241,0.5942836,"\int \frac{\sqrt{f+g x} \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) (a e g+c d (f+2 g x))-\sqrt{a e+c d x} (c d f-a e g)^{5/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{4 g^{3/2} (c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","-\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{4 c^{3/2} d^{3/2} g^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 g \sqrt{d+e x}}+\frac{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(\frac{a e}{c d}-\frac{f}{g}\right)}{4 \sqrt{d+e x}}",1,"(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(a*e*g + c*d*(f + 2*g*x)) - (c*d*f - a*e*g)^(5/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(4*(c*d)^(5/2)*g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
736,1,173,167,0.8139831,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{g} (f+g x)-\frac{\sqrt{c} \sqrt{d} (c d f-a e g)^{3/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{(c d)^{3/2} \sqrt{a e+c d x}}\right)}{g^{3/2} \sqrt{d+e x} \sqrt{f+g x}}","\frac{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g \sqrt{d+e x}}-\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{\sqrt{c} \sqrt{d} g^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[g]*(f + g*x) - (Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^(3/2)*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/((c*d)^(3/2)*Sqrt[a*e + c*d*x])))/(g^(3/2)*Sqrt[d + e*x]*Sqrt[f + g*x])","A",1
737,1,169,158,0.7892558,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{3/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(3/2)),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{\sqrt{c} \sqrt{d} \sqrt{c d f-a e g} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{\sqrt{c d} \sqrt{a e+c d x}}-\sqrt{g}\right)}{g^{3/2} \sqrt{d+e x} \sqrt{f+g x}}","\frac{2 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{g^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g \sqrt{d+e x} \sqrt{f+g x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-Sqrt[g] + (Sqrt[c]*Sqrt[d]*Sqrt[c*d*f - a*e*g]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x])))/(g^(3/2)*Sqrt[d + e*x]*Sqrt[f + g*x])","A",1
738,1,52,63,0.0330351,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{5/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(5/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(3/2))","A",1
739,1,69,129,0.0543499,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{7/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(7/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+2 g x)-3 a e g)}{15 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e*g + c*d*(5*f + 2*g*x)))/(15*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2))","A",1
740,1,105,198,0.0921504,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{9/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(9/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left(35 f^2+28 f g x+8 g^2 x^2\right)\right)}{105 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(15*a^2*e^2*g^2 - 6*a*c*d*e*g*(7*f + 2*g*x) + c^2*d^2*(35*f^2 + 28*f*g*x + 8*g^2*x^2)))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(7/2))","A",1
741,1,152,267,0.1392379,"\int \frac{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{11/2}} \, dx","Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(11/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{3/2} \left(-35 a^3 e^3 g^3+15 a^2 c d e^2 g^2 (9 f+2 g x)-3 a c^2 d^2 e g \left(63 f^2+36 f g x+8 g^2 x^2\right)+c^3 d^3 \left(105 f^3+126 f^2 g x+72 f g^2 x^2+16 g^3 x^3\right)\right)}{315 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{315 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{21 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{9 (d+e x)^{3/2} (f+g x)^{9/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-35*a^3*e^3*g^3 + 15*a^2*c*d*e^2*g^2*(9*f + 2*g*x) - 3*a*c^2*d^2*e*g*(63*f^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(105*f^3 + 126*f^2*g*x + 72*f*g^2*x^2 + 16*g^3*x^3)))/(315*(c*d*f - a*e*g)^4*(d + e*x)^(3/2)*(f + g*x)^(9/2))","A",1
742,1,302,382,1.170419,"\int \frac{(f+g x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{\sqrt{c d} \sqrt{d+e x} \left(3 \sqrt{a e+c d x} (c d f-a e g)^{9/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)-\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left(3 a^3 e^3 g^3-a^2 c d e^2 g^2 (11 f+2 g x)-a c^2 d^2 e g \left(11 f^2+44 f g x+24 g^2 x^2\right)+c^3 d^3 \left(3 f^3-2 f^2 g x-24 f g^2 x^2-16 g^3 x^3\right)\right)\right)}{64 c^{7/2} d^{7/2} g^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{64 c^{5/2} d^{5/2} g^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3}{64 c^2 d^2 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{32 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{8 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{4 g (d+e x)^{3/2}}",1,"(Sqrt[c*d]*Sqrt[d + e*x]*(-(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(3*a^3*e^3*g^3 - a^2*c*d*e^2*g^2*(11*f + 2*g*x) - a*c^2*d^2*e*g*(11*f^2 + 44*f*g*x + 24*g^2*x^2) + c^3*d^3*(3*f^3 - 2*f^2*g*x - 24*f*g^2*x^2 - 16*g^3*x^3))) + 3*(c*d*f - a*e*g)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(64*c^(7/2)*d^(7/2)*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
743,1,254,310,0.8285793,"\int \frac{\sqrt{f+g x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left(3 a^2 e^2 g^2+2 a c d e g (4 f+7 g x)+c^2 d^2 \left(-3 f^2+2 f g x+8 g^2 x^2\right)\right)+3 \sqrt{a e+c d x} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{24 g^{5/2} (c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{8 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g (d+e x)^{3/2}}",1,"(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(3*a^2*e^2*g^2 + 2*a*c*d*e*g*(4*f + 7*g*x) + c^2*d^2*(-3*f^2 + 2*f*g*x + 8*g^2*x^2)) + 3*(c*d*f - a*e*g)^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(24*(c*d)^(5/2)*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
744,1,193,238,0.7712015,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} \sqrt{f+g x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*Sqrt[f + g*x]),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{g} (f+g x) (5 a e g+c d (2 g x-3 f))+\frac{3 \sqrt{c} \sqrt{d} (c d f-a e g)^{5/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{(c d)^{3/2} \sqrt{a e+c d x}}\right)}{4 g^{5/2} \sqrt{d+e x} \sqrt{f+g x}}","\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{4 \sqrt{c} \sqrt{d} g^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{3 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 g (d+e x)^{3/2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[g]*(f + g*x)*(5*a*e*g + c*d*(-3*f + 2*g*x)) + (3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^(5/2)*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/((c*d)^(3/2)*Sqrt[a*e + c*d*x])))/(4*g^(5/2)*Sqrt[d + e*x]*Sqrt[f + g*x])","A",1
745,1,102,222,0.1575137,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(3/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 c d (d+e x)^{5/2} (f+g x)^{3/2}}","-\frac{3 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{g^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{3 c d \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g^2 \sqrt{d+e x}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{g (d+e x)^{3/2} \sqrt{f+g x}}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*c*d*(d + e*x)^(5/2)*(f + g*x)^(3/2))","C",1
746,1,188,214,1.0634361,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(5/2)),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{3 \sqrt{c} \sqrt{d} (c d f-a e g)^{3/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{\sqrt{c d} \sqrt{a e+c d x}}-\sqrt{g} (a e g+c d (3 f+4 g x))\right)}{3 g^{5/2} \sqrt{d+e x} (f+g x)^{3/2}}","\frac{2 c^{3/2} d^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{g^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g^2 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g (d+e x)^{3/2} (f+g x)^{3/2}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[g]*(a*e*g + c*d*(3*f + 4*g*x))) + (3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^(3/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^(3/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x])))/(3*g^(5/2)*Sqrt[d + e*x]*(f + g*x)^(3/2))","A",1
747,1,52,63,0.0517752,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(7/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(5/2))","A",1
748,1,69,129,0.0850386,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(9/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+2 g x)-5 a e g)}{35 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-5*a*e*g + c*d*(7*f + 2*g*x)))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2))","A",1
749,1,105,198,0.1294769,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(11/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(35 a^2 e^2 g^2-10 a c d e g (9 f+2 g x)+c^2 d^2 \left(63 f^2+36 f g x+8 g^2 x^2\right)\right)}{315 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(35*a^2*e^2*g^2 - 10*a*c*d*e*g*(9*f + 2*g*x) + c^2*d^2*(63*f^2 + 36*f*g*x + 8*g^2*x^2)))/(315*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(9/2))","A",1
750,1,152,267,0.1763988,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2} (f+g x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(13/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{5/2} \left(-105 a^3 e^3 g^3+35 a^2 c d e^2 g^2 (11 f+2 g x)-5 a c^2 d^2 e g \left(99 f^2+44 f g x+8 g^2 x^2\right)+c^3 d^3 \left(231 f^3+198 f^2 g x+88 f g^2 x^2+16 g^3 x^3\right)\right)}{1155 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{33 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^3*g^3 + 35*a^2*c*d*e^2*g^2*(11*f + 2*g*x) - 5*a*c^2*d^2*e*g*(99*f^2 + 44*f*g*x + 8*g^2*x^2) + c^3*d^3*(231*f^3 + 198*f^2*g*x + 88*f*g^2*x^2 + 16*g^3*x^3)))/(1155*(c*d*f - a*e*g)^4*(d + e*x)^(5/2)*(f + g*x)^(11/2))","A",1
751,1,285,448,6.0134449,"\int \frac{(f+g x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{\sqrt{f+g x} ((d+e x) (a e+c d x))^{7/2} \left(-\frac{15 \sqrt{c} \sqrt{d} \sqrt{c d} (c d f-a e g)^{9/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{g^{7/2} (a e+c d x)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}}}+\frac{15 c d (c d f-a e g)^4}{g^3 (a e+c d x)^3}-\frac{10 c d (c d f-a e g)^3}{g^2 (a e+c d x)^2}+\frac{8 c d (c d f-a e g)^2}{g (a e+c d x)}+48 c d (c d f-a e g)+128 c^2 d^2 (f+g x)\right)}{640 c^3 d^3 (d+e x)^{7/2}}","-\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^5 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{3 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^4}{128 c^2 d^2 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{16 g^3 \sqrt{d+e x}}-\frac{(f+g x)^{5/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{5/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 g (d+e x)^{5/2}}",1,"(((a*e + c*d*x)*(d + e*x))^(7/2)*Sqrt[f + g*x]*(48*c*d*(c*d*f - a*e*g) + (15*c*d*(c*d*f - a*e*g)^4)/(g^3*(a*e + c*d*x)^3) - (10*c*d*(c*d*f - a*e*g)^3)/(g^2*(a*e + c*d*x)^2) + (8*c*d*(c*d*f - a*e*g)^2)/(g*(a*e + c*d*x)) + 128*c^2*d^2*(f + g*x) - (15*Sqrt[c]*Sqrt[d]*Sqrt[c*d]*(c*d*f - a*e*g)^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/(g^(7/2)*(a*e + c*d*x)^(7/2)*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)])))/(640*c^3*d^3*(d + e*x)^(7/2))","A",1
752,1,299,376,1.1347051,"\int \frac{\sqrt{f+g x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x} \left(\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left(15 a^3 e^3 g^3+a^2 c d e^2 g^2 (73 f+118 g x)+a c^2 d^2 e g \left(-55 f^2+36 f g x+136 g^2 x^2\right)+c^3 d^3 \left(15 f^3-10 f^2 g x+8 f g^2 x^2+48 g^3 x^3\right)\right)-15 \sqrt{a e+c d x} (c d f-a e g)^{9/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)\right)}{192 g^{7/2} (c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}}","-\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt{d+e x}}+\frac{5 (f+g x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{32 g^3 \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{24 g^2 (d+e x)^{3/2}}+\frac{(f+g x)^{3/2} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{4 g (d+e x)^{5/2}}",1,"(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(15*a^3*e^3*g^3 + a^2*c*d*e^2*g^2*(73*f + 118*g*x) + a*c^2*d^2*e*g*(-55*f^2 + 36*f*g*x + 136*g^2*x^2) + c^3*d^3*(15*f^3 - 10*f^2*g*x + 8*f*g^2*x^2 + 48*g^3*x^3)) - 15*(c*d*f - a*e*g)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(192*(c*d)^(5/2)*g^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])","A",1
753,1,229,304,0.9983823,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} \sqrt{f+g x}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\sqrt{g} (f+g x) \left(33 a^2 e^2 g^2+2 a c d e g (13 g x-20 f)+c^2 d^2 \left(15 f^2-10 f g x+8 g^2 x^2\right)\right)-\frac{15 \sqrt{c} \sqrt{d} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{(c d)^{3/2} \sqrt{a e+c d x}}\right)}{24 g^{7/2} \sqrt{d+e x} \sqrt{f+g x}}","-\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2}{8 g^3 \sqrt{d+e x}}-\frac{5 \sqrt{f+g x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3 g (d+e x)^{5/2}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[g]*(f + g*x)*(33*a^2*e^2*g^2 + 2*a*c*d*e*g*(-20*f + 13*g*x) + c^2*d^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2)) - (15*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^(7/2)*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/((c*d)^(3/2)*Sqrt[a*e + c*d*x])))/(24*g^(7/2)*Sqrt[d + e*x]*Sqrt[f + g*x])","A",1
754,1,112,294,0.1041194,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(3/2)),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 c d \sqrt{d+e x} (f+g x)^{3/2}}","\frac{15 \sqrt{c} \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{4 g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{15 c d \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}{4 g^3 \sqrt{d+e x}}+\frac{5 c d \sqrt{f+g x} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{g (d+e x)^{5/2} \sqrt{f+g x}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*((c*d*(f + g*x))/(c*d*f - a*e*g))^(3/2)*Hypergeometric2F1[3/2, 7/2, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*c*d*Sqrt[d + e*x]*(f + g*x)^(3/2))","C",1
755,1,112,284,0.1228806,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{5/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(5/2)),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{5/2} \, _2F_1\left(\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 c d \sqrt{d+e x} (f+g x)^{5/2}}","-\frac{5 c^{3/2} d^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}+\frac{5 c^2 d^2 \sqrt{f+g x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g^3 \sqrt{d+e x}}-\frac{10 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g^2 (d+e x)^{3/2} \sqrt{f+g x}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^{3/2}}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*((c*d*(f + g*x))/(c*d*f - a*e*g))^(5/2)*Hypergeometric2F1[5/2, 7/2, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*c*d*Sqrt[d + e*x]*(f + g*x)^(5/2))","C",1
756,1,224,274,1.3687332,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{7/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(7/2)),x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(\frac{15 \sqrt{c} \sqrt{d} (c d f-a e g)^{5/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right)}{\sqrt{c d} \sqrt{a e+c d x}}-\sqrt{g} \left(3 a^2 e^2 g^2+a c d e g (5 f+11 g x)+c^2 d^2 \left(15 f^2+35 f g x+23 g^2 x^2\right)\right)\right)}{15 g^{7/2} \sqrt{d+e x} (f+g x)^{5/2}}","\frac{2 c^{5/2} d^{5/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right)}{g^{7/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}-\frac{2 c^2 d^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g^3 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2}}{3 g^2 (d+e x)^{3/2} (f+g x)^{3/2}}-\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^{5/2}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[g]*(3*a^2*e^2*g^2 + a*c*d*e*g*(5*f + 11*g*x) + c^2*d^2*(15*f^2 + 35*f*g*x + 23*g^2*x^2))) + (15*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^(5/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^(5/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x])))/(15*g^(7/2)*Sqrt[d + e*x]*(f + g*x)^(5/2))","A",1
757,1,52,63,0.0771447,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(9/2)),x]","\frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)}","\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))","A",1
758,1,79,129,0.0818027,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(11/2)),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} (c d (9 f+2 g x)-7 a e g)}{63 \sqrt{d+e x} (f+g x)^{9/2} (c d f-a e g)^2}","\frac{4 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-7*a*e*g + c*d*(9*f + 2*g*x)))/(63*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(9/2))","A",1
759,1,115,198,0.1075399,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(13/2)),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(63 a^2 e^2 g^2-14 a c d e g (11 f+2 g x)+c^2 d^2 \left(99 f^2+44 f g x+8 g^2 x^2\right)\right)}{693 \sqrt{d+e x} (f+g x)^{11/2} (c d f-a e g)^3}","\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{693 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac{8 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{99 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^2*g^2 - 14*a*c*d*e*g*(11*f + 2*g*x) + c^2*d^2*(99*f^2 + 44*f*g*x + 8*g^2*x^2)))/(693*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(11/2))","A",1
760,1,162,267,0.1408383,"\int \frac{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx","Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(15/2)),x]","\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(-231 a^3 e^3 g^3+63 a^2 c d e^2 g^2 (13 f+2 g x)-7 a c^2 d^2 e g \left(143 f^2+52 f g x+8 g^2 x^2\right)+c^3 d^3 \left(429 f^3+286 f^2 g x+104 f g^2 x^2+16 g^3 x^3\right)\right)}{3003 \sqrt{d+e x} (f+g x)^{13/2} (c d f-a e g)^4}","\frac{32 c^3 d^3 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac{12 c d \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)^2}+\frac{2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^3*g^3 + 63*a^2*c*d*e^2*g^2*(13*f + 2*g*x) - 7*a*c^2*d^2*e*g*(143*f^2 + 52*f*g*x + 8*g^2*x^2) + c^3*d^3*(429*f^3 + 286*f^2*g*x + 104*f*g^2*x^2 + 16*g^3*x^3)))/(3003*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f + g*x)^(13/2))","A",1
761,1,100,104,0.0834794,"\int \frac{(d+e x)^{5/2} (f+g x)^n}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}} \, dx","Integrate[((d + e*x)^(5/2)*(f + g*x)^n)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]","-\frac{2 (d+e x)^{3/2} (f+g x)^n \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(-\frac{3}{2},-n;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 c d ((d+e x) (a e+c d x))^{3/2}}","-\frac{(d+e x)^{5/2} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left(1,n-\frac{1}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} (c d f-a e g)}",1,"(-2*(d + e*x)^(3/2)*(f + g*x)^n*Hypergeometric2F1[-3/2, -n, -1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
762,1,98,104,0.0430302,"\int \frac{(d+e x)^{3/2} (f+g x)^n}{\left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^n)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]","-\frac{2 \sqrt{d+e x} (f+g x)^n \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(-\frac{1}{2},-n;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{c d \sqrt{(d+e x) (a e+c d x)}}","-\frac{(d+e x)^{3/2} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left(1,n+\frac{1}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} (c d f-a e g)}",1,"(-2*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[-1/2, -n, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*Sqrt[(a*e + c*d*x)*(d + e*x)]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
763,1,98,104,0.0430039,"\int \frac{\sqrt{d+e x} (f+g x)^n}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(f + g*x)^n)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 (f+g x)^n \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{c d \sqrt{d+e x}}","-\frac{\sqrt{d+e x} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left(1,n+\frac{3}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^n*Hypergeometric2F1[1/2, -n, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*Sqrt[d + e*x]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
764,1,100,104,0.0663069,"\int \frac{(f+g x)^n \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}}{\sqrt{d+e x}} \, dx","Integrate[((f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]","\frac{2 (f+g x)^n ((d+e x) (a e+c d x))^{3/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(\frac{3}{2},-n;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{3 c d (d+e x)^{3/2}}","-\frac{(f+g x)^{n+1} (a e+c d x) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \, _2F_1\left(1,n+\frac{5}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) \sqrt{d+e x} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(f + g*x)^n*Hypergeometric2F1[3/2, -n, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*(d + e*x)^(3/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
765,1,100,104,0.1009619,"\int \frac{(f+g x)^n \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{3/2}}{(d+e x)^{3/2}} \, dx","Integrate[((f + g*x)^n*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]","\frac{2 (f+g x)^n ((d+e x) (a e+c d x))^{5/2} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(\frac{5}{2},-n;\frac{7}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{5 c d (d+e x)^{5/2}}","-\frac{(f+g x)^{n+1} (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{3/2} \, _2F_1\left(1,n+\frac{7}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) (d+e x)^{3/2} (c d f-a e g)}",1,"(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(f + g*x)^n*Hypergeometric2F1[5/2, -n, 7/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(5*c*d*(d + e*x)^(5/2)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
766,1,110,104,0.0711948,"\int \frac{(f+g x)^n \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{5/2}}{(d+e x)^{5/2}} \, dx","Integrate[((f + g*x)^n*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]","\frac{2 (f+g x)^n (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(\frac{7}{2},-n;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{7 c d \sqrt{d+e x}}","-\frac{(f+g x)^{n+1} (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{5/2} \, _2F_1\left(1,n+\frac{9}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) (d+e x)^{5/2} (c d f-a e g)}",1,"(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^n*Hypergeometric2F1[7/2, -n, 9/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(7*c*d*Sqrt[d + e*x]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)","A",1
767,1,95,103,0.0517332,"\int (d+e x)^m (f+g x)^n \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(f + g*x)^n)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","\frac{(d+e x)^m (f+g x)^{n+1} ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(m,n+1;n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{g (n+1)}","-\frac{(d+e x)^m (f+g x)^{n+1} (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \, _2F_1\left(1,-m+n+2;n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{(n+1) (c d f-a e g)}",1,"(((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*(f + g*x)^(1 + n)*Hypergeometric2F1[m, 1 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*(1 + n)*((a*e + c*d*x)*(d + e*x))^m)","A",1
768,1,134,343,0.1724604,"\int (d+e x)^m (f+g x)^3 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left(\frac{3 g^2 (a e+c d x)^2 (a e g-c d f)}{m-3}-\frac{3 g (a e+c d x) (c d f-a e g)^2}{m-2}-\frac{(c d f-a e g)^3}{m-1}-\frac{g^3 (a e+c d x)^3}{m-4}\right)}{c^4 d^4}","-\frac{6 (d+e x)^{m-1} (c d f-a e g)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m} \left(a e^2 g+c d (d g (1-m)-e f (2-m))\right)}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac{6 g (d+e x)^m (c d f-a e g)^2 \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac{3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac{(f+g x)^3 (d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c d (4-m)}",1,"((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*(-((c*d*f - a*e*g)^3/(-1 + m)) - (3*g*(c*d*f - a*e*g)^2*(a*e + c*d*x))/(-2 + m) + (3*g^2*(-(c*d*f) + a*e*g)*(a*e + c*d*x)^2)/(-3 + m) - (g^3*(a*e + c*d*x)^3)/(-4 + m)))/(c^4*d^4)","A",1
769,1,131,246,0.1081272,"\int (d+e x)^m (f+g x)^2 \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(f + g*x)^2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","-\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left(2 a^2 e^2 g^2+2 a c d e g (f (m-3)+g (m-1) x)+c^2 d^2 \left(f^2 \left(m^2-5 m+6\right)+2 f g \left(m^2-4 m+3\right) x+g^2 \left(m^2-3 m+2\right) x^2\right)\right)}{c^3 d^3 (m-3) (m-2) (m-1)}","-\frac{2 (d+e x)^{m-1} (c d f-a e g) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m} \left(a e^2 g+c d (d g (1-m)-e f (2-m))\right)}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac{2 g (d+e x)^m (c d f-a e g) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac{(f+g x)^2 (d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c d (3-m)}",1,"-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*(2*a^2*e^2*g^2 + 2*a*c*d*e*g*(f*(-3 + m) + g*(-1 + m)*x) + c^2*d^2*(f^2*(6 - 5*m + m^2) + 2*f*g*(3 - 4*m + m^2)*x + g^2*(2 - 3*m + m^2)*x^2)))/(c^3*d^3*(-3 + m)*(-2 + m)*(-1 + m)))","A",1
770,1,67,150,0.0615897,"\int (d+e x)^m (f+g x) \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(f + g*x))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","-\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} (a e g+c d (f (m-2)+g (m-1) x))}{c^2 d^2 (m-2) (m-1)}","\frac{g (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m} \left(a e^2 g+c d (d g (1-m)-e f (2-m))\right)}{c^2 d^2 e (1-m) (2-m)}",1,"-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*(a*e*g + c*d*(f*(-2 + m) + g*(-1 + m)*x)))/(c^2*d^2*(-2 + m)*(-1 + m)))","A",1
771,1,42,54,0.0219678,"\int (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","-\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)}","\frac{(d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m}}{c d (1-m)}",1,"-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m))/(c*d*(-1 + m)))","A",1
772,1,82,99,0.0265266,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{f+g x} \, dx","Integrate[(d + e*x)^m/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","-\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \, _2F_1\left(1,1-m;2-m;\frac{g (a e+c d x)}{a e g-c d f}\right)}{(m-1) (c d f-a e g)}","\frac{(d+e x)^m (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \, _2F_1\left(1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right)}{(1-m) (c d f-a e g)}",1,"-(((d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*Hypergeometric2F1[1, 1 - m, 2 - m, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)*(-1 + m)))","A",1
773,1,84,101,0.0329688,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{(f+g x)^2} \, dx","Integrate[(d + e*x)^m/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","-\frac{c d (d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \, _2F_1\left(2,1-m;2-m;\frac{g (a e+c d x)}{a e g-c d f}\right)}{(m-1) (c d f-a e g)^2}","\frac{c d (d+e x)^m (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \, _2F_1\left(2,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right)}{(1-m) (c d f-a e g)^2}",1,"-((c*d*(d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*Hypergeometric2F1[2, 1 - m, 2 - m, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^2*(-1 + m)))","A",1
774,1,88,105,0.036197,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{(f+g x)^3} \, dx","Integrate[(d + e*x)^m/((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","-\frac{c^2 d^2 (d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \, _2F_1\left(3,1-m;2-m;\frac{g (a e+c d x)}{a e g-c d f}\right)}{(m-1) (c d f-a e g)^3}","\frac{c^2 d^2 (d+e x)^m (a e+c d x) \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \, _2F_1\left(3,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right)}{(1-m) (c d f-a e g)^3}",1,"-((c^2*d^2*(d + e*x)^(-1 + m)*((a*e + c*d*x)*(d + e*x))^(1 - m)*Hypergeometric2F1[3, 1 - m, 2 - m, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^3*(-1 + m)))","A",1
775,1,93,105,0.0529462,"\int (d+e x)^m (f+g x)^{3/2} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(f + g*x)^(3/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","\frac{2 (f+g x)^{5/2} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{5 g}","\frac{2 (f+g x)^{5/2} (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \left(-\frac{g (a e+c d x)}{c d f-a e g}\right)^m \, _2F_1\left(\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{5 g}",1,"(2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*(f + g*x)^(5/2)*Hypergeometric2F1[5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(5*g*((a*e + c*d*x)*(d + e*x))^m)","A",1
776,1,93,105,0.0385588,"\int (d+e x)^m \sqrt{f+g x} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","\frac{2 (f+g x)^{3/2} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(\frac{3}{2},m;\frac{5}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{3 g}","\frac{2 (f+g x)^{3/2} (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \left(-\frac{g (a e+c d x)}{c d f-a e g}\right)^m \, _2F_1\left(\frac{3}{2},m;\frac{5}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{3 g}",1,"(2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*(f + g*x)^(3/2)*Hypergeometric2F1[3/2, m, 5/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(3*g*((a*e + c*d*x)*(d + e*x))^m)","A",1
777,1,91,103,0.0326399,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{\sqrt{f+g x}} \, dx","Integrate[(d + e*x)^m/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","\frac{2 \sqrt{f+g x} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(\frac{1}{2},m;\frac{3}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{g}","\frac{2 \sqrt{f+g x} (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \left(-\frac{g (a e+c d x)}{c d f-a e g}\right)^m \, _2F_1\left(\frac{1}{2},m;\frac{3}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{g}",1,"(2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*Sqrt[f + g*x]*Hypergeometric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*((a*e + c*d*x)*(d + e*x))^m)","A",1
778,1,91,103,0.0342664,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{(f+g x)^{3/2}} \, dx","Integrate[(d + e*x)^m/((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","-\frac{2 (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(-\frac{1}{2},m;\frac{1}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{g \sqrt{f+g x}}","-\frac{2 (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \left(-\frac{g (a e+c d x)}{c d f-a e g}\right)^m \, _2F_1\left(-\frac{1}{2},m;\frac{1}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{g \sqrt{f+g x}}",1,"(-2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*Hypergeometric2F1[-1/2, m, 1/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*((a*e + c*d*x)*(d + e*x))^m*Sqrt[f + g*x])","A",1
779,1,93,105,0.0374233,"\int \frac{(d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m}}{(f+g x)^{5/2}} \, dx","Integrate[(d + e*x)^m/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]","-\frac{2 (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left(\frac{g (a e+c d x)}{a e g-c d f}\right)^m \, _2F_1\left(-\frac{3}{2},m;-\frac{1}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{3 g (f+g x)^{3/2}}","-\frac{2 (d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \left(-\frac{g (a e+c d x)}{c d f-a e g}\right)^m \, _2F_1\left(-\frac{3}{2},m;-\frac{1}{2};\frac{c d (f+g x)}{c d f-a e g}\right)}{3 g (f+g x)^{3/2}}",1,"(-2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*Hypergeometric2F1[-3/2, m, -1/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(3*g*((a*e + c*d*x)*(d + e*x))^m*(f + g*x)^(3/2))","A",1
780,1,53,65,0.0294442,"\int (a e+c d x)^n (d+e x)^m \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((a*e + c*d*x)^n*(d + e*x)^m)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","\frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} (a e+c d x)^{n+1}}{-c d m+c d n+c d}","\frac{(d+e x)^{m-1} \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{1-m} (a e+c d x)^n}{c d (-m+n+1)}",1,"((a*e + c*d*x)^(1 + n)*(d + e*x)^m)/((c*d - c*d*m + c*d*n)*((a*e + c*d*x)*(d + e*x))^m)","A",1
781,1,64,78,0.0349389,"\int (d+e x)^m \left(c d^2 e g-e \left(c d^2+a e^2\right) g-c d e^2 g x\right)^{-1+m} \left(a d e+\left(c d^2+a e^2\right) x+c d e x^2\right)^{-m} \, dx","Integrate[((d + e*x)^m*(c*d^2*e*g - e*(c*d^2 + a*e^2)*g - c*d*e^2*g*x)^(-1 + m))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]","-\frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} \log (a e+c d x) \left(-e^2 g (a e+c d x)\right)^m}{c d e^2 g}","-\frac{(d+e x)^m \left(x \left(a e^2+c d^2\right)+a d e+c d e x^2\right)^{-m} \log (a e+c d x) \left(-a e^3 g-c d e^2 g x\right)^m}{c d e^2 g}",1,"-(((-(e^2*g*(a*e + c*d*x)))^m*(d + e*x)^m*Log[a*e + c*d*x])/(c*d*e^2*g*((a*e + c*d*x)*(d + e*x))^m))","A",1
782,1,145,213,0.1250512,"\int \frac{(d+e x)^{3/2} (f+g x)^n}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^n)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{(f+g x)^n \sqrt{(d+e x) (a e+c d x)} \left(\left(c d (d g (2 n+3)-e f)-2 a e^2 g (n+1)\right) \left(\frac{c d (f+g x)}{c d f-a e g}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)+c d e (f+g x)\right)}{c^2 d^2 g \left(n+\frac{3}{2}\right) \sqrt{d+e x}}","\frac{\sqrt{d+e x} (f+g x)^{n+1} (a e+c d x) \left(2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right) \, _2F_1\left(1,n+\frac{3}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right)}{c d g (n+1) (2 n+3) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)}+\frac{2 e (f+g x)^{n+1} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d g (2 n+3) \sqrt{d+e x}}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^n*(c*d*e*(f + g*x) + ((-2*a*e^2*g*(1 + n) + c*d*(-(e*f) + d*g*(3 + 2*n)))*Hypergeometric2F1[1/2, -n, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*(f + g*x))/(c*d*f - a*e*g))^n))/(c^2*d^2*g*(3/2 + n)*Sqrt[d + e*x])","A",1
783,1,246,501,0.4185058,"\int \frac{(d+e x)^{3/2} (f+g x)^4}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(3465 \left(c d^2-a e^2\right) (c d f-a e g)^4-385 g^3 (a e+c d x)^4 \left(5 a e^2 g-c d (d g+4 e f)\right)+990 g^2 (a e+c d x)^3 (c d f-a e g) \left(c d (2 d g+3 e f)-5 a e^2 g\right)+1386 g (a e+c d x)^2 (c d f-a e g)^2 \left(c d (3 d g+2 e f)-5 a e^2 g\right)+1155 (a e+c d x) (c d f-a e g)^3 \left(c d (4 d g+e f)-5 a e^2 g\right)+315 e g^4 (a e+c d x)^5\right)}{3465 c^6 d^6 \sqrt{d+e x}}","\frac{128 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3 \left(10 a e^2 g+c d (e f-11 d g)\right) \left(2 a e^2 g-c d (3 e f-d g)\right)}{3465 c^6 d^6 e g \sqrt{d+e x}}-\frac{128 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^3 \left(10 a e^2 g+c d (e f-11 d g)\right)}{3465 c^5 d^5 e}-\frac{32 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2 \left(10 a e^2 g+c d (e f-11 d g)\right)}{1155 c^4 d^4 g \sqrt{d+e x}}-\frac{16 (f+g x)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(10 a e^2 g+c d (e f-11 d g)\right)}{693 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(10 a e^2 g+c d (e f-11 d g)\right)}{99 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^5 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{11 c d g \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3465*(c*d^2 - a*e^2)*(c*d*f - a*e*g)^4 + 1155*(c*d*f - a*e*g)^3*(-5*a*e^2*g + c*d*(e*f + 4*d*g))*(a*e + c*d*x) + 1386*g*(c*d*f - a*e*g)^2*(-5*a*e^2*g + c*d*(2*e*f + 3*d*g))*(a*e + c*d*x)^2 + 990*g^2*(c*d*f - a*e*g)*(-5*a*e^2*g + c*d*(3*e*f + 2*d*g))*(a*e + c*d*x)^3 - 385*g^3*(5*a*e^2*g - c*d*(4*e*f + d*g))*(a*e + c*d*x)^4 + 315*e*g^4*(a*e + c*d*x)^5))/(3465*c^6*d^6*Sqrt[d + e*x])","A",1
784,1,264,412,0.2726171,"\int \frac{(d+e x)^{3/2} (f+g x)^3}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^3)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(128 a^4 e^5 g^3-16 a^3 c d e^3 g^2 (9 d g+27 e f+4 e g x)+24 a^2 c^2 d^2 e^2 g \left(3 d g (7 f+g x)+e \left(21 f^2+9 f g x+2 g^2 x^2\right)\right)-2 a c^3 d^3 e \left(9 d g \left(35 f^2+14 f g x+3 g^2 x^2\right)+e \left(105 f^3+126 f^2 g x+81 f g^2 x^2+20 g^3 x^3\right)\right)+c^4 d^4 \left(9 d \left(35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right)+e x \left(105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right)\right)\right)}{315 c^5 d^5 \sqrt{d+e x}}","\frac{16 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2 \left(8 a e^2 g+c d (e f-9 d g)\right) \left(2 a e^2 g-c d (3 e f-d g)\right)}{315 c^5 d^5 e g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g)^2 \left(8 a e^2 g+c d (e f-9 d g)\right)}{315 c^4 d^4 e}-\frac{4 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(8 a e^2 g+c d (e f-9 d g)\right)}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(8 a e^2 g+c d (e f-9 d g)\right)}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^5*g^3 - 16*a^3*c*d*e^3*g^2*(27*e*f + 9*d*g + 4*e*g*x) + 24*a^2*c^2*d^2*e^2*g*(3*d*g*(7*f + g*x) + e*(21*f^2 + 9*f*g*x + 2*g^2*x^2)) - 2*a*c^3*d^3*e*(9*d*g*(35*f^2 + 14*f*g*x + 3*g^2*x^2) + e*(105*f^3 + 126*f^2*g*x + 81*f*g^2*x^2 + 20*g^3*x^3)) + c^4*d^4*(9*d*(35*f^3 + 35*f^2*g*x + 21*f*g^2*x^2 + 5*g^3*x^3) + e*x*(105*f^3 + 189*f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3))))/(315*c^5*d^5*Sqrt[d + e*x])","A",1
785,1,169,321,0.1799895,"\int \frac{(d+e x)^{3/2} (f+g x)^2}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x)^2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(-48 a^3 e^4 g^2+8 a^2 c d e^2 g (7 d g+14 e f+3 e g x)-2 a c^2 d^2 e \left(14 d g (5 f+g x)+e \left(35 f^2+28 f g x+9 g^2 x^2\right)\right)+c^3 d^3 \left(7 d \left(15 f^2+10 f g x+3 g^2 x^2\right)+e x \left(35 f^2+42 f g x+15 g^2 x^2\right)\right)\right)}{105 c^4 d^4 \sqrt{d+e x}}","\frac{8 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(6 a e^2 g+c d (e f-7 d g)\right) \left(2 a e^2 g-c d (3 e f-d g)\right)}{105 c^4 d^4 e g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} (c d f-a e g) \left(6 a e^2 g+c d (e f-7 d g)\right)}{105 c^3 d^3 e}-\frac{2 (f+g x)^2 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(6 a e^2 g+c d (e f-7 d g)\right)}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-48*a^3*e^4*g^2 + 8*a^2*c*d*e^2*g*(14*e*f + 7*d*g + 3*e*g*x) - 2*a*c^2*d^2*e*(14*d*g*(5*f + g*x) + e*(35*f^2 + 28*f*g*x + 9*g^2*x^2)) + c^3*d^3*(7*d*(15*f^2 + 10*f*g*x + 3*g^2*x^2) + e*x*(35*f^2 + 42*f*g*x + 15*g^2*x^2))))/(105*c^4*d^4*Sqrt[d + e*x])","A",1
786,1,96,209,0.0961001,"\int \frac{(d+e x)^{3/2} (f+g x)}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(f + g*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(8 a^2 e^3 g-2 a c d e (5 d g+5 e f+2 e g x)+c^2 d^2 (5 d (3 f+g x)+e x (5 f+3 g x))\right)}{15 c^3 d^3 \sqrt{d+e x}}","-\frac{4 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(4 a e^2 g-c d (5 e f-d g)\right)}{15 c^3 d^3 e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(4 a e^2 g-c d (5 e f-d g)\right)}{15 c^2 d^2 e}+\frac{2 g (d+e x)^{3/2} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{5 c d e}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^3*g - 2*a*c*d*e*(5*e*f + 5*d*g + 2*e*g*x) + c^2*d^2*(5*d*(3*f + g*x) + e*x*(5*f + 3*g*x))))/(15*c^3*d^3*Sqrt[d + e*x])","A",1
787,1,54,109,0.0424361,"\int \frac{(d+e x)^{3/2}}{\sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]","\frac{2 \sqrt{(d+e x) (a e+c d x)} \left(c d (3 d+e x)-2 a e^2\right)}{3 c^2 d^2 \sqrt{d+e x}}","\frac{4 \left(c d^2-a e^2\right) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 c d}",1,"(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(3*d + e*x)))/(3*c^2*d^2*Sqrt[d + e*x])","A",1
788,1,140,139,0.107913,"\int \frac{(d+e x)^{3/2}}{(f+g x) \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{2 \sqrt{d+e x} \left(e \sqrt{g} (a e+c d x) \sqrt{c d f-a e g}+c d (d g-e f) \sqrt{a e+c d x} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)\right)}{c d g^{3/2} \sqrt{(d+e x) (a e+c d x)} \sqrt{c d f-a e g}}","\frac{2 e \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{2 (e f-d g) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{3/2} \sqrt{c d f-a e g}}",1,"(2*Sqrt[d + e*x]*(e*Sqrt[g]*Sqrt[c*d*f - a*e*g]*(a*e + c*d*x) + c*d*(-(e*f) + d*g)*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(c*d*g^(3/2)*Sqrt[c*d*f - a*e*g]*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
789,1,155,170,0.1471923,"\int \frac{(d+e x)^{3/2}}{(f+g x)^2 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{d+e x} \left(-\frac{\sqrt{a e+c d x} \left(c d (d g+e f)-2 a e^2 g\right) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{c d f-a e g}}-\frac{\sqrt{g} (d g-e f) (a e+c d x)}{f+g x}\right)}{g^{3/2} \sqrt{(d+e x) (a e+c d x)} (a e g-c d f)}","-\frac{\left(2 a e^2 g-c d (d g+e f)\right) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{g^{3/2} (c d f-a e g)^{3/2}}-\frac{(e f-d g) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x) (c d f-a e g)}",1,"(Sqrt[d + e*x]*(-((Sqrt[g]*(-(e*f) + d*g)*(a*e + c*d*x))/(f + g*x)) - ((-2*a*e^2*g + c*d*(e*f + d*g))*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/Sqrt[c*d*f - a*e*g]))/(g^(3/2)*(-(c*d*f) + a*e*g)*Sqrt[(a*e + c*d*x)*(d + e*x)])","A",1
790,1,189,261,0.4232419,"\int \frac{(d+e x)^{3/2}}{(f+g x)^3 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{c d \left(2 a e^2 g-\frac{1}{2} c d (3 d g+e f)\right) \left(\frac{c d f-a e g}{c d f+c d g x}+\frac{\sqrt{c d f-a e g} \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right)}{\sqrt{g} \sqrt{a e+c d x}}\right)}{(c d f-a e g)^2}+\frac{e f-d g}{(f+g x)^2}\right)}{2 g \sqrt{d+e x} (a e g-c d f)}","-\frac{c d \left(4 a e^2 g-c d (3 d g+e f)\right) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac{(e f-d g) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(4 a e^2 g-c d (3 d g+e f)\right)}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((e*f - d*g)/(f + g*x)^2 + (c*d*(2*a*e^2*g - (c*d*(e*f + 3*d*g))/2)*((c*d*f - a*e*g)/(c*d*f + c*d*g*x) + (Sqrt[c*d*f - a*e*g]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[g]*Sqrt[a*e + c*d*x])))/(c*d*f - a*e*g)^2))/(2*g*(-(c*d*f) + a*e*g)*Sqrt[d + e*x])","A",1
791,1,132,351,0.1032409,"\int \frac{(d+e x)^{3/2}}{(f+g x)^4 \sqrt{a d e+\left(c d^2+a e^2\right) x+c d e x^2}} \, dx","Integrate[(d + e*x)^(3/2)/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]","\frac{\sqrt{(d+e x) (a e+c d x)} \left(\frac{e f-d g}{(f+g x)^3}-\frac{c^2 d^2 \left(c d (5 d g+e f)-6 a e^2 g\right) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right)}{(c d f-a e g)^3}\right)}{3 g \sqrt{d+e x} (a e g-c d f)}","-\frac{c^2 d^2 \left(6 a e^2 g-c d (5 d g+e f)\right) \tan ^{-1}\left(\frac{\sqrt{g} \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right)}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac{c d \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(6 a e^2 g-c d (5 d g+e f)\right)}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{\sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2} \left(6 a e^2 g-c d (5 d g+e f)\right)}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{(e f-d g) \sqrt{x \left(a e^2+c d^2\right)+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)}",1,"(Sqrt[(a*e + c*d*x)*(d + e*x)]*((e*f - d*g)/(f + g*x)^3 - (c^2*d^2*(-6*a*e^2*g + c*d*(e*f + 5*d*g))*Hypergeometric2F1[1/2, 3, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*f - a*e*g)^3))/(3*g*(-(c*d*f) + a*e*g)*Sqrt[d + e*x])","C",1
792,1,229,324,0.2641846,"\int \frac{\left(a+b x+c x^2\right)^3}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)^3/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{15 \sin ^{-1}(d x) \left(16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right)-d \sqrt{1-d^2 x^2} \left(48 b \left(15 a^2 d^4+10 a c d^2 \left(d^2 x^2+2\right)+c^2 \left(3 d^4 x^4+4 d^2 x^2+8\right)\right)+5 c x \left(72 a^2 d^4+18 a c d^2 \left(2 d^2 x^2+3\right)+c^2 \left(8 d^4 x^4+10 d^2 x^2+15\right)\right)+90 b^2 d^2 x \left(4 a d^2+c \left(2 d^2 x^2+3\right)\right)+80 b^3 d^2 \left(d^2 x^2+2\right)\right)}{240 d^7}","-\frac{b \sqrt{1-d^2 x^2} \left(45 a^2 d^4+60 a c d^2+10 b^2 d^2+24 c^2\right)}{15 d^6}-\frac{x \sqrt{1-d^2 x^2} \left(24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right)}{16 d^6}+\frac{\sin ^{-1}(d x) \left(16 a^3 d^6+24 a^2 c d^4+24 a b^2 d^4+18 a c^2 d^2+18 b^2 c d^2+5 c^3\right)}{16 d^7}-\frac{b x^2 \sqrt{1-d^2 x^2} \left(30 a c d^2+5 b^2 d^2+12 c^2\right)}{15 d^4}-\frac{c x^3 \sqrt{1-d^2 x^2} \left(18 a c d^2+18 b^2 d^2+5 c^2\right)}{24 d^4}-\frac{3 b c^2 x^4 \sqrt{1-d^2 x^2}}{5 d^2}-\frac{c^3 x^5 \sqrt{1-d^2 x^2}}{6 d^2}",1,"(-(d*Sqrt[1 - d^2*x^2]*(80*b^3*d^2*(2 + d^2*x^2) + 90*b^2*d^2*x*(4*a*d^2 + c*(3 + 2*d^2*x^2)) + 48*b*(15*a^2*d^4 + 10*a*c*d^2*(2 + d^2*x^2) + c^2*(8 + 4*d^2*x^2 + 3*d^4*x^4)) + 5*c*x*(72*a^2*d^4 + 18*a*c*d^2*(3 + 2*d^2*x^2) + c^2*(15 + 10*d^2*x^2 + 8*d^4*x^4)))) + 15*(5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^3*d^6)*ArcSin[d*x])/(240*d^7)","A",1
793,1,114,166,0.1158425,"\int \frac{\left(a+b x+c x^2\right)^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)^2/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{3 \sin ^{-1}(d x) \left(8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right)-d \sqrt{1-d^2 x^2} \left(16 b \left(3 a d^2+c d^2 x^2+2 c\right)+3 c x \left(8 a d^2+2 c d^2 x^2+3 c\right)+12 b^2 d^2 x\right)}{24 d^5}","\frac{\sin ^{-1}(d x) \left(8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right)}{8 d^5}-\frac{x \sqrt{1-d^2 x^2} \left(c \left(8 a+\frac{3 c}{d^2}\right)+4 b^2\right)}{8 d^2}-\frac{2 b \sqrt{1-d^2 x^2} \left(3 a d^2+2 c\right)}{3 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}",1,"(-(d*Sqrt[1 - d^2*x^2]*(12*b^2*d^2*x + 16*b*(2*c + 3*a*d^2 + c*d^2*x^2) + 3*c*x*(3*c + 8*a*d^2 + 2*c*d^2*x^2))) + 3*(3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*ArcSin[d*x])/(24*d^5)","A",1
794,1,45,63,0.0326952,"\int \frac{a+b x+c x^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{\left(2 a d^2+c\right) \sin ^{-1}(d x)-d \sqrt{1-d^2 x^2} (2 b+c x)}{2 d^3}","\frac{\left(2 a d^2+c\right) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2}",1,"(-(d*(2*b + c*x)*Sqrt[1 - d^2*x^2]) + (c + 2*a*d^2)*ArcSin[d*x])/(2*d^3)","A",1
795,1,260,282,0.5584789,"\int \frac{1}{\sqrt{1-d x} \sqrt{1+d x} \left(a+b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(a + b*x + c*x^2)),x]","\frac{2 \sqrt{2} c \left(\frac{\tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{-2 b d^2 \left(\sqrt{b^2-4 a c}+b\right)+4 a c d^2+4 c^2}}\right)}{2 \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}-\frac{\tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{2 b d^2 \left(\sqrt{b^2-4 a c}-b\right)+4 a c d^2+4 c^2}}\right)}{2 \sqrt{b d^2 \left(\sqrt{b^2-4 a c}-b\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{b^2-4 a c}}","\frac{\sqrt{2} c \tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}-\frac{\sqrt{2} c \tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{b^2-4 a c} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2}}",1,"(2*Sqrt[2]*c*(-1/2*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 + 2*b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])]/Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2] + ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])]/(2*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2])))/Sqrt[b^2 - 4*a*c]","A",1
796,1,508,571,1.2822085,"\int \frac{1}{\sqrt{1-d x} \sqrt{1+d x} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(a + b*x + c*x^2)^2),x]","\frac{\frac{c \left(c d^2 \left(8 a^2 d^2+b \sqrt{b^2-4 a c}-5 b^2\right)-a b d^4 \left(\sqrt{b^2-4 a c}+b\right)+12 a c^2 d^2+4 c^3\right) \tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{2 b d^2 \left(\sqrt{b^2-4 a c}-b\right)+4 a c d^2+4 c^2}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{b d^2 \left(\sqrt{b^2-4 a c}-b\right)+2 a c d^2+2 c^2}}+\frac{c \left(c d^2 \left(-8 a^2 d^2+b \sqrt{b^2-4 a c}+5 b^2\right)+a b d^4 \left(b-\sqrt{b^2-4 a c}\right)-12 a c^2 d^2-4 c^3\right) \tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{-2 b d^2 \left(\sqrt{b^2-4 a c}+b\right)+4 a c d^2+4 c^2}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}+\frac{\sqrt{1-d^2 x^2} \left(-b c \left(3 a d^2+c\right)-2 c^2 x \left(a d^2+c\right)+b^3 d^2+b^2 c d^2 x\right)}{a+x (b+c x)}}{\left(b^2-4 a c\right) \left(\left(a d^2+c\right)^2-b^2 d^2\right)}","-\frac{c \left(-c d^2 \left(-8 a^2 d^2-b \sqrt{b^2-4 a c}+5 b^2\right)-a b d^4 \left(\sqrt{b^2-4 a c}+b\right)+12 a c^2 d^2+4 c^3\right) \tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2} \left(b^2 d^2-\left(a d^2+c\right)^2\right)}+\frac{c \left(-4 c d^2 \left(b^2-2 a^2 d^2\right)-b d^2 \left(\sqrt{b^2-4 a c}+b\right) \left(c-a d^2\right)-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right) \tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2} \left(b^2 d^2-\left(a d^2+c\right)^2\right)}-\frac{\sqrt{1-d^2 x^2} \left(b \left(b^2 d^2-c \left(3 a d^2+c\right)\right)-c x \left(2 a c d^2-b^2 d^2+2 c^2\right)\right)}{\left(b^2-4 a c\right) \left(b^2 d^2-\left(a d^2+c\right)^2\right) \left(a+b x+c x^2\right)}",1,"(((b^3*d^2 - b*c*(c + 3*a*d^2) + b^2*c*d^2*x - 2*c^2*(c + a*d^2)*x)*Sqrt[1 - d^2*x^2])/(a + x*(b + c*x)) + (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 + c*d^2*(-5*b^2 + b*Sqrt[b^2 - 4*a*c] + 8*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 + 2*b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2]) + (c*(-4*c^3 - 12*a*c^2*d^2 + a*b*(b - Sqrt[b^2 - 4*a*c])*d^4 + c*d^2*(5*b^2 + b*Sqrt[b^2 - 4*a*c] - 8*a^2*d^2))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]))/((b^2 - 4*a*c)*(-(b^2*d^2) + (c + a*d^2)^2))","A",1
797,1,239,276,0.2412414,"\int \frac{\left(a+b x+c x^2\right)^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^3/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)),x]","\frac{-3 \sqrt{1-d^2 x^2} \sin ^{-1}(d x) \left(8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right)-8 b \left(-3 a^2 d^5+6 a c d^3 \left(d^2 x^2-2\right)+c^2 d \left(d^4 x^4+4 d^2 x^2-8\right)\right)+d x \left(8 a^3 d^6+24 a^2 c d^4-12 a c^2 d^2 \left(d^2 x^2-3\right)+c^3 \left(-2 d^4 x^4-5 d^2 x^2+15\right)\right)-12 b^2 d^3 x \left(c \left(d^2 x^2-3\right)-2 a d^2\right)-8 b^3 d^3 \left(d^2 x^2-2\right)}{8 d^7 \sqrt{1-d^2 x^2}}","\frac{x \left(a d^2+c\right) \left(a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right)+b d^4 \left(3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right)}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left(8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right)}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left(12 a c d^2+12 b^2 d^2+7 c^2\right)}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left(6 a c d^2+b^2 d^2+5 c^2\right)}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}",1,"(-8*b^3*d^3*(-2 + d^2*x^2) - 12*b^2*d^3*x*(-2*a*d^2 + c*(-3 + d^2*x^2)) + d*x*(24*a^2*c*d^4 + 8*a^3*d^6 - 12*a*c^2*d^2*(-3 + d^2*x^2) + c^3*(15 - 5*d^2*x^2 - 2*d^4*x^4)) - 8*b*(-3*a^2*d^5 + 6*a*c*d^3*(-2 + d^2*x^2) + c^2*d*(-8 + 4*d^2*x^2 + d^4*x^4)) - 3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*Sqrt[1 - d^2*x^2]*ArcSin[d*x])/(8*d^7*Sqrt[1 - d^2*x^2])","A",1
798,1,127,135,0.1166182,"\int \frac{\left(a+b x+c x^2\right)^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)^2/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)),x]","\frac{d x \left(2 a^2 d^4+4 a c d^2+c^2 \left(3-d^2 x^2\right)\right)-\sqrt{1-d^2 x^2} \sin ^{-1}(d x) \left(4 a c d^2+2 b^2 d^2+3 c^2\right)+4 b d \left(a d^2+c \left(2-d^2 x^2\right)\right)+2 b^2 d^3 x}{2 d^5 \sqrt{1-d^2 x^2}}","\frac{x \left(a^2 d^4+2 a c d^2+b^2 d^2+c^2\right)+2 b d^2 \left(a+\frac{c}{d^2}\right)}{d^4 \sqrt{1-d^2 x^2}}-\frac{\sin ^{-1}(d x) \left(c \left(4 a+\frac{3 c}{d^2}\right)+2 b^2\right)}{2 d^3}+\frac{2 b c \sqrt{1-d^2 x^2}}{d^4}+\frac{c^2 x \sqrt{1-d^2 x^2}}{2 d^4}",1,"(2*b^2*d^3*x + 4*b*d*(a*d^2 + c*(2 - d^2*x^2)) + d*x*(4*a*c*d^2 + 2*a^2*d^4 + c^2*(3 - d^2*x^2)) - (3*c^2 + 2*b^2*d^2 + 4*a*c*d^2)*Sqrt[1 - d^2*x^2]*ArcSin[d*x])/(2*d^5*Sqrt[1 - d^2*x^2])","A",1
799,1,39,40,0.0476244,"\int \frac{a+b x+c x^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)),x]","\frac{\frac{d \left(x \left(a d^2+c\right)+b\right)}{\sqrt{1-d^2 x^2}}-c \sin ^{-1}(d x)}{d^3}","\frac{x \left(a d^2+c\right)+b}{d^2 \sqrt{1-d^2 x^2}}-\frac{c \sin ^{-1}(d x)}{d^3}",1,"((d*(b + (c + a*d^2)*x))/Sqrt[1 - d^2*x^2] - c*ArcSin[d*x])/d^3","A",1
800,1,335,443,2.393562,"\int \frac{1}{(1-d x)^{3/2} (1+d x)^{3/2} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)*(a + b*x + c*x^2)),x]","\frac{d^2 \left(x \left(a d^2+c\right)-b\right)}{\sqrt{1-d^2 x^2} \left(a^2 d^4+2 a c d^2-b^2 d^2+c^2\right)}-\frac{2 \sqrt{2} c^3 \tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{2 b d^2 \left(\sqrt{b^2-4 a c}-b\right)+4 a c d^2+4 c^2}}\right)}{\sqrt{b^2-4 a c} \left(b d^2 \left(\sqrt{b^2-4 a c}-b\right)+2 a c d^2+2 c^2\right)^{3/2}}+\frac{2 \sqrt{2} c^3 \tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{1-d^2 x^2} \sqrt{-2 b d^2 \left(\sqrt{b^2-4 a c}+b\right)+4 a c d^2+4 c^2}}\right)}{\sqrt{b^2-4 a c} \left(-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2\right)^{3/2}}","\frac{c \left(-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2\right) \tanh ^{-1}\left(\frac{d^2 x \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2} \left(b^2 d^2-\left(a d^2+c\right)^2\right)}-\frac{c \left(-b d^2 \left(b-\sqrt{b^2-4 a c}\right)+2 a c d^2+2 c^2\right) \tanh ^{-1}\left(\frac{d^2 x \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2}}\right)}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left(\sqrt{b^2-4 a c}+b\right)+2 a c d^2+2 c^2} \left(b^2 d^2-\left(a d^2+c\right)^2\right)}+\frac{d^2 \left(b-x \left(a d^2+c\right)\right)}{\sqrt{1-d^2 x^2} \left(b^2 d^2-\left(a d^2+c\right)^2\right)}",1,"(d^2*(-b + (c + a*d^2)*x))/((c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*Sqrt[1 - d^2*x^2]) - (2*Sqrt[2]*c^3*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 + 2*b*(-b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2)) + (2*Sqrt[2]*c^3*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2))","A",1
801,1,890,939,9.2187526,"\int \frac{1}{(1-d x)^{3/2} (1+d x)^{3/2} \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)*(a + b*x + c*x^2)^2),x]","\frac{\frac{\sqrt{2} \left(-3 a b \left(b+\sqrt{b^2-4 a c}\right) d^4+20 a c^2 d^2-c \left(7 b^2-3 \sqrt{b^2-4 a c} b-16 a^2 d^2\right) d^2+4 c^3\right) \tanh ^{-1}\left(\frac{\left(b-\sqrt{b^2-4 a c}\right) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left(b-\sqrt{b^2-4 a c}\right) d^2} \sqrt{1-d^2 x^2}}\right) c^3}{\sqrt{b^2-4 a c} \left(2 c^2+2 a d^2 c-b \left(b-\sqrt{b^2-4 a c}\right) d^2\right)^{3/2}}-\frac{\sqrt{2} \left(-3 a b \left(b-\sqrt{b^2-4 a c}\right) d^4+20 a c^2 d^2-c \left(7 b^2+3 \sqrt{b^2-4 a c} b-16 a^2 d^2\right) d^2+4 c^3\right) \tanh ^{-1}\left(\frac{\left(b+\sqrt{b^2-4 a c}\right) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left(b+\sqrt{b^2-4 a c}\right) d^2} \sqrt{1-d^2 x^2}}\right) c^3}{\sqrt{b^2-4 a c} \left(2 c^2+2 a d^2 c-b \left(b+\sqrt{b^2-4 a c}\right) d^2\right)^{3/2}}-\frac{\left(3 b \left(c-a d^2\right) d^2+\frac{-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt{b^2-4 a c}}\right) \left(2 c-\left(b-\sqrt{b^2-4 a c}\right) d^2 x\right) c}{\left(4 c^2-\left(b-\sqrt{b^2-4 a c}\right)^2 d^2\right) \sqrt{1-d^2 x^2}}-\frac{\left(3 b d^2 \left(c-a d^2\right)-\frac{-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt{b^2-4 a c}}\right) \left(2 c-\left(b+\sqrt{b^2-4 a c}\right) d^2 x\right) c}{\left(4 c^2-\left(b+\sqrt{b^2-4 a c}\right)^2 d^2\right) \sqrt{1-d^2 x^2}}+\frac{2 d^2 \left(2 c^2+2 a d^2 c-b^2 d^2\right) x}{\sqrt{1-d^2 x^2}}}{\left(b^2-4 a c\right) \left(\left(a d^2+c\right)^2-b^2 d^2\right)}-\frac{-d^2 b^3+c \left(3 a d^2+c\right) b+c \left(2 c^2+2 a d^2 c-b^2 d^2\right) x}{\left(b^2-4 a c\right) \left(\left(a d^2+c\right)^2-b^2 d^2\right) \left(c x^2+b x+a\right) \sqrt{1-d^2 x^2}}","-\frac{\left(b \left(3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right)-\left(2 c^4-d^2 \left(b^2+6 a^2 d^2\right) c^2-\left(4 a^3 d^6+6 a b^2 d^4\right) c+b^2 d^4 \left(2 b^2+a^2 d^2\right)\right) x\right) d^2}{\left(b^2-4 a c\right) \left(a d^2-b d+c\right)^2 \left(a d^2+b d+c\right)^2 \sqrt{1-d^2 x^2}}+\frac{c \left(3 a b^3 \left(b+\sqrt{b^2-4 a c}\right) d^6-2 a c^2 \left(7 b^2+5 \sqrt{b^2-4 a c} b-8 a^2 d^2\right) d^4+b c \left(2 b^3+2 \sqrt{b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt{b^2-4 a c} d^2\right) d^4+24 a c^4 d^2-c^3 \left(9 b^2-\sqrt{b^2-4 a c} b-36 a^2 d^2\right) d^2+4 c^5\right) \tanh ^{-1}\left(\frac{\left(b-\sqrt{b^2-4 a c}\right) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left(b-\sqrt{b^2-4 a c}\right) d^2} \sqrt{1-d^2 x^2}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left(b-\sqrt{b^2-4 a c}\right) d^2} \left(a^2 d^4-b^2 d^2+2 a c d^2+c^2\right)^2}-\frac{b \left(b^2 d^2-c \left(3 a d^2+c\right)\right)-c \left(2 c^2+2 a d^2 c-b^2 d^2\right) x}{\left(b^2-4 a c\right) \left(b^2 d^2-\left(a d^2+c\right)^2\right) \left(c x^2+b x+a\right) \sqrt{1-d^2 x^2}}+\frac{c \left(b \left(b+\sqrt{b^2-4 a c}\right) d^4 \left(3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right)-2 \left(3 a b^4 d^8+2 b^2 c \left(b^2-7 a^2 d^2\right) d^6+12 a c^4 d^4+2 c^5 d^2-c^3 \left(4 b^2 d^4-18 a^2 d^6\right)-4 c^2 \left(3 a b^2 d^6-2 a^3 d^8\right)\right)\right) \tanh ^{-1}\left(\frac{\left(b+\sqrt{b^2-4 a c}\right) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left(b+\sqrt{b^2-4 a c}\right) d^2} \sqrt{1-d^2 x^2}}\right)}{\sqrt{2} \left(b^2-4 a c\right)^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left(b+\sqrt{b^2-4 a c}\right) d^2} \left(a^2 d^4-b^2 d^2+2 a c d^2+c^2\right)^2 d^2}",1,"-((-(b^3*d^2) + b*c*(c + 3*a*d^2) + c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/((b^2 - 4*a*c)*(-(b^2*d^2) + (c + a*d^2)^2)*(a + b*x + c*x^2)*Sqrt[1 - d^2*x^2])) + ((2*d^2*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/Sqrt[1 - d^2*x^2] - (c*(3*b*d^2*(c - a*d^2) + (4*c^3 - 7*b^2*c*d^2 + 20*a*c^2*d^2 - 3*a*b^2*d^4 + 16*a^2*c*d^4)/Sqrt[b^2 - 4*a*c])*(2*c - (b - Sqrt[b^2 - 4*a*c])*d^2*x))/((4*c^2 - (b - Sqrt[b^2 - 4*a*c])^2*d^2)*Sqrt[1 - d^2*x^2]) - (c*(3*b*d^2*(c - a*d^2) - (4*c^3 - 7*b^2*c*d^2 + 20*a*c^2*d^2 - 3*a*b^2*d^4 + 16*a^2*c*d^4)/Sqrt[b^2 - 4*a*c])*(2*c - (b + Sqrt[b^2 - 4*a*c])*d^2*x))/((4*c^2 - (b + Sqrt[b^2 - 4*a*c])^2*d^2)*Sqrt[1 - d^2*x^2]) + (Sqrt[2]*c^3*(4*c^3 + 20*a*c^2*d^2 - 3*a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(7*b^2 - 3*b*Sqrt[b^2 - 4*a*c] - 16*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2)^(3/2)) - (Sqrt[2]*c^3*(4*c^3 + 20*a*c^2*d^2 - 3*a*b*(b - Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(7*b^2 + 3*b*Sqrt[b^2 - 4*a*c] - 16*a^2*d^2))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2)))/((b^2 - 4*a*c)*(-(b^2*d^2) + (c + a*d^2)^2))","A",1
802,1,167,54,0.2026643,"\int (1-e x)^m (1+e x)^m \left(a+c x^2\right)^p \, dx","Integrate[(1 - e*x)^m*(1 + e*x)^m*(a + c*x^2)^p,x]","\frac{3 a x \left(1-e^2 x^2\right)^m \left(a+c x^2\right)^p F_1\left(\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right)}{2 x^2 \left(c p F_1\left(\frac{3}{2};1-p,-m;\frac{5}{2};-\frac{c x^2}{a},e^2 x^2\right)-a e^2 m F_1\left(\frac{3}{2};-p,1-m;\frac{5}{2};-\frac{c x^2}{a},e^2 x^2\right)\right)+3 a F_1\left(\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right)}","x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right)",1,"(3*a*x*(a + c*x^2)^p*(1 - e^2*x^2)^m*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2])/(3*a*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2] + 2*x^2*(c*p*AppellF1[3/2, 1 - p, -m, 5/2, -((c*x^2)/a), e^2*x^2] - a*e^2*m*AppellF1[3/2, -p, 1 - m, 5/2, -((c*x^2)/a), e^2*x^2]))","B",0
803,0,0,89,0.0932145,"\int (d-e x)^m (d+e x)^m \left(a+c x^2\right)^p \, dx","Integrate[(d - e*x)^m*(d + e*x)^m*(a + c*x^2)^p,x]","\int (d-e x)^m (d+e x)^m \left(a+c x^2\right)^p \, dx","x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} (d-e x)^m (d+e x)^m \left(1-\frac{e^2 x^2}{d^2}\right)^{-m} F_1\left(\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)",1,"Integrate[(d - e*x)^m*(d + e*x)^m*(a + c*x^2)^p, x]","F",-1
804,0,0,92,0.0799818,"\int (d+e x)^m (d f-e f x)^m \left(a+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p,x]","\int (d+e x)^m (d f-e f x)^m \left(a+c x^2\right)^p \, dx","x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} (d+e x)^m \left(1-\frac{e^2 x^2}{d^2}\right)^{-m} (d f-e f x)^m F_1\left(\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},\frac{e^2 x^2}{d^2}\right)",1,"Integrate[(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p, x]","F",-1
805,1,249,275,0.3128125,"\int (d+e x)^3 (f+g x)^n \left(a+2 c d x+c e x^2\right) \, dx","Integrate[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]","\frac{(f+g x)^{n+1} \left(\frac{e^2 (f+g x)^3 \left(a e g^2+c \left(9 d^2 g^2-20 d e f g+10 e^2 f^2\right)\right)}{n+4}-\frac{e (f+g x)^2 (e f-d g) \left(3 a e g^2+c \left(7 d^2 g^2-20 d e f g+10 e^2 f^2\right)\right)}{n+3}+\frac{(f+g x) (e f-d g)^2 \left(3 a e g^2+c \left(2 d^2 g^2-10 d e f g+5 e^2 f^2\right)\right)}{n+2}-\frac{(e f-d g)^3 \left(a g^2+c f (e f-2 d g)\right)}{n+1}-\frac{5 c e^3 (f+g x)^4 (e f-d g)}{n+5}+\frac{c e^4 (f+g x)^5}{n+6}\right)}{g^6}","\frac{(e f-d g)^2 (f+g x)^{n+2} \left(3 a e g^2+c \left(2 d^2 g^2-10 d e f g+5 e^2 f^2\right)\right)}{g^6 (n+2)}-\frac{e (e f-d g) (f+g x)^{n+3} \left(3 a e g^2+c \left(7 d^2 g^2-20 d e f g+10 e^2 f^2\right)\right)}{g^6 (n+3)}+\frac{e^2 (f+g x)^{n+4} \left(a e g^2+c \left(9 d^2 g^2-20 d e f g+10 e^2 f^2\right)\right)}{g^6 (n+4)}-\frac{(e f-d g)^3 (f+g x)^{n+1} \left(a g^2+c f (e f-2 d g)\right)}{g^6 (n+1)}-\frac{5 c e^3 (e f-d g) (f+g x)^{n+5}}{g^6 (n+5)}+\frac{c e^4 (f+g x)^{n+6}}{g^6 (n+6)}",1,"((f + g*x)^(1 + n)*(-(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g)))/(1 + n)) + ((e*f - d*g)^2*(3*a*e*g^2 + c*(5*e^2*f^2 - 10*d*e*f*g + 2*d^2*g^2))*(f + g*x))/(2 + n) - (e*(e*f - d*g)*(3*a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g + 7*d^2*g^2))*(f + g*x)^2)/(3 + n) + (e^2*(a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g + 9*d^2*g^2))*(f + g*x)^3)/(4 + n) - (5*c*e^3*(e*f - d*g)*(f + g*x)^4)/(5 + n) + (c*e^4*(f + g*x)^5)/(6 + n)))/g^6","A",1
806,1,187,208,0.1987847,"\int (d+e x)^2 (f+g x)^n \left(a+2 c d x+c e x^2\right) \, dx","Integrate[(d + e*x)^2*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]","\frac{(f+g x)^{n+1} \left(\frac{e (f+g x)^2 \left(a e g^2+c \left(5 d^2 g^2-12 d e f g+6 e^2 f^2\right)\right)}{n+3}-\frac{2 (f+g x) (e f-d g) \left(a e g^2+c \left(d^2 g^2-4 d e f g+2 e^2 f^2\right)\right)}{n+2}+\frac{(e f-d g)^2 \left(a g^2+c f (e f-2 d g)\right)}{n+1}-\frac{4 c e^2 (f+g x)^3 (e f-d g)}{n+4}+\frac{c e^3 (f+g x)^4}{n+5}\right)}{g^5}","-\frac{2 (e f-d g) (f+g x)^{n+2} \left(a e g^2+c \left(d^2 g^2-4 d e f g+2 e^2 f^2\right)\right)}{g^5 (n+2)}+\frac{e (f+g x)^{n+3} \left(a e g^2+c \left(5 d^2 g^2-12 d e f g+6 e^2 f^2\right)\right)}{g^5 (n+3)}+\frac{(e f-d g)^2 (f+g x)^{n+1} \left(a g^2+c f (e f-2 d g)\right)}{g^5 (n+1)}-\frac{4 c e^2 (e f-d g) (f+g x)^{n+4}}{g^5 (n+4)}+\frac{c e^3 (f+g x)^{n+5}}{g^5 (n+5)}",1,"((f + g*x)^(1 + n)*(((e*f - d*g)^2*(a*g^2 + c*f*(e*f - 2*d*g)))/(1 + n) - (2*(e*f - d*g)*(a*e*g^2 + c*(2*e^2*f^2 - 4*d*e*f*g + d^2*g^2))*(f + g*x))/(2 + n) + (e*(a*e*g^2 + c*(6*e^2*f^2 - 12*d*e*f*g + 5*d^2*g^2))*(f + g*x)^2)/(3 + n) - (4*c*e^2*(e*f - d*g)*(f + g*x)^3)/(4 + n) + (c*e^3*(f + g*x)^4)/(5 + n)))/g^5","A",1
807,1,141,146,0.2796042,"\int (d+e x) (f+g x)^n \left(a+2 c d x+c e x^2\right) \, dx","Integrate[(d + e*x)*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]","\frac{(f+g x)^{n+1} \left(\frac{2 (f+g x) \left(a e g^2 (n+3)+c \left(-d^2 g^2 n-6 d e f g+3 e^2 f^2\right)\right)}{g^2 (n+2)}+\frac{6 (d g-e f) \left(a g^2+c f (e f-2 d g)\right)}{g^2 (n+1)}+(a+c x (2 d+e x)) (d g (n+6)-3 e f+e g (n+3) x)\right)}{g^2 (n+3) (n+4)}","\frac{(f+g x)^{n+2} \left(a e g^2+c \left(2 d^2 g^2-6 d e f g+3 e^2 f^2\right)\right)}{g^4 (n+2)}-\frac{(e f-d g) (f+g x)^{n+1} \left(a g^2+c f (e f-2 d g)\right)}{g^4 (n+1)}-\frac{3 c e (e f-d g) (f+g x)^{n+3}}{g^4 (n+3)}+\frac{c e^2 (f+g x)^{n+4}}{g^4 (n+4)}",1,"((f + g*x)^(1 + n)*((6*(-(e*f) + d*g)*(a*g^2 + c*f*(e*f - 2*d*g)))/(g^2*(1 + n)) + (2*(a*e*g^2*(3 + n) + c*(3*e^2*f^2 - 6*d*e*f*g - d^2*g^2*n))*(f + g*x))/(g^2*(2 + n)) + (-3*e*f + d*g*(6 + n) + e*g*(3 + n)*x)*(a + c*x*(2*d + e*x))))/(g^2*(3 + n)*(4 + n))","A",1
808,1,73,84,0.0998755,"\int (f+g x)^n \left(a+2 c d x+c e x^2\right) \, dx","Integrate[(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]","\frac{(f+g x)^{n+1} \left(\frac{a g^2+c f (e f-2 d g)}{n+1}-\frac{2 c (f+g x) (e f-d g)}{n+2}+\frac{c e (f+g x)^2}{n+3}\right)}{g^3}","\frac{(f+g x)^{n+1} \left(a g^2+c f (e f-2 d g)\right)}{g^3 (n+1)}-\frac{2 c (e f-d g) (f+g x)^{n+2}}{g^3 (n+2)}+\frac{c e (f+g x)^{n+3}}{g^3 (n+3)}",1,"((f + g*x)^(1 + n)*((a*g^2 + c*f*(e*f - 2*d*g))/(1 + n) - (2*c*(e*f - d*g)*(f + g*x))/(2 + n) + (c*e*(f + g*x)^2)/(3 + n)))/g^3","A",1
809,1,93,114,0.1503138,"\int \frac{(f+g x)^n \left(a+2 c d x+c e x^2\right)}{d+e x} \, dx","Integrate[((f + g*x)^n*(a + 2*c*d*x + c*e*x^2))/(d + e*x),x]","\frac{(f+g x)^{n+1} \left(\frac{\left(c d^2-a e\right) \, _2F_1\left(1,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)}{e f-d g}+\frac{c (d g (n+2)-e f+e g (n+1) x)}{g^2 (n+2)}\right)}{e (n+1)}","\frac{\left(c d^2-a e\right) (f+g x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)}{e (n+1) (e f-d g)}-\frac{c (e f-d g) (f+g x)^{n+1}}{e g^2 (n+1)}+\frac{c (f+g x)^{n+2}}{g^2 (n+2)}",1,"((f + g*x)^(1 + n)*((c*(-(e*f) + d*g*(2 + n) + e*g*(1 + n)*x))/(g^2*(2 + n)) + ((c*d^2 - a*e)*Hypergeometric2F1[1, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)])/(e*f - d*g)))/(e*(1 + n))","A",1
810,1,83,88,0.0905611,"\int \frac{(f+g x)^n \left(a+2 c d x+c e x^2\right)}{(d+e x)^2} \, dx","Integrate[((f + g*x)^n*(a + 2*c*d*x + c*e*x^2))/(d + e*x)^2,x]","\frac{(f+g x)^{n+1} \left(g^2 \left(a e-c d^2\right) \, _2F_1\left(2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)+c (e f-d g)^2\right)}{e g (n+1) (e f-d g)^2}","\frac{c (f+g x)^{n+1}}{e g (n+1)}-\frac{g \left(c d^2-a e\right) (f+g x)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)}{e (n+1) (e f-d g)^2}",1,"((f + g*x)^(1 + n)*(c*(e*f - d*g)^2 + (-(c*d^2) + a*e)*g^2*Hypergeometric2F1[2, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)]))/(e*g*(e*f - d*g)^2*(1 + n))","A",1
811,1,106,193,0.0950878,"\int \frac{(f+g x)^n \left(a+2 c d x+c e x^2\right)}{(d+e x)^3} \, dx","Integrate[((f + g*x)^n*(a + 2*c*d*x + c*e*x^2))/(d + e*x)^3,x]","-\frac{(f+g x)^{n+1} \left(g^2 \left(a e-c d^2\right) \, _2F_1\left(3,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)+c (e f-d g)^2 \, _2F_1\left(1,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)\right)}{e (n+1) (e f-d g)^3}","\frac{(f+g x)^{n+1} \left(a e g^2 (1-n) n-c \left(d^2 g^2 \left(-n^2+n+2\right)-4 d e f g+2 e^2 f^2\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)}{2 e (n+1) (e f-d g)^3}-\frac{g (1-n) \left(c d^2-a e\right) (f+g x)^{n+1}}{2 e (d+e x) (e f-d g)^2}-\frac{\left(a-\frac{c d^2}{e}\right) (f+g x)^{n+1}}{2 (d+e x)^2 (e f-d g)}",1,"-(((f + g*x)^(1 + n)*(c*(e*f - d*g)^2*Hypergeometric2F1[1, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)] + (-(c*d^2) + a*e)*g^2*Hypergeometric2F1[3, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)]))/(e*(e*f - d*g)^3*(1 + n)))","A",1
812,1,106,197,0.1040523,"\int \frac{(f+g x)^n \left(a+2 c d x+c e x^2\right)}{(d+e x)^4} \, dx","Integrate[((f + g*x)^n*(a + 2*c*d*x + c*e*x^2))/(d + e*x)^4,x]","\frac{g (f+g x)^{n+1} \left(g^2 \left(a e-c d^2\right) \, _2F_1\left(4,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)+c (e f-d g)^2 \, _2F_1\left(2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)\right)}{e (n+1) (e f-d g)^4}","\frac{g (f+g x)^{n+1} \left(a e g^2 \left(n^2-3 n+2\right)+c \left(d^2 g^2 \left(-n^2+3 n+4\right)-12 d e f g+6 e^2 f^2\right)\right) \, _2F_1\left(2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right)}{6 e (n+1) (e f-d g)^4}-\frac{g (2-n) \left(c d^2-a e\right) (f+g x)^{n+1}}{6 e (d+e x)^2 (e f-d g)^2}-\frac{\left(a-\frac{c d^2}{e}\right) (f+g x)^{n+1}}{3 (d+e x)^3 (e f-d g)}",1,"(g*(f + g*x)^(1 + n)*(c*(e*f - d*g)^2*Hypergeometric2F1[2, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)] + (-(c*d^2) + a*e)*g^2*Hypergeometric2F1[4, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)]))/(e*(e*f - d*g)^4*(1 + n))","A",1
813,1,179,231,0.1847906,"\int (d+e x)^m (f+g x)^n \left(a+2 c d x+c e x^2\right) \, dx","Integrate[(d + e*x)^m*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]","\frac{(d+e x)^{m+1} (f+g x)^n \left(\frac{e (f+g x)}{e f-d g}\right)^{-n} \left(e \left(a g^2+c f (e f-2 d g)\right) \, _2F_1\left(m+1,-n;m+2;\frac{g (d+e x)}{d g-e f}\right)+c (e f-d g)^2 \, _2F_1\left(m+1,-n-2;m+2;\frac{g (d+e x)}{d g-e f}\right)-2 c (e f-d g)^2 \, _2F_1\left(m+1,-n-1;m+2;\frac{g (d+e x)}{d g-e f}\right)\right)}{e^2 g^2 (m+1)}","\frac{(d+e x)^{m+1} (f+g x)^n \left(\frac{e (f+g x)}{e f-d g}\right)^{-n} (g (m+n+2) (a e g (m+n+3)-c d (d g (n+1)+e f (m+2)))+c (m+2) (e f-d g) (d g (n+1)+e f (m+1))) \, _2F_1\left(m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right)}{e^2 g^2 (m+1) (m+n+2) (m+n+3)}-\frac{c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)}",1,"((d + e*x)^(1 + m)*(f + g*x)^n*(c*(e*f - d*g)^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] - 2*c*(e*f - d*g)^2*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e*(a*g^2 + c*f*(e*f - 2*d*g))*Hypergeometric2F1[1 + m, -n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)]))/(e^2*g^2*(1 + m)*((e*(f + g*x))/(e*f - d*g))^n)","A",1
814,1,85,83,0.0493654,"\int \frac{a+b x+c x^2}{(d+e x) (f+g x)} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)*(f + g*x)),x]","-\frac{\log (d+e x) \left(-a e^2+b d e-c d^2\right)}{e^2 (e f-d g)}-\frac{\log (f+g x) \left(a g^2-b f g+c f^2\right)}{g^2 (e f-d g)}+\frac{c x}{e g}","\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)}{e^2 (e f-d g)}-\frac{\log (f+g x) \left(a g^2-b f g+c f^2\right)}{g^2 (e f-d g)}+\frac{c x}{e g}",1,"(c*x)/(e*g) - ((-(c*d^2) + b*d*e - a*e^2)*Log[d + e*x])/(e^2*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)*Log[f + g*x])/(g^2*(e*f - d*g))","A",1
815,1,177,184,0.1480544,"\int \frac{\left(a+b x+c x^2\right)^2}{(d+e x) (f+g x)} \, dx","Integrate[(a + b*x + c*x^2)^2/((d + e*x)*(f + g*x)),x]","-\frac{e g x (d g-e f) \left(6 c e g (2 a e g+b (-2 d g-2 e f+e g x))+6 b^2 e^2 g^2+c^2 \left(6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left(6 f^2-3 f g x+2 g^2 x^2\right)\right)\right)-6 g^4 \log (d+e x) \left(e (a e-b d)+c d^2\right)^2+6 e^4 \log (f+g x) \left(g (a g-b f)+c f^2\right)^2}{6 e^4 g^4 (e f-d g)}","\frac{x \left(-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)}{e^3 g^3}+\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)^2}{e^4 (e f-d g)}-\frac{\log (f+g x) \left(a g^2-b f g+c f^2\right)^2}{g^4 (e f-d g)}-\frac{c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac{c^2 x^3}{3 e g}",1,"-1/6*(e*g*(-(e*f) + d*g)*x*(6*b^2*e^2*g^2 + 6*c*e*g*(2*a*e*g + b*(-2*e*f - 2*d*g + e*g*x)) + c^2*(6*d^2*g^2 - 3*d*e*g*(-2*f + g*x) + e^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2))) - 6*(c*d^2 + e*(-(b*d) + a*e))^2*g^4*Log[d + e*x] + 6*e^4*(c*f^2 + g*(-(b*f) + a*g))^2*Log[f + g*x])/(e^4*g^4*(e*f - d*g))","A",1
816,1,476,531,0.4231998,"\int \frac{\left(a+b x+c x^2\right)^3}{(d+e x) (f+g x)} \, dx","Integrate[(a + b*x + c*x^2)^3/((d + e*x)*(f + g*x)),x]","-\frac{e g x \left(-30 c e^2 g^2 (e f-d g) \left(6 a^2 e^2 g^2+6 a b e g (-2 d g-2 e f+e g x)+b^2 \left(6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left(6 f^2-3 f g x+2 g^2 x^2\right)\right)\right)-30 b^2 e^3 g^3 (e f-d g) (6 a e g+b (-2 d g-2 e f+e g x))+15 c^2 e g \left(b \left(-12 d^4 g^4+6 d^3 e g^4 x-4 d^2 e^2 g^4 x^2+3 d e^3 g^4 x^3+e^4 f \left(12 f^3-6 f^2 g x+4 f g^2 x^2-3 g^3 x^3\right)\right)-2 a e g (e f-d g) \left(6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left(6 f^2-3 f g x+2 g^2 x^2\right)\right)\right)+c^3 \left(60 d^5 g^5-30 d^4 e g^5 x+20 d^3 e^2 g^5 x^2-15 d^2 e^3 g^5 x^3+12 d e^4 g^5 x^4+e^5 f \left(-60 f^4+30 f^3 g x-20 f^2 g^2 x^2+15 f g^3 x^3-12 g^4 x^4\right)\right)\right)-60 g^6 \log (d+e x) \left(e (a e-b d)+c d^2\right)^3+60 e^6 \log (f+g x) \left(g (a g-b f)+c f^2\right)^3}{60 e^6 g^6 (e f-d g)}","-\frac{x \left(-3 c e^2 g^2 \left(a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)+b^2 e^3 g^3 (-3 a e g+b d g+b e f)-3 c^2 e g \left(a e g \left(d^2 g^2+d e f g+e^2 f^2\right)-b \left(d^3 g^3+d^2 e f g^2+d e^2 f^2 g+e^3 f^3\right)\right)-\left(c^3 \left(d^4 g^4+d^3 e f g^3+d^2 e^2 f^2 g^2+d e^3 f^3 g+e^4 f^4\right)\right)\right)}{e^5 g^5}+\frac{x^2 \left(-3 c^2 e g \left(a e g (d g+e f)-b \left(d^2 g^2+d e f g+e^2 f^2\right)\right)-3 b c e^2 g^2 (-2 a e g+b d g+b e f)+b^3 e^3 g^3-\left(c^3 \left(d^3 g^3+d^2 e f g^2+d e^2 f^2 g+e^3 f^3\right)\right)\right)}{2 e^4 g^4}+\frac{c x^3 \left(-3 c e g (-a e g+b d g+b e f)+3 b^2 e^2 g^2+c^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)}{3 e^3 g^3}+\frac{\log (d+e x) \left(a e^2-b d e+c d^2\right)^3}{e^6 (e f-d g)}-\frac{\log (f+g x) \left(a g^2-b f g+c f^2\right)^3}{g^6 (e f-d g)}-\frac{c^2 x^4 (-3 b e g+c d g+c e f)}{4 e^2 g^2}+\frac{c^3 x^5}{5 e g}",1,"-1/60*(e*g*x*(-30*b^2*e^3*g^3*(e*f - d*g)*(6*a*e*g + b*(-2*e*f - 2*d*g + e*g*x)) + c^3*(60*d^5*g^5 - 30*d^4*e*g^5*x + 20*d^3*e^2*g^5*x^2 - 15*d^2*e^3*g^5*x^3 + 12*d*e^4*g^5*x^4 + e^5*f*(-60*f^4 + 30*f^3*g*x - 20*f^2*g^2*x^2 + 15*f*g^3*x^3 - 12*g^4*x^4)) - 30*c*e^2*g^2*(e*f - d*g)*(6*a^2*e^2*g^2 + 6*a*b*e*g*(-2*e*f - 2*d*g + e*g*x) + b^2*(6*d^2*g^2 - 3*d*e*g*(-2*f + g*x) + e^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2))) + 15*c^2*e*g*(-2*a*e*g*(e*f - d*g)*(6*d^2*g^2 - 3*d*e*g*(-2*f + g*x) + e^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2)) + b*(-12*d^4*g^4 + 6*d^3*e*g^4*x - 4*d^2*e^2*g^4*x^2 + 3*d*e^3*g^4*x^3 + e^4*f*(12*f^3 - 6*f^2*g*x + 4*f*g^2*x^2 - 3*g^3*x^3)))) - 60*(c*d^2 + e*(-(b*d) + a*e))^3*g^6*Log[d + e*x] + 60*e^6*(c*f^2 + g*(-(b*f) + a*g))^3*Log[f + g*x])/(e^6*g^6*(e*f - d*g))","A",1
817,1,246,246,0.3237551,"\int \frac{1}{(d+e x) (f+g x) \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)*(f + g*x)*(a + b*x + c*x^2)),x]","\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right)}{\sqrt{4 a c-b^2} \left(e (a e-b d)+c d^2\right) \left(g (a g-b f)+c f^2\right)}+\frac{e^2 \log (d+e x)}{(e f-d g) \left(e (a e-b d)+c d^2\right)}-\frac{\log (a+x (b+c x)) (-b e g+c d g+c e f)}{2 \left(e (a e-b d)+c d^2\right) \left(g (a g-b f)+c f^2\right)}-\frac{g^2 \log (f+g x)}{(e f-d g) \left(g (a g-b f)+c f^2\right)}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right) \left(c f^2-g (b f-a g)\right)}-\frac{\log \left(a+b x+c x^2\right) (-b e g+c d g+c e f)}{2 \left(a e^2-b d e+c d^2\right) \left(c f^2-g (b f-a g)\right)}+\frac{e^2 \log (d+e x)}{(e f-d g) \left(a e^2-b d e+c d^2\right)}-\frac{g^2 \log (f+g x)}{(e f-d g) \left(a g^2-b f g+c f^2\right)}",1,"((2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c]*(c*d^2 + e*(-(b*d) + a*e))*(c*f^2 + g*(-(b*f) + a*g))) + (e^2*Log[d + e*x])/((c*d^2 + e*(-(b*d) + a*e))*(e*f - d*g)) - (g^2*Log[f + g*x])/((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))) - ((c*e*f + c*d*g - b*e*g)*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))*(c*f^2 + g*(-(b*f) + a*g)))","A",1
818,1,710,644,2.6315412,"\int \frac{1}{(d+e x) (f+g x) \left(a+b x+c x^2\right)^2} \, dx","Integrate[1/((d + e*x)*(f + g*x)*(a + b*x + c*x^2)^2),x]","\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-2 c^3 \left(2 a^2 e g \left(d^2 g^2-5 d e f g+e^2 f^2\right)+a b \left(3 d^3 g^3+11 d^2 e f g^2+11 d e^2 f^2 g+3 e^3 f^3\right)-4 b^2 d^2 e f^2 g\right)-2 b^2 c e g \left(-6 a^2 e^2 g^2+2 a b e g (d g+e f)+b^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)+c^2 \left(-12 a^3 e^3 g^3-6 a^2 b e^2 g^2 (d g+e f)+12 a b^2 e g \left(d^2 g^2+d e f g+e^2 f^2\right)+b^3 \left(d^3 g^3+d^2 e f g^2+d e^2 f^2 g+e^3 f^3\right)\right)+b^4 e^2 g^2 (-2 a e g+b d g+b e f)+2 c^4 d f \left(2 a \left(3 d^2 g^2+d e f g+3 e^2 f^2\right)-3 b d f (d g+e f)\right)+4 c^5 d^3 f^3\right)}{\left(4 a c-b^2\right)^{3/2} \left(e (a e-b d)+c d^2\right)^2 \left(g (a g-b f)+c f^2\right)^2}+\frac{b c (3 a e g+c (-d f+d g x+e f x))-2 c^2 (a d g+a e (f-g x)+c d f x)+b^3 (-e) g+b^2 c (d g+e (f-g x))}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (b d-a e)-c d^2\right) \left(g (b f-a g)-c f^2\right)}+\frac{e^4 \log (d+e x)}{(e f-d g) \left(e (a e-b d)+c d^2\right)^2}-\frac{\log (a+x (b+c x)) (-b e g+c d g+c e f) \left(e g (2 a e g-b (d g+e f))+c \left(d^2 g^2+e^2 f^2\right)\right)}{2 \left(e (a e-b d)+c d^2\right)^2 \left(g (a g-b f)+c f^2\right)^2}-\frac{g^4 \log (f+g x)}{(e f-d g) \left(g (a g-b f)+c f^2\right)^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 c e g \left(a^2 e^2 g^2+a b e g (d g+e f)-b^2 (d g+e f)^2\right)+b^2 e^2 g^2 (-2 a e g+b d g+b e f)-c^2 \left(4 a d e^2 f g^2-b \left(d^3 g^3+5 d^2 e f g^2+5 d e^2 f^2 g+e^3 f^3\right)\right)-2 c^3 d f \left(d^2 g^2+d e f g+e^2 f^2\right)\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)^2 \left(c f^2-g (b f-a g)\right)^2}+\frac{2 c \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right)}{\left(b^2-4 a c\right)^{3/2} \left(a e^2-b d e+c d^2\right) \left(c f^2-g (b f-a g)\right)}-\frac{c x \left(-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right)+b c (c d f-3 a e g)+2 a c^2 (d g+e f)+b^3 e g-b^2 c (d g+e f)}{\left(b^2-4 a c\right) \left(a+b x+c x^2\right) \left(a e^2-b d e+c d^2\right) \left(c f^2-g (b f-a g)\right)}-\frac{\log \left(a+b x+c x^2\right) (-b e g+c d g+c e f) \left(e g (2 a e g-b (d g+e f))+c \left(d^2 g^2+e^2 f^2\right)\right)}{2 \left(a e^2-b d e+c d^2\right)^2 \left(c f^2-g (b f-a g)\right)^2}+\frac{e^4 \log (d+e x)}{(e f-d g) \left(a e^2-b d e+c d^2\right)^2}-\frac{g^4 \log (f+g x)}{(e f-d g) \left(a g^2-b f g+c f^2\right)^2}",1,"(-(b^3*e*g) + b^2*c*(d*g + e*(f - g*x)) - 2*c^2*(a*d*g + c*d*f*x + a*e*(f - g*x)) + b*c*(3*a*e*g + c*(-(d*f) + e*f*x + d*g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(-(c*f^2) + g*(b*f - a*g))*(a + x*(b + c*x))) + ((4*c^5*d^3*f^3 + b^4*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*b^2*c*e*g*(-6*a^2*e^2*g^2 + 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) + 2*c^4*d*f*(-3*b*d*f*(e*f + d*g) + 2*a*(3*e^2*f^2 + d*e*f*g + 3*d^2*g^2)) + c^2*(-12*a^3*e^3*g^3 - 6*a^2*b*e^2*g^2*(e*f + d*g) + 12*a*b^2*e*g*(e^2*f^2 + d*e*f*g + d^2*g^2) + b^3*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)) - 2*c^3*(-4*b^2*d^2*e*f^2*g + 2*a^2*e*g*(e^2*f^2 - 5*d*e*f*g + d^2*g^2) + a*b*(3*e^3*f^3 + 11*d*e^2*f^2*g + 11*d^2*e*f*g^2 + 3*d^3*g^3)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2) + (e^4*Log[d + e*x])/((c*d^2 + e*(-(b*d) + a*e))^2*(e*f - d*g)) - (g^4*Log[f + g*x])/((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - ((c*e*f + c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2*(c*f^2 + g*(-(b*f) + a*g))^2)","A",1
819,1,249,287,0.4123346,"\int \frac{(d+e x)^3 \left(a+b x+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^3*(a + b*x + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(-495 e (f+g x)^3 \left(c \left(-3 d^2 g^2+12 d e f g-10 e^2 f^2\right)-e g (a e g+3 b d g-4 b e f)\right)+693 (f+g x)^2 (e f-d g) \left(-3 e g (a e g+b d g-2 b e f)-c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)-3465 (e f-d g)^3 \left(g (a g-b f)+c f^2\right)+1155 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-385 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+315 c e^3 (f+g x)^5\right)}{3465 g^6}","-\frac{2 e (f+g x)^{7/2} \left(e g (-a e g-3 b d g+4 b e f)-c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)}{7 g^6}+\frac{2 (f+g x)^{5/2} (e f-d g) \left(3 e g (-a e g-b d g+2 b e f)-c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)}{5 g^6}-\frac{2 \sqrt{f+g x} (e f-d g)^3 \left(a g^2-b f g+c f^2\right)}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac{2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6}",1,"(2*Sqrt[f + g*x]*(-3465*(e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g)) + 1155*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) + g*(-4*b*e*f + b*d*g + 3*a*e*g))*(f + g*x) + 693*(e*f - d*g)*(-3*e*g*(-2*b*e*f + b*d*g + a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 495*e*(-(e*g*(-4*b*e*f + 3*b*d*g + a*e*g)) + c*(-10*e^2*f^2 + 12*d*e*f*g - 3*d^2*g^2))*(f + g*x)^3 - 385*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^4 + 315*c*e^3*(f + g*x)^5))/(3465*g^6)","A",1
820,1,184,212,0.3408339,"\int \frac{(d+e x)^2 \left(a+b x+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^2*(a + b*x + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(-63 (f+g x)^2 \left(-e g (a e g+2 b d g-3 b e f)-c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)+315 (e f-d g)^2 \left(g (a g-b f)+c f^2\right)-105 (f+g x) (e f-d g) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))-45 e (f+g x)^3 (-b e g-2 c d g+4 c e f)+35 c e^2 (f+g x)^4\right)}{315 g^5}","-\frac{2 (f+g x)^{5/2} \left(e g (-a e g-2 b d g+3 b e f)-c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)}{5 g^5}+\frac{2 \sqrt{f+g x} (e f-d g)^2 \left(a g^2-b f g+c f^2\right)}{g^5}-\frac{2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac{2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac{2 c e^2 (f+g x)^{9/2}}{9 g^5}",1,"(2*Sqrt[f + g*x]*(315*(e*f - d*g)^2*(c*f^2 + g*(-(b*f) + a*g)) - 105*(e*f - d*g)*(2*c*f*(2*e*f - d*g) + g*(-3*b*e*f + b*d*g + 2*a*e*g))*(f + g*x) - 63*(-(e*g*(-3*b*e*f + 2*b*d*g + a*e*g)) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 45*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^3 + 35*c*e^2*(f + g*x)^4))/(315*g^5)","A",1
821,1,131,137,0.1934038,"\int \frac{(d+e x) \left(a+b x+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)*(a + b*x + c*x^2))/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(7 g \left(5 a g (3 d g-2 e f+e g x)+5 b d g (g x-2 f)+b e \left(8 f^2-4 f g x+3 g^2 x^2\right)\right)+c \left(7 d g \left(8 f^2-4 f g x+3 g^2 x^2\right)-3 e \left(16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right)\right)\right)}{105 g^4}","-\frac{2 \sqrt{f+g x} (e f-d g) \left(a g^2-b f g+c f^2\right)}{g^4}+\frac{2 (f+g x)^{3/2} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{3 g^4}-\frac{2 (f+g x)^{5/2} (-b e g-c d g+3 c e f)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4}",1,"(2*Sqrt[f + g*x]*(7*g*(5*b*d*g*(-2*f + g*x) + 5*a*g*(-2*e*f + 3*d*g + e*g*x) + b*e*(8*f^2 - 4*f*g*x + 3*g^2*x^2)) + c*(7*d*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*e*(16*f^3 - 8*f^2*g*x + 6*f*g^2*x^2 - 5*g^3*x^3))))/(105*g^4)","A",1
822,1,54,73,0.0482352,"\int \frac{a+b x+c x^2}{\sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/Sqrt[f + g*x],x]","\frac{2 \sqrt{f+g x} \left(5 g (3 a g-2 b f+b g x)+c \left(8 f^2-4 f g x+3 g^2 x^2\right)\right)}{15 g^3}","\frac{2 \sqrt{f+g x} \left(a g^2-b f g+c f^2\right)}{g^3}-\frac{2 (f+g x)^{3/2} (2 c f-b g)}{3 g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}",1,"(2*Sqrt[f + g*x]*(5*g*(-2*b*f + 3*a*g + b*g*x) + c*(8*f^2 - 4*f*g*x + 3*g^2*x^2)))/(15*g^3)","A",1
823,1,118,116,0.1916511,"\int \frac{a+b x+c x^2}{(d+e x) \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)*Sqrt[f + g*x]),x]","\frac{2 \left(-\frac{g^2 \left(c d^2-e (b d-a e)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{5/2} \sqrt{e f-d g}}+\frac{\sqrt{f+g x} (b e g-c (d g+e f))}{e^2}+\frac{c (f+g x)^{3/2}}{3 e}\right)}{g^2}","-\frac{2 \left(a e^2-b d e+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{5/2} \sqrt{e f-d g}}+\frac{2 \sqrt{f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2}",1,"(2*(((b*e*g - c*(e*f + d*g))*Sqrt[f + g*x])/e^2 + (c*(f + g*x)^(3/2))/(3*e) - ((c*d^2 - e*(b*d - a*e))*g^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*Sqrt[e*f - d*g])))/g^2","A",1
824,1,150,140,0.5604101,"\int \frac{a+b x+c x^2}{(d+e x)^2 \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^2*Sqrt[f + g*x]),x]","\frac{\sqrt{f+g x} \left(e g (b d-a e)+c \left(-3 d^2 g+2 d e (f-g x)+2 e^2 f x\right)\right)}{e^2 g (d+e x) (e f-d g)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) (e (-a e g-b d g+2 b e f)+c d (3 d g-4 e f))}{e^{5/2} (e f-d g)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}-\frac{\sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{(d+e x) (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e^2 g}",1,"(Sqrt[f + g*x]*(e*(b*d - a*e)*g + c*(-3*d^2*g + 2*e^2*f*x + 2*d*e*(f - g*x))))/(e^2*g*(e*f - d*g)*(d + e*x)) - ((c*d*(-4*e*f + 3*d*g) + e*(2*b*e*f - b*d*g - a*e*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*(e*f - d*g)^(3/2))","A",1
825,1,297,206,0.6460293,"\int \frac{a+b x+c x^2}{(d+e x)^3 \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^3*Sqrt[f + g*x]),x]","\frac{-\frac{2 \sqrt{e} \sqrt{f+g x} \left(e (a e-b d)+c d^2\right)}{(d+e x)^2 (e f-d g)}-\frac{3 g \left(e (a e-b d)+c d^2\right) \left(g (d+e x) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)-\sqrt{e} \sqrt{f+g x} \sqrt{e f-d g}\right)}{(d+e x) (e f-d g)^{5/2}}-\frac{4 \sqrt{e} \sqrt{f+g x} (b e-2 c d)}{(d+e x) (e f-d g)}-\frac{4 g (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{(e f-d g)^{3/2}}-\frac{8 c \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{\sqrt{e f-d g}}}{4 e^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) \left(e g (-3 a e g-b d g+4 b e f)-c \left(3 d^2 g^2-8 d e f g+8 e^2 f^2\right)\right)}{4 e^{5/2} (e f-d g)^{5/2}}-\frac{\sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{2 (d+e x)^2 (e f-d g)}+\frac{\sqrt{f+g x} (c d (8 e f-5 d g)-e (-3 a e g-b d g+4 b e f))}{4 e^2 (d+e x) (e f-d g)^2}",1,"((-2*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[f + g*x])/((e*f - d*g)*(d + e*x)^2) - (4*Sqrt[e]*(-2*c*d + b*e)*Sqrt[f + g*x])/((e*f - d*g)*(d + e*x)) - (4*(2*c*d - b*e)*g*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e*f - d*g)^(3/2) - (8*c*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/Sqrt[e*f - d*g] - (3*(c*d^2 + e*(-(b*d) + a*e))*g*(-(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[f + g*x]) + g*(d + e*x)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]))/((e*f - d*g)^(5/2)*(d + e*x)))/(4*e^(5/2))","A",1
826,1,249,285,0.7286969,"\int \frac{(d+e x)^3 \left(a+b x+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)^3*(a + b*x + c*x^2))/(f + g*x)^(3/2),x]","\frac{2 \left(-63 e (f+g x)^3 \left(c \left(-3 d^2 g^2+12 d e f g-10 e^2 f^2\right)-e g (a e g+3 b d g-4 b e f)\right)+105 (f+g x)^2 (e f-d g) \left(-3 e g (a e g+b d g-2 b e f)-c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)+315 (e f-d g)^3 \left(g (a g-b f)+c f^2\right)+315 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-45 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+35 c e^3 (f+g x)^5\right)}{315 g^6 \sqrt{f+g x}}","-\frac{2 e (f+g x)^{5/2} \left(e g (-a e g-3 b d g+4 b e f)-c \left(3 d^2 g^2-12 d e f g+10 e^2 f^2\right)\right)}{5 g^6}+\frac{2 (f+g x)^{3/2} (e f-d g) \left(3 e g (-a e g-b d g+2 b e f)-c \left(d^2 g^2-8 d e f g+10 e^2 f^2\right)\right)}{3 g^6}+\frac{2 (e f-d g)^3 \left(a g^2-b f g+c f^2\right)}{g^6 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac{2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{7 g^6}+\frac{2 c e^3 (f+g x)^{9/2}}{9 g^6}",1,"(2*(315*(e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g)) + 315*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) + g*(-4*b*e*f + b*d*g + 3*a*e*g))*(f + g*x) + 105*(e*f - d*g)*(-3*e*g*(-2*b*e*f + b*d*g + a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 63*e*(-(e*g*(-4*b*e*f + 3*b*d*g + a*e*g)) + c*(-10*e^2*f^2 + 12*d*e*f*g - 3*d^2*g^2))*(f + g*x)^3 - 45*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^4 + 35*c*e^3*(f + g*x)^5))/(315*g^6*Sqrt[f + g*x])","A",1
827,1,184,210,0.3555153,"\int \frac{(d+e x)^2 \left(a+b x+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)^2*(a + b*x + c*x^2))/(f + g*x)^(3/2),x]","\frac{2 \left(-35 (f+g x)^2 \left(-e g (a e g+2 b d g-3 b e f)-c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)-105 (e f-d g)^2 \left(g (a g-b f)+c f^2\right)-105 (f+g x) (e f-d g) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))-21 e (f+g x)^3 (-b e g-2 c d g+4 c e f)+15 c e^2 (f+g x)^4\right)}{105 g^5 \sqrt{f+g x}}","-\frac{2 (f+g x)^{3/2} \left(e g (-a e g-2 b d g+3 b e f)-c \left(d^2 g^2-6 d e f g+6 e^2 f^2\right)\right)}{3 g^5}-\frac{2 (e f-d g)^2 \left(a g^2-b f g+c f^2\right)}{g^5 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{g^5}-\frac{2 e (f+g x)^{5/2} (-b e g-2 c d g+4 c e f)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5}",1,"(2*(-105*(e*f - d*g)^2*(c*f^2 + g*(-(b*f) + a*g)) - 105*(e*f - d*g)*(2*c*f*(2*e*f - d*g) + g*(-3*b*e*f + b*d*g + 2*a*e*g))*(f + g*x) - 35*(-(e*g*(-3*b*e*f + 2*b*d*g + a*e*g)) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 21*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^3 + 15*c*e^2*(f + g*x)^4))/(105*g^5*Sqrt[f + g*x])","A",1
828,1,128,135,0.1789567,"\int \frac{(d+e x) \left(a+b x+c x^2\right)}{(f+g x)^{3/2}} \, dx","Integrate[((d + e*x)*(a + b*x + c*x^2))/(f + g*x)^(3/2),x]","\frac{2 \left(5 g \left(3 a g (-d g+2 e f+e g x)+3 b d g (2 f+g x)+b e \left(-8 f^2-4 f g x+g^2 x^2\right)\right)+c \left(5 d g \left(-8 f^2-4 f g x+g^2 x^2\right)+3 e \left(16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right)\right)\right)}{15 g^4 \sqrt{f+g x}}","\frac{2 (e f-d g) \left(a g^2-b f g+c f^2\right)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{g^4}-\frac{2 (f+g x)^{3/2} (-b e g-c d g+3 c e f)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4}",1,"(2*(5*g*(3*b*d*g*(2*f + g*x) + 3*a*g*(2*e*f - d*g + e*g*x) + b*e*(-8*f^2 - 4*f*g*x + g^2*x^2)) + c*(5*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 3*e*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3))))/(15*g^4*Sqrt[f + g*x])","A",1
829,1,54,71,0.0550591,"\int \frac{a+b x+c x^2}{(f+g x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/(f + g*x)^(3/2),x]","\frac{6 g (-a g+2 b f+b g x)+2 c \left(-8 f^2-4 f g x+g^2 x^2\right)}{3 g^3 \sqrt{f+g x}}","-\frac{2 \left(a g^2-b f g+c f^2\right)}{g^3 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (2 c f-b g)}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3}",1,"(6*g*(2*b*f - a*g + b*g*x) + 2*c*(-8*f^2 - 4*f*g*x + g^2*x^2))/(3*g^3*Sqrt[f + g*x])","A",1
830,1,124,122,0.2911004,"\int \frac{a+b x+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)*(f + g*x)^(3/2)),x]","\frac{2 \left(-\frac{g^2 \left(c d^2-e (b d-a e)\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{3/2} (e f-d g)^{3/2}}+\frac{c f^2-g (b f-a g)}{\sqrt{f+g x} (e f-d g)}+\frac{c \sqrt{f+g x}}{e}\right)}{g^2}","-\frac{2 \left(a e^2-b d e+c d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{e^{3/2} (e f-d g)^{3/2}}+\frac{2 \left(a g^2-b f g+c f^2\right)}{g^2 \sqrt{f+g x} (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e g^2}",1,"(2*((c*f^2 - g*(b*f - a*g))/((e*f - d*g)*Sqrt[f + g*x]) + (c*Sqrt[f + g*x])/e - ((c*d^2 - e*(b*d - a*e))*g^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(3/2))))/g^2","A",1
831,1,176,165,0.4071675,"\int \frac{a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^2*(f + g*x)^(3/2)),x]","-\frac{e g (2 a d g+a e (f+3 g x)-b (3 d f+d g x+2 e f x))+c \left(d^2 g (f+g x)+2 d e f^2+2 e^2 f^2 x\right)}{e g (d+e x) \sqrt{f+g x} (e f-d g)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) (e (-3 a e g+b d g+2 b e f)+c d (d g-4 e f))}{e^{3/2} (e f-d g)^{5/2}}","-\frac{\sqrt{f+g x} \left(a e^2-b d e+c d^2\right)}{e (d+e x) (e f-d g)^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) (c d (4 e f-d g)-e (-3 a e g+b d g+2 b e f))}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left(a g^2-b f g+c f^2\right)}{g \sqrt{f+g x} (e f-d g)^2}",1,"-((c*(2*d*e*f^2 + 2*e^2*f^2*x + d^2*g*(f + g*x)) + e*g*(2*a*d*g + a*e*(f + 3*g*x) - b*(3*d*f + 2*e*f*x + d*g*x)))/(e*g*(e*f - d*g)^2*(d + e*x)*Sqrt[f + g*x])) - ((c*d*(-4*e*f + d*g) + e*(2*b*e*f + b*d*g - 3*a*e*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(5/2))","A",1
832,1,290,248,1.1016999,"\int \frac{a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)),x]","\frac{1}{4} \left(-\frac{2 \sqrt{f+g x} \left(e (a e-b d)+c d^2\right)}{e (d+e x)^2 (e f-d g)^2}+\frac{g \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) (e (-7 a e g+3 b d g+4 b e f)+c d (d g-8 e f))}{e^{3/2} (e f-d g)^{7/2}}+\frac{8 \left(g (a g-b f)+c f^2\right)}{\sqrt{f+g x} (e f-d g)^3}-\frac{8 \sqrt{e} \left(g (a g-b f)+c f^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right)}{(e f-d g)^{7/2}}-\frac{\sqrt{f+g x} (e (-7 a e g+3 b d g+4 b e f)+c d (d g-8 e f))}{e (d+e x) (e f-d g)^3}\right)","-\frac{\sqrt{f+g x} \left(a e^2-b d e+c d^2\right)}{2 e (d+e x)^2 (e f-d g)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right) \left(3 e g (5 a e g-b (d g+4 e f))+c \left(-d^2 g^2+8 d e f g+8 e^2 f^2\right)\right)}{4 e^{3/2} (e f-d g)^{7/2}}+\frac{2 \left(a g^2-b f g+c f^2\right)}{\sqrt{f+g x} (e f-d g)^3}+\frac{\sqrt{f+g x} (c d (8 e f-d g)-e (-7 a e g+3 b d g+4 b e f))}{4 e (d+e x) (e f-d g)^3}",1,"((8*(c*f^2 + g*(-(b*f) + a*g)))/((e*f - d*g)^3*Sqrt[f + g*x]) - (2*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[f + g*x])/(e*(e*f - d*g)^2*(d + e*x)^2) - ((c*d*(-8*e*f + d*g) + e*(4*b*e*f + 3*b*d*g - 7*a*e*g))*Sqrt[f + g*x])/(e*(e*f - d*g)^3*(d + e*x)) - (8*Sqrt[e]*(c*f^2 + g*(-(b*f) + a*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e*f - d*g)^(7/2) + (g*(c*d*(-8*e*f + d*g) + e*(4*b*e*f + 3*b*d*g - 7*a*e*g))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(3/2)*(e*f - d*g)^(7/2)))/4","A",1
833,1,113,91,0.2120986,"\int \frac{\sqrt{-1+x} \sqrt{1+x}}{1+x-x^2} \, dx","Integrate[(Sqrt[-1 + x]*Sqrt[1 + x])/(1 + x - x^2),x]","-\frac{1}{5} \sqrt{\sqrt{5}-2} \left(5+\sqrt{5}\right) \tan ^{-1}\left(\sqrt{\sqrt{5}-2} \sqrt{\frac{x-1}{x+1}}\right)-2 \tanh ^{-1}\left(\sqrt{\frac{x-1}{x+1}}\right)-\frac{1}{5} \left(\sqrt{5}-5\right) \sqrt{2+\sqrt{5}} \tanh ^{-1}\left(\sqrt{2+\sqrt{5}} \sqrt{\frac{x-1}{x+1}}\right)","\sqrt{\frac{2}{5} \left(\sqrt{5}-1\right)} \tan ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{\sqrt{5}-2} \sqrt{x-1}}\right)-\cosh ^{-1}(x)+\sqrt{\frac{2}{5} \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{2+\sqrt{5}} \sqrt{x-1}}\right)",1,"-1/5*(Sqrt[-2 + Sqrt[5]]*(5 + Sqrt[5])*ArcTan[Sqrt[-2 + Sqrt[5]]*Sqrt[(-1 + x)/(1 + x)]]) - 2*ArcTanh[Sqrt[(-1 + x)/(1 + x)]] - ((-5 + Sqrt[5])*Sqrt[2 + Sqrt[5]]*ArcTanh[Sqrt[2 + Sqrt[5]]*Sqrt[(-1 + x)/(1 + x)]])/5","A",0
834,1,173,164,0.7749761,"\int \frac{a+b x+c x^2}{\sqrt{d+e x} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]","\frac{\sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right) \left(4 e g (2 a e g-b (d g+e f))+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right)+e \sqrt{g} \sqrt{d+e x} (f+g x) (4 b e g+c (-3 d g-3 e f+2 e g x))}{4 e^3 g^{5/2} \sqrt{f+g x}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right) \left(4 e g (2 a e g-b (d g+e f))+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right)}{4 e^{5/2} g^{5/2}}-\frac{\sqrt{d+e x} \sqrt{f+g x} (-4 b e g+5 c d g+3 c e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g}",1,"(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(4*b*e*g + c*(-3*e*f - 3*d*g + 2*e*g*x)) + Sqrt[e*f - d*g]*(c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 4*e*g*(2*a*e*g - b*(e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(4*e^3*g^(5/2)*Sqrt[f + g*x])","A",1
835,1,313,333,1.5383626,"\int \frac{(d+e x)^{3/2} \left(a+b x+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^(3/2)*(a + b*x + c*x^2))/Sqrt[f + g*x],x]","\frac{3 (e f-d g)^{5/2} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right) \left(8 e g (6 a e g-b (d g+5 e f))+c \left(3 d^2 g^2+10 d e f g+35 e^2 f^2\right)\right)-e \sqrt{g} \sqrt{d+e x} (f+g x) \left(c \left(9 d^3 g^3+3 d^2 e g^2 (5 f-2 g x)+d e^2 g \left(-145 f^2+92 f g x-72 g^2 x^2\right)+e^3 \left(105 f^3-70 f^2 g x+56 f g^2 x^2-48 g^3 x^3\right)\right)-8 e g \left(6 a e g (5 d g-3 e f+2 e g x)+b \left(3 d^2 g^2+2 d e g (7 g x-11 f)+e^2 \left(15 f^2-10 f g x+8 g^2 x^2\right)\right)\right)\right)}{192 e^3 g^{9/2} \sqrt{f+g x}}","-\frac{\sqrt{d+e x} \sqrt{f+g x} (e f-d g) \left(8 e g (6 a e g-b (d g+5 e f))+c \left(3 d^2 g^2+10 d e f g+35 e^2 f^2\right)\right)}{64 e^2 g^4}+\frac{(d+e x)^{3/2} \sqrt{f+g x} \left(8 e g (6 a e g-b (d g+5 e f))+c \left(3 d^2 g^2+10 d e f g+35 e^2 f^2\right)\right)}{96 e^2 g^3}+\frac{(e f-d g)^2 \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right) \left(8 e g (6 a e g-b (d g+5 e f))+c \left(3 d^2 g^2+10 d e f g+35 e^2 f^2\right)\right)}{64 e^{5/2} g^{9/2}}-\frac{(d+e x)^{5/2} \sqrt{f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g}",1,"(-(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(c*(9*d^3*g^3 + 3*d^2*e*g^2*(5*f - 2*g*x) + d*e^2*g*(-145*f^2 + 92*f*g*x - 72*g^2*x^2) + e^3*(105*f^3 - 70*f^2*g*x + 56*f*g^2*x^2 - 48*g^3*x^3)) - 8*e*g*(6*a*e*g*(-3*e*f + 5*d*g + 2*e*g*x) + b*(3*d^2*g^2 + 2*d*e*g*(-11*f + 7*g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2))))) + 3*(e*f - d*g)^(5/2)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(192*e^3*g^(9/2)*Sqrt[f + g*x])","A",1
836,1,225,246,1.0078021,"\int \frac{\sqrt{d+e x} \left(a+b x+c x^2\right)}{\sqrt{f+g x}} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*x + c*x^2))/Sqrt[f + g*x],x]","\frac{-e \sqrt{g} \sqrt{d+e x} (f+g x) \left(c \left(3 d^2 g^2-2 d e g (g x-2 f)+e^2 \left(-15 f^2+10 f g x-8 g^2 x^2\right)\right)-6 e g (4 a e g+b (d g-3 e f+2 e g x))\right)-3 (e f-d g)^{3/2} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right) \left(2 e g (4 a e g-b (d g+3 e f))+c \left(d^2 g^2+2 d e f g+5 e^2 f^2\right)\right)}{24 e^3 g^{7/2} \sqrt{f+g x}}","\frac{\sqrt{d+e x} \sqrt{f+g x} \left(2 e g (4 a e g-b (d g+3 e f))+c \left(d^2 g^2+2 d e f g+5 e^2 f^2\right)\right)}{8 e^2 g^3}-\frac{(e f-d g) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right) \left(2 e g (4 a e g-b (d g+3 e f))+c \left(d^2 g^2+2 d e f g+5 e^2 f^2\right)\right)}{8 e^{5/2} g^{7/2}}-\frac{(d+e x)^{3/2} \sqrt{f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g}",1,"(-(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(-6*e*g*(4*a*e*g + b*(-3*e*f + d*g + 2*e*g*x)) + c*(3*d^2*g^2 - 2*d*e*g*(-2*f + g*x) + e^2*(-15*f^2 + 10*f*g*x - 8*g^2*x^2)))) - 3*(e*f - d*g)^(3/2)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(24*e^3*g^(7/2)*Sqrt[f + g*x])","A",1
837,1,173,164,0.6224838,"\int \frac{a+b x+c x^2}{\sqrt{d+e x} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]","\frac{\sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right) \left(4 e g (2 a e g-b (d g+e f))+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right)+e \sqrt{g} \sqrt{d+e x} (f+g x) (4 b e g+c (-3 d g-3 e f+2 e g x))}{4 e^3 g^{5/2} \sqrt{f+g x}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right) \left(4 e g (2 a e g-b (d g+e f))+c \left(3 d^2 g^2+2 d e f g+3 e^2 f^2\right)\right)}{4 e^{5/2} g^{5/2}}-\frac{\sqrt{d+e x} \sqrt{f+g x} (-4 b e g+5 c d g+3 c e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g}",1,"(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(4*b*e*g + c*(-3*e*f - 3*d*g + 2*e*g*x)) + Sqrt[e*f - d*g]*(c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 4*e*g*(2*a*e*g - b*(e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(4*e^3*g^(5/2)*Sqrt[f + g*x])","A",1
838,1,222,129,0.5918685,"\int \frac{a+b x+c x^2}{(d+e x)^{3/2} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^(3/2)*Sqrt[f + g*x]),x]","-\frac{2 \sqrt{f+g x} \left(e \sqrt{e f-d g} \sqrt{\frac{e (f+g x)}{e f-d g}} \left(g^2 (a e-b d)+c f (2 d g-e f)\right)+e \sqrt{g} \sqrt{d+e x} (2 c f-b g) (e f-d g) \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)+c (e f-d g)^{5/2} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{g (d+e x)}{d g-e f}\right)\right)}{e^2 g^2 \sqrt{d+e x} (e f-d g)^{3/2} \sqrt{\frac{e (f+g x)}{e f-d g}}}","-\frac{2 \sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{\sqrt{d+e x} (e f-d g)}-\frac{(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{e^{5/2} g^{3/2}}+\frac{c \sqrt{d+e x} \sqrt{f+g x}}{e^2 g}",1,"(-2*Sqrt[f + g*x]*(e*Sqrt[e*f - d*g]*((-(b*d) + a*e)*g^2 + c*f*(-(e*f) + 2*d*g))*Sqrt[(e*(f + g*x))/(e*f - d*g)] + e*Sqrt[g]*(2*c*f - b*g)*(e*f - d*g)*Sqrt[d + e*x]*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]] + c*(e*f - d*g)^(5/2)*Hypergeometric2F1[-3/2, -1/2, 1/2, (g*(d + e*x))/(-(e*f) + d*g)]))/(e^2*g^2*(e*f - d*g)^(3/2)*Sqrt[d + e*x]*Sqrt[(e*(f + g*x))/(e*f - d*g)])","C",1
839,1,173,160,0.2297722,"\int \frac{a+b x+c x^2}{(d+e x)^{5/2} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]","\frac{2 \sqrt{f+g x} \left(2 g (d+e x) \left(g (a g-b f)+c f^2\right)-(e f-d g) \left(g (a g-b f)+c f^2\right)+(f+g x) (2 c f-b g) (e f-d g)-\frac{c (e f-d g)^3 \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{g (d+e x)}{d g-e f}\right)}{e^2 \sqrt{\frac{e (f+g x)}{e f-d g}}}\right)}{3 g^2 (d+e x)^{3/2} (e f-d g)^2}","\frac{2 \sqrt{f+g x} \left(c \left(6 d e f-4 d^2 g\right)-e (-2 a e g-b d g+3 b e f)\right)}{3 e^2 \sqrt{d+e x} (e f-d g)^2}-\frac{2 \sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{3 (d+e x)^{3/2} (e f-d g)}+\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{e^{5/2} \sqrt{g}}",1,"(2*Sqrt[f + g*x]*(-((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))) + 2*g*(c*f^2 + g*(-(b*f) + a*g))*(d + e*x) + (2*c*f - b*g)*(e*f - d*g)*(f + g*x) - (c*(e*f - d*g)^3*Hypergeometric2F1[-3/2, -3/2, -1/2, (g*(d + e*x))/(-(e*f) + d*g)])/(e^2*Sqrt[(e*(f + g*x))/(e*f - d*g)])))/(3*g^2*(e*f - d*g)^2*(d + e*x)^(3/2))","C",1
840,1,178,198,0.2283511,"\int \frac{a+b x+c x^2}{(d+e x)^{7/2} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^(7/2)*Sqrt[f + g*x]),x]","-\frac{2 \sqrt{f+g x} \left(a \left(15 d^2 g^2-10 d e g (f-2 g x)+e^2 \left(3 f^2-4 f g x+8 g^2 x^2\right)\right)+b \left(5 d^2 g (g x-2 f)+2 d e \left(f^2-13 f g x+g^2 x^2\right)+5 e^2 f x (f-2 g x)\right)+c \left(d^2 \left(8 f^2-4 f g x+3 g^2 x^2\right)+10 d e f x (2 f-g x)+15 e^2 f^2 x^2\right)\right)}{15 (d+e x)^{5/2} (e f-d g)^3}","\frac{2 \sqrt{f+g x} \left(2 e g (-4 a e g-b d g+5 b e f)-c \left(3 d^2 g^2-10 d e f g+15 e^2 f^2\right)\right)}{15 e^2 \sqrt{d+e x} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{5 (d+e x)^{5/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (5 e f-3 d g)-e (-4 a e g-b d g+5 b e f))}{15 e^2 (d+e x)^{3/2} (e f-d g)^2}",1,"(-2*Sqrt[f + g*x]*(b*(5*e^2*f*x*(f - 2*g*x) + 5*d^2*g*(-2*f + g*x) + 2*d*e*(f^2 - 13*f*g*x + g^2*x^2)) + c*(15*e^2*f^2*x^2 + 10*d*e*f*x*(2*f - g*x) + d^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2)) + a*(15*d^2*g^2 - 10*d*e*g*(f - 2*g*x) + e^2*(3*f^2 - 4*f*g*x + 8*g^2*x^2))))/(15*(e*f - d*g)^3*(d + e*x)^(5/2))","A",1
841,1,332,281,0.3532553,"\int \frac{a+b x+c x^2}{(d+e x)^{9/2} \sqrt{f+g x}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^(9/2)*Sqrt[f + g*x]),x]","\frac{2 \sqrt{f+g x} \left(3 a \left(35 d^3 g^3-35 d^2 e g^2 (f-2 g x)+7 d e^2 g \left(3 f^2-4 f g x+8 g^2 x^2\right)+e^3 \left(-5 f^3+6 f^2 g x-8 f g^2 x^2+16 g^3 x^3\right)\right)+b \left(35 d^3 g^2 (g x-2 f)+7 d^2 e g \left(4 f^2-37 f g x+4 g^2 x^2\right)+d e^2 \left(-6 f^3+101 f^2 g x-200 f g^2 x^2+8 g^3 x^3\right)-7 e^3 f x \left(3 f^2-4 f g x+8 g^2 x^2\right)\right)+c \left(7 d^3 g \left(8 f^2-4 f g x+3 g^2 x^2\right)+d^2 e \left(-8 f^3+200 f^2 g x-101 f g^2 x^2+6 g^3 x^3\right)-7 d e^2 f x \left(4 f^2-37 f g x+4 g^2 x^2\right)-35 e^3 f^2 x^2 (f-2 g x)\right)\right)}{105 (d+e x)^{7/2} (e f-d g)^4}","-\frac{4 g \sqrt{f+g x} \left(4 e g (-6 a e g-b d g+7 b e f)-c \left(3 d^2 g^2-14 d e f g+35 e^2 f^2\right)\right)}{105 e^2 \sqrt{d+e x} (e f-d g)^4}+\frac{2 \sqrt{f+g x} \left(4 e g (-6 a e g-b d g+7 b e f)-c \left(3 d^2 g^2-14 d e f g+35 e^2 f^2\right)\right)}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left(a+\frac{d (c d-b e)}{e^2}\right)}{7 (d+e x)^{7/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2}",1,"(2*Sqrt[f + g*x]*(c*(-35*e^3*f^2*x^2*(f - 2*g*x) + 7*d^3*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 7*d*e^2*f*x*(4*f^2 - 37*f*g*x + 4*g^2*x^2) + d^2*e*(-8*f^3 + 200*f^2*g*x - 101*f*g^2*x^2 + 6*g^3*x^3)) + b*(35*d^3*g^2*(-2*f + g*x) + 7*d^2*e*g*(4*f^2 - 37*f*g*x + 4*g^2*x^2) - 7*e^3*f*x*(3*f^2 - 4*f*g*x + 8*g^2*x^2) + d*e^2*(-6*f^3 + 101*f^2*g*x - 200*f*g^2*x^2 + 8*g^3*x^3)) + 3*a*(35*d^3*g^3 - 35*d^2*e*g^2*(f - 2*g*x) + 7*d*e^2*g*(3*f^2 - 4*f*g*x + 8*g^2*x^2) + e^3*(-5*f^3 + 6*f^2*g*x - 8*f*g^2*x^2 + 16*g^3*x^3))))/(105*(e*f - d*g)^4*(d + e*x)^(7/2))","A",1
842,1,196,249,1.0911511,"\int \frac{\sqrt{d+e x} \left(a+b x+c x^2\right)}{(e+f x)^{3/2}} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*x + c*x^2))/(e + f*x)^(3/2),x]","\frac{\frac{\sqrt{e^2-d f} \sqrt{\frac{e (e+f x)}{e^2-d f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{e^2-d f}}\right) \left(4 e f \left(2 a e f+b d f-3 b e^2\right)+c \left(-d^2 f^2-6 d e^2 f+15 e^4\right)\right)}{e}+\sqrt{f} \sqrt{d+e x} \left(4 e f (-2 a f+3 b e+b f x)+c \left(e f \left(d+2 f x^2\right)+d f^2 x-15 e^3-5 e^2 f x\right)\right)}{4 e f^{7/2} \sqrt{e+f x}}","\frac{\sqrt{d+e x} \sqrt{e+f x} \left(4 e f \left(-2 a e f-b d f+3 b e^2\right)-c \left(-d^2 f^2-6 d e^2 f+15 e^4\right)\right)}{4 e f^3 \left(e^2-d f\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{e} \sqrt{e+f x}}\right) \left(4 e f \left(-2 a e f-b d f+3 b e^2\right)-c \left(-d^2 f^2-6 d e^2 f+15 e^4\right)\right)}{4 e^{3/2} f^{7/2}}+\frac{2 (d+e x)^{3/2} \left(a+\frac{e (c e-b f)}{f^2}\right)}{\left(e^2-d f\right) \sqrt{e+f x}}+\frac{c (d+e x)^{3/2} \sqrt{e+f x}}{2 e f^2}",1,"(Sqrt[f]*Sqrt[d + e*x]*(4*e*f*(3*b*e - 2*a*f + b*f*x) + c*(-15*e^3 - 5*e^2*f*x + d*f^2*x + e*f*(d + 2*f*x^2))) + (Sqrt[e^2 - d*f]*(4*e*f*(-3*b*e^2 + b*d*f + 2*a*e*f) + c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*Sqrt[(e*(e + f*x))/(e^2 - d*f)]*ArcSinh[(Sqrt[f]*Sqrt[d + e*x])/Sqrt[e^2 - d*f]])/e)/(4*e*f^(7/2)*Sqrt[e + f*x])","A",1
843,1,204,240,0.8481341,"\int \frac{(d+e x)^{3/2} \left(15 d^2+20 d e x+8 e^2 x^2\right)}{\sqrt{a+b x}} \, dx","Integrate[((d + e*x)^(3/2)*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]","\frac{\sqrt{d+e x} \left(\frac{3 \left(35 a^2 e^2-90 a b d e+73 b^2 d^2\right) (b d-a e)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right)}{\sqrt{e} \sqrt{\frac{b (d+e x)}{b d-a e}}}+\sqrt{a+b x} \left(-105 a^3 e^3+5 a^2 b e^2 (89 d+14 e x)-a b^2 e \left(725 d^2+292 d e x+56 e^2 x^2\right)+b^3 \left(501 d^3+466 d^2 e x+232 d e^2 x^2+48 e^3 x^3\right)\right)\right)}{24 b^4}","\frac{(b d-a e)^2 \left(35 a^2 e^2-90 a b d e+73 b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right)}{8 b^{9/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) \left(35 a^2 e^2-90 a b d e+73 b^2 d^2\right)}{8 b^4}+\frac{\sqrt{a+b x} (d+e x)^{3/2} \left(35 a^2 e^2-90 a b d e+73 b^2 d^2\right)}{12 b^3}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2}",1,"(Sqrt[d + e*x]*(Sqrt[a + b*x]*(-105*a^3*e^3 + 5*a^2*b*e^2*(89*d + 14*e*x) - a*b^2*e*(725*d^2 + 292*d*e*x + 56*e^2*x^2) + b^3*(501*d^3 + 466*d^2*e*x + 232*d*e^2*x^2 + 48*e^3*x^3)) + (3*(b*d - a*e)^(3/2)*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/(Sqrt[e]*Sqrt[(b*(d + e*x))/(b*d - a*e)])))/(24*b^4)","A",1
844,1,163,176,0.4370838,"\int \frac{\sqrt{d+e x} \left(15 d^2+20 d e x+8 e^2 x^2\right)}{\sqrt{a+b x}} \, dx","Integrate[(Sqrt[d + e*x]*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]","\frac{\sqrt{d+e x} \left(\sqrt{a+b x} \left(15 a^2 e^2-a b e (49 d+10 e x)+b^2 \left(57 d^2+32 d e x+8 e^2 x^2\right)\right)+\frac{3 \sqrt{b d-a e} \left(5 a^2 e^2-13 a b d e+11 b^2 d^2\right) \sinh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right)}{\sqrt{e} \sqrt{\frac{b (d+e x)}{b d-a e}}}\right)}{3 b^3}","\frac{(b d-a e) \left(5 a^2 e^2-13 a b d e+11 b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right)}{b^{7/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} \left(5 a^2 e^2-13 a b d e+11 b^2 d^2\right)}{b^3}+\frac{8 e (a+b x)^{3/2} (d+e x)^{3/2}}{3 b^2}+\frac{2 \sqrt{a+b x} (d+e x)^{3/2} (4 b d-3 a e)}{b^2}",1,"(Sqrt[d + e*x]*(Sqrt[a + b*x]*(15*a^2*e^2 - a*b*e*(49*d + 10*e*x) + b^2*(57*d^2 + 32*d*e*x + 8*e^2*x^2)) + (3*Sqrt[b*d - a*e]*(11*b^2*d^2 - 13*a*b*d*e + 5*a^2*e^2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/(Sqrt[e]*Sqrt[(b*(d + e*x))/(b*d - a*e)])))/(3*b^3)","A",1
845,1,135,122,0.4083814,"\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} \sqrt{d+e x}} \, dx","Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*Sqrt[d + e*x]),x]","\frac{2 \left(\frac{\sqrt{b d-a e} \left(3 a^2 e^2-8 a b d e+8 b^2 d^2\right) \sqrt{\frac{b (d+e x)}{b d-a e}} \sinh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right)}{\sqrt{e}}+b \sqrt{a+b x} (d+e x) (-3 a e+7 b d+2 b e x)\right)}{b^3 \sqrt{d+e x}}","\frac{2 \left(3 a^2 e^2-8 a b d e+8 b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right)}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2}",1,"(2*(b*Sqrt[a + b*x]*(d + e*x)*(7*b*d - 3*a*e + 2*b*e*x) + (Sqrt[b*d - a*e]*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*Sqrt[(b*(d + e*x))/(b*d - a*e)]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/Sqrt[e]))/(b^3*Sqrt[d + e*x])","A",1
846,1,134,108,0.3366924,"\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{3/2}} \, dx","Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(3/2)),x]","\frac{2 \left(\frac{b \sqrt{a+b x} (b d (7 d+4 e x)-4 a e (d+e x))}{b d-a e}+\frac{4 \sqrt{b d-a e} (2 b d-a e) \sqrt{\frac{b (d+e x)}{b d-a e}} \sinh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right)}{\sqrt{e}}\right)}{b^2 \sqrt{d+e x}}","\frac{8 (2 b d-a e) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right)}{b^{3/2} \sqrt{e}}+\frac{6 d^2 \sqrt{a+b x}}{\sqrt{d+e x} (b d-a e)}+\frac{8 \sqrt{a+b x} \sqrt{d+e x}}{b}",1,"(2*((b*Sqrt[a + b*x]*(-4*a*e*(d + e*x) + b*d*(7*d + 4*e*x)))/(b*d - a*e) + (4*Sqrt[b*d - a*e]*(2*b*d - a*e)*Sqrt[(b*(d + e*x))/(b*d - a*e)]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/Sqrt[e]))/(b^2*Sqrt[d + e*x])","A",1
847,1,128,116,0.2956539,"\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{5/2}} \, dx","Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(5/2)),x]","\frac{2 \left(\frac{8 (b d-a e)^{3/2} \left(\frac{b (d+e x)}{b d-a e}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right)}{b^2 \sqrt{e}}+\frac{d \sqrt{a+b x} (b d (7 d+6 e x)-a e (5 d+4 e x))}{(b d-a e)^2}\right)}{(d+e x)^{3/2}}","\frac{2 d^2 \sqrt{a+b x}}{(d+e x)^{3/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (3 b d-2 a e)}{\sqrt{d+e x} (b d-a e)^2}+\frac{16 \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right)}{\sqrt{b} \sqrt{e}}",1,"(2*((d*Sqrt[a + b*x]*(-(a*e*(5*d + 4*e*x)) + b*d*(7*d + 6*e*x)))/(b*d - a*e)^2 + (8*(b*d - a*e)^(3/2)*((b*(d + e*x))/(b*d - a*e))^(3/2)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/(b^2*Sqrt[e])))/(d + e*x)^(3/2)","A",1
848,1,110,133,0.0835524,"\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{7/2}} \, dx","Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]","\frac{2 \sqrt{a+b x} \left(a^2 e^2 \left(149 d^2+260 d e x+120 e^2 x^2\right)-2 a b d e \left(175 d^2+306 d e x+140 e^2 x^2\right)+b^2 d^2 \left(225 d^2+400 d e x+184 e^2 x^2\right)\right)}{15 (d+e x)^{5/2} (b d-a e)^3}","\frac{16 \sqrt{a+b x} \left(15 a^2 e^2-35 a b d e+23 b^2 d^2\right)}{15 \sqrt{d+e x} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{5 (d+e x)^{5/2} (b d-a e)}+\frac{8 d \sqrt{a+b x} (8 b d-5 a e)}{15 (d+e x)^{3/2} (b d-a e)^2}",1,"(2*Sqrt[a + b*x]*(a^2*e^2*(149*d^2 + 260*d*e*x + 120*e^2*x^2) - 2*a*b*d*e*(175*d^2 + 306*d*e*x + 140*e^2*x^2) + b^2*d^2*(225*d^2 + 400*d*e*x + 184*e^2*x^2)))/(15*(b*d - a*e)^3*(d + e*x)^(5/2))","A",1
849,1,173,189,0.1214542,"\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{a+b x} (d+e x)^{9/2}} \, dx","Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]","\frac{2 \sqrt{a+b x} \left(-a^3 e^3 \left(409 d^2+644 d e x+280 e^2 x^2\right)+a^2 b e^2 \left(1953 d^3+3890 d^2 e x+2632 d e^2 x^2+560 e^3 x^3\right)-a b^2 d e \left(2975 d^3+6664 d^2 e x+5168 d e^2 x^2+1344 e^3 x^3\right)+b^3 d^2 \left(1575 d^3+3850 d^2 e x+3248 d e^2 x^2+928 e^3 x^3\right)\right)}{105 (d+e x)^{7/2} (b d-a e)^4}","\frac{32 b \sqrt{a+b x} \left(35 a^2 e^2-84 a b d e+58 b^2 d^2\right)}{105 \sqrt{d+e x} (b d-a e)^4}+\frac{16 \sqrt{a+b x} \left(35 a^2 e^2-84 a b d e+58 b^2 d^2\right)}{105 (d+e x)^{3/2} (b d-a e)^3}+\frac{6 d^2 \sqrt{a+b x}}{7 (d+e x)^{7/2} (b d-a e)}+\frac{4 d \sqrt{a+b x} (23 b d-14 a e)}{35 (d+e x)^{5/2} (b d-a e)^2}",1,"(2*Sqrt[a + b*x]*(-(a^3*e^3*(409*d^2 + 644*d*e*x + 280*e^2*x^2)) + a^2*b*e^2*(1953*d^3 + 3890*d^2*e*x + 2632*d*e^2*x^2 + 560*e^3*x^3) + b^3*d^2*(1575*d^3 + 3850*d^2*e*x + 3248*d*e^2*x^2 + 928*e^3*x^3) - a*b^2*d*e*(2975*d^3 + 6664*d^2*e*x + 5168*d*e^2*x^2 + 1344*e^3*x^3)))/(105*(b*d - a*e)^4*(d + e*x)^(7/2))","A",1
850,1,401,417,1.7891918,"\int \frac{(d+e x)^{3/2}}{\sqrt{f+g x} \left(a+b x+c x^2\right)} \, dx","Integrate[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a + b*x + c*x^2)),x]","\frac{\left(e \left(b-\sqrt{b^2-4 a c}\right)-2 c d\right)^{3/2} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d}}\right)-\left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)^{3/2} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}\right)}{c \sqrt{b^2-4 a c} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}+\frac{2 (e f-d g)^{3/2} \left(\frac{e (f+g x)}{e f-d g}\right)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right)}{c \sqrt{g} (f+g x)^{3/2}}","-\frac{2 \left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{c \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}-\frac{2 \left(e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{c \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{2 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right)}{c \sqrt{g}}",1,"(2*(e*f - d*g)^(3/2)*((e*(f + g*x))/(e*f - d*g))^(3/2)*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(c*Sqrt[g]*(f + g*x)^(3/2)) + ((-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]*ArcTanh[(Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])] - (-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*ArcTanh[(Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(c*Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g])","A",1
851,1,266,285,0.8735225,"\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \left(a+b x+c x^2\right)} \, dx","Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*(a + b*x + c*x^2)),x]","\frac{2 \left(\frac{\sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}\right)}{\sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}-\frac{\sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d}}\right)}{\sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}\right)}{\sqrt{b^2-4 a c}}","\frac{2 \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{2 \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}",1,"(2*(-((Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]) + (Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]))/Sqrt[b^2 - 4*a*c]","A",1
852,1,267,287,0.7519067,"\int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \left(a+b x+c x^2\right)} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*(a + b*x + c*x^2)),x]","\frac{4 c \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d}}\right)}{\sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}\right)}{\sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}\right)}{\sqrt{b^2-4 a c}}","\frac{4 c \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}-\frac{4 c \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}",1,"(4*c*(ArcTanh[(Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])]/(Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]) - ArcTanh[(Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])]/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g])))/Sqrt[b^2 - 4*a*c]","A",1
853,1,334,429,2.062227,"\int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x} \left(a+b x+c x^2\right)} \, dx","Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + b*x + c*x^2)),x]","-\frac{8 c^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(b-\sqrt{b^2-4 a c}\right)-2 c d}}\right)}{\sqrt{b^2-4 a c} \left(e \left(b-\sqrt{b^2-4 a c}\right)-2 c d\right)^{3/2} \sqrt{g \left(b-\sqrt{b^2-4 a c}\right)-2 c f}}+\frac{8 c^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}{\sqrt{f+g x} \sqrt{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}\right)}{\sqrt{b^2-4 a c} \left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right)^{3/2} \sqrt{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}+\frac{2 e^2 \sqrt{f+g x}}{\sqrt{d+e x} (d g-e f) \left(e (a e-b d)+c d^2\right)}","-\frac{8 c^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}}\right)}{\sqrt{b^2-4 a c} \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)^{3/2} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}}+\frac{8 c^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}{\sqrt{f+g x} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}\right)}{\sqrt{b^2-4 a c} \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)^{3/2} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}+\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right)}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right)}",1,"(2*e^2*Sqrt[f + g*x])/((c*d^2 + e*(-(b*d) + a*e))*(-(e*f) + d*g)*Sqrt[d + e*x]) - (8*c^2*ArcTanh[(Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[-2*c*f + (b - Sqrt[b^2 - 4*a*c])*g]) + (8*c^2*ArcTanh[(Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g])","A",1
854,1,559,532,1.0129016,"\int \frac{(f+g x)^3 \sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[((f + g*x)^3*Sqrt[a + b*x + c*x^2])/(d + e*x),x]","\frac{-\frac{24 e^2 g (b g-2 c f) (e f-d g) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{c^{5/2}}-\frac{48 e g \left(b^2-4 a c\right) (e f-d g)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}+\frac{e^3 g \left(3 \left(-4 c g (a g+4 b f)+5 b^2 g^2+16 c^2 f^2\right) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)+80 c^{3/2} g (a+x (b+c x))^{3/2} (2 c f-b g)\right)}{c^{7/2}}+\frac{192 (e f-d g)^3 \left(2 \sqrt{c} \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{\sqrt{c} e}+\frac{128 e^2 g^2 (a+x (b+c x))^{3/2} (e f-d g)}{c}+\frac{96 e g (b+2 c x) \sqrt{a+x (b+c x)} (e f-d g)^2}{c}+384 \sqrt{a+x (b+c x)} (e f-d g)^3+\frac{96 e^3 g^2 (f+g x) (a+x (b+c x))^{3/2}}{c}}{384 e^4}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c e (2 c d-b e) \left(-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right)-2 g \left(-2 c e (b d-a e)-\frac{b^2 e^2}{2}+4 c^2 d^2\right) \left(-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left(d^2 g^2-3 d e f g+3 e^2 f^2\right)\right)\right)}{128 c^{7/2} e^5}+\frac{\sqrt{a+b x+c x^2} \left(2 c e g x \left(-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left(d^2 g^2-3 d e f g+3 e^2 f^2\right)\right)-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left(d^2 g^2-3 d e f g+3 e^2 f^2\right)+64 c^3 (e f-d g)^3\right)}{64 c^3 e^4}+\frac{g^2 \left(a+b x+c x^2\right)^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac{(e f-d g)^3 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^5}+\frac{g^3 (d+e x) \left(a+b x+c x^2\right)^{3/2}}{4 c e^2}",1,"(384*(e*f - d*g)^3*Sqrt[a + x*(b + c*x)] + (96*e*g*(e*f - d*g)^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])/c + (128*e^2*g^2*(e*f - d*g)*(a + x*(b + c*x))^(3/2))/c + (96*e^3*g^2*(f + g*x)*(a + x*(b + c*x))^(3/2))/c - (48*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) - (24*e^2*g*(-2*c*f + b*g)*(e*f - d*g)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (e^3*g*(80*c^(3/2)*g*(2*c*f - b*g)*(a + x*(b + c*x))^(3/2) + 3*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/c^(7/2) + (192*(e*f - d*g)^3*((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*e))/(384*e^4)","A",1
855,1,372,325,0.4145643,"\int \frac{(f+g x)^2 \sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[((f + g*x)^2*Sqrt[a + b*x + c*x^2])/(d + e*x),x]","\frac{-\frac{6 e g \left(b^2-4 a c\right) (e f-d g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}-\frac{3 e^2 g (b g-2 c f) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{c^{5/2}}+\frac{24 (e f-d g)^2 \left(2 \sqrt{c} \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{\sqrt{c} e}+\frac{12 e g (b+2 c x) \sqrt{a+x (b+c x)} (e f-d g)}{c}+48 \sqrt{a+x (b+c x)} (e f-d g)^2+\frac{16 e^2 g^2 (a+x (b+c x))^{3/2}}{c}}{48 e^3}","-\frac{\sqrt{a+b x+c x^2} \left(b^2 e^2 g^2-2 c e g x (-b e g-2 c d g+4 c e f)-2 b c e g (2 e f-d g)-8 c^2 (e f-d g)^2\right)}{8 c^2 e^3}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(g \left(-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right) (-b e g-2 c d g+4 c e f)-4 c e (2 c d-b e) \left(2 c e f^2-b d g^2\right)\right)}{16 c^{5/2} e^4}+\frac{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^4}+\frac{g^2 \left(a+b x+c x^2\right)^{3/2}}{3 c e}",1,"(48*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)] + (12*e*g*(e*f - d*g)*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])/c + (16*e^2*g^2*(a + x*(b + c*x))^(3/2))/c - (6*(b^2 - 4*a*c)*e*g*(e*f - d*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) - (3*e^2*g*(-2*c*f + b*g)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (24*(e*f - d*g)^2*((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*e))/(48*e^3)","A",1
856,1,216,219,0.346738,"\int \frac{(f+g x) \sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x),x]","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(4 c e (a e g-b d g+b e f)-b^2 e^2 g+8 c^2 d (d g-e f)\right)+2 \sqrt{c} \left(4 c (d g-e f) \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+e \sqrt{a+x (b+c x)} (b e g+2 c (-2 d g+2 e f+e g x))\right)}{8 c^{3/2} e^3}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c e (a e g-b d g+b e f)+b^2 e^2 g+8 c^2 d (e f-d g)\right)}{8 c^{3/2} e^3}+\frac{(e f-d g) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^3}+\frac{\sqrt{a+b x+c x^2} (b e g-4 c d g+4 c e f+2 c e g x)}{4 c e^2}",1,"((-(b^2*e^2*g) + 8*c^2*d*(-(e*f) + d*g) + 4*c*e*(b*e*f - b*d*g + a*e*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(b*e*g + 2*c*(2*e*f - 2*d*g + e*g*x)) + 4*c*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(-(e*f) + d*g)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(8*c^(3/2)*e^3)","A",1
857,1,145,152,0.1545527,"\int \frac{\sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x),x]","\frac{-2 \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+\frac{(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+2 e \sqrt{a+x (b+c x)}}{2 e^2}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^2}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} e^2}+\frac{\sqrt{a+b x+c x^2}}{e}",1,"(2*e*Sqrt[a + x*(b + c*x)] + ((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] - 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(2*e^2)","A",1
858,1,218,228,0.3413905,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)),x]","\frac{g \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b d-b e x+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}}\right)+\sqrt{c} (e f-d g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-e \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left(\frac{-2 a g+b f-b g x+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{a g^2-b f g+c f^2}}\right)}{e g (e f-d g)}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e (e f-d g)}-\frac{\sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{g (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{e g}",1,"(Sqrt[c]*(e*f - d*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + Sqrt[c*d^2 - b*d*e + a*e^2]*g*ArcTanh[(b*d - 2*a*e + 2*c*d*x - b*e*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + x*(b + c*x)])] - e*Sqrt[c*f^2 - b*f*g + a*g^2]*ArcTanh[(b*f - 2*a*g + 2*c*f*x - b*g*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + x*(b + c*x)])])/(e*g*(e*f - d*g))","A",1
859,1,222,490,0.5165936,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)^2} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^2),x]","\frac{2 \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)-\frac{(2 a e g-b (d g+e f)+2 c d f) \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\sqrt{g (a g-b f)+c f^2}}+\frac{2 \sqrt{a+x (b+c x)} (e f-d g)}{f+g x}}{2 (e f-d g)^2}","\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g)^2}-\frac{e \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{g (e f-d g)^2}+\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 g (e f-d g) \sqrt{a g^2-b f g+c f^2}}+\frac{\sqrt{a+b x+c x^2}}{(f+g x) (e f-d g)}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{g (e f-d g)}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} (e f-d g)^2}+\frac{e (2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} g (e f-d g)^2}",1,"((2*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/(f + g*x) + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] - ((2*c*d*f + 2*a*e*g - b*(e*f + d*g))*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*f^2 + g*(-(b*f) + a*g)])/(2*(e*f - d*g)^2)","A",1
860,1,609,673,1.3732038,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)^3} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^3),x]","\frac{\frac{g \left(b^2-4 a c\right) (e f-d g)^2 \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{3/2}}+8 e \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+\frac{2 g \sqrt{a+x (b+c x)} (e f-d g)^2 (2 a g-b f+b g x-2 c f x)}{(f+g x)^2 \left(g (a g-b f)+c f^2\right)}-\frac{4 e (e f-d g) \left(2 \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\sqrt{g (a g-b f)+c f^2}}\right)}{g}+\frac{8 e \sqrt{a+x (b+c x)} (e f-d g)}{f+g x}+\frac{4 e (b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+\frac{4 e^2 \left((2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} \sqrt{g (a g-b f)+c f^2} \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)\right)}{\sqrt{c} g}}{8 (e f-d g)^3}","\frac{g \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{8 (e f-d g) \left(a g^2-b f g+c f^2\right)^{3/2}}+\frac{e \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g)^3}-\frac{e^2 \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{g (e f-d g)^3}+\frac{e^2 (2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} g (e f-d g)^3}-\frac{g \sqrt{a+b x+c x^2} (-2 a g+x (2 c f-b g)+b f)}{4 (f+g x)^2 (e f-d g) \left(a g^2-b f g+c f^2\right)}+\frac{e (2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 g (e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}+\frac{e \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g)^2}-\frac{\sqrt{c} e \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{g (e f-d g)^2}-\frac{e (2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} (e f-d g)^3}",1,"((8*e*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/(f + g*x) + (2*g*(e*f - d*g)^2*(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^2) + (4*e*(-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 8*e*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] + ((b^2 - 4*a*c)*g*(e*f - d*g)^2*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2) - (4*e*(e*f - d*g)*(2*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - ((2*c*f - b*g)*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*f^2 + g*(-(b*f) + a*g)]))/g + (4*e^2*((2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c]*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*g))/(8*(e*f - d*g)^3)","A",1
861,1,858,933,4.1593674,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*(f + g*x)^4),x]","\frac{\frac{24 \left((2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} \sqrt{c f^2+g (a g-b f)} \tanh ^{-1}\left(\frac{-2 a g+2 c f x+b (f-g x)}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)\right) e^3}{\sqrt{c} g}+24 \left(\frac{(b e-2 c d) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+2 \sqrt{c d^2+e (a e-b d)} \tanh ^{-1}\left(\frac{-2 a e+2 c d x+b (d-e x)}{2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}}\right)\right) e^2-\frac{24 (e f-d g) \left(2 \sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+2 c f x+b (f-g x)}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c f^2+g (a g-b f)}}\right) e^2}{g}+\frac{48 (e f-d g) \sqrt{a+x (b+c x)} e^2}{f+g x}+\frac{6 \left(b^2-4 a c\right) g (e f-d g)^2 \tanh ^{-1}\left(\frac{-2 a g+2 c f x+b (f-g x)}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right) e}{\left(c f^2+g (a g-b f)\right)^{3/2}}+\frac{12 g (e f-d g)^2 (-b f-2 c x f+2 a g+b g x) \sqrt{a+x (b+c x)} e}{\left(c f^2+g (a g-b f)\right) (f+g x)^2}-\frac{16 g^2 (d g-e f)^3 (a+x (b+c x))^{3/2}}{\left(c f^2+g (a g-b f)\right) (f+g x)^3}-\frac{3 g (2 c f-b g) (e f-d g)^3 \left(\frac{2 \sqrt{a+x (b+c x)} (-2 a g+2 c f x+b (f-g x))}{\left(c f^2+g (a g-b f)\right) (f+g x)^2}+\frac{\left(4 a c-b^2\right) \tanh ^{-1}\left(\frac{-2 a g+2 c f x+b (f-g x)}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)}{\left(c f^2+g (a g-b f)\right)^{3/2}}\right)}{c f^2+g (a g-b f)}}{48 (e f-d g)^4}","\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) e^3}{2 \sqrt{c} g (e f-d g)^4}-\frac{\sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e^3}{g (e f-d g)^4}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) e^2}{g (e f-d g)^3}-\frac{(2 c d-b e) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) e^2}{2 \sqrt{c} (e f-d g)^4}+\frac{\sqrt{c d^2-b e d+a e^2} \tanh ^{-1}\left(\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right) e^2}{(e f-d g)^4}+\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e^2}{2 g (e f-d g)^3 \sqrt{c f^2-b g f+a g^2}}+\frac{\sqrt{c x^2+b x+a} e^2}{(e f-d g)^3 (f+g x)}+\frac{\left(b^2-4 a c\right) g \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e}{8 (e f-d g)^2 \left(c f^2-b g f+a g^2\right)^{3/2}}-\frac{g (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a} e}{4 (e f-d g)^2 \left(c f^2-b g f+a g^2\right) (f+g x)^2}+\frac{g^2 \left(c x^2+b x+a\right)^{3/2}}{3 (e f-d g) \left(c f^2-b g f+a g^2\right) (f+g x)^3}+\frac{\left(b^2-4 a c\right) g (2 c f-b g) \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right)}{16 (e f-d g) \left(c f^2-b g f+a g^2\right)^{5/2}}-\frac{g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt{c x^2+b x+a}}{8 (e f-d g) \left(c f^2-b g f+a g^2\right)^2 (f+g x)^2}",1,"((48*e^2*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/(f + g*x) + (12*e*g*(e*f - d*g)^2*(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^2) - (16*g^2*(-(e*f) + d*g)^3*(a + x*(b + c*x))^(3/2))/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^3) + 24*e^2*(((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]) + (6*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2) - (3*g*(2*c*f - b*g)*(e*f - d*g)^3*((2*Sqrt[a + x*(b + c*x)]*(-2*a*g + 2*c*f*x + b*(f - g*x)))/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^2) + ((-b^2 + 4*a*c)*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2)))/(c*f^2 + g*(-(b*f) + a*g)) - (24*e^2*(e*f - d*g)*(2*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - ((2*c*f - b*g)*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*f^2 + g*(-(b*f) + a*g)]))/g + (24*e^3*((2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c]*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*g))/(48*(e*f - d*g)^4)","A",1
862,1,743,1098,2.3848856,"\int \frac{(f+g x)^3 \left(a+b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[((f + g*x)^3*(a + b*x + c*x^2)^(3/2))/(d + e*x),x]","\frac{-\frac{60 e^2 g (b g-2 c f) (e f-d g) \left(2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left(4 c \left(5 a+2 c x^2\right)-3 b^2+8 b c x\right)+3 \left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{c^{7/2}}+\frac{360 e g \left(b^2-4 a c\right) (e f-d g)^2 \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right)}{c^{5/2}}+\frac{960 (e f-d g)^3 \left(-\left((2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)-2 \sqrt{c} \left(e \sqrt{a+x (b+c x)} \left(-2 c e (4 a e-5 b d+b e x)-b^2 e^2+4 c^2 d (e x-2 d)\right)+8 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)\right)}{c^{3/2} e^3}+\frac{e^3 g \left(5 \left(-4 c g (a g+6 b f)+7 b^2 g^2+24 c^2 f^2\right) \left(\frac{3 \left(b^2-4 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right)}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right)+1792 g (a+x (b+c x))^{5/2} (2 c f-b g)\right)}{c^2}+\frac{3072 e^2 g^2 (a+x (b+c x))^{5/2} (e f-d g)}{c}+\frac{1920 e g (b+2 c x) (a+x (b+c x))^{3/2} (e f-d g)^2}{c}+5120 (a+x (b+c x))^{3/2} (e f-d g)^3+\frac{2560 e^3 g^2 (f+g x) (a+x (b+c x))^{5/2}}{c}}{15360 e^4}","\frac{(d+e x) \left(c x^2+b x+a\right)^{5/2} g^3}{6 c e^2}+\frac{(36 c e f-22 c d g-7 b e g) \left(c x^2+b x+a\right)^{5/2} g^2}{60 c^2 e^2}+\frac{\left(7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) g+2 c e \left(24 \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right) x g+64 c^3 (e f-d g)^3\right) \left(c x^2+b x+a\right)^{3/2}}{192 c^3 e^4}+\frac{\left(4 c e (2 c d-b e) \left(8 c e (b d-2 a e) \left(24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right)-d \left(-3 e b^2+8 c d b-4 a c e\right) g \left(24 \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right)\right)-2 \left(4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right) \left(8 c e (2 c d-b e) \left(24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right)-2 \left(8 c^2 d^2-4 b c e d-\frac{3 b^2 e^2}{2}+6 a c e^2\right) g \left(24 \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{3072 c^{9/2} e^7}+\frac{\left(c d^2-b e d+a e^2\right)^{3/2} (e f-d g)^3 \tanh ^{-1}\left(\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right)}{e^7}-\frac{\left(3 \left(-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left(2 b (e f-d g)^3+3 a e g \left(3 e^2 f^2-3 d e g f+d^2 g^2\right)\right) c^3+8 b e^3 g \left(3 \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right)+2 c e \left(8 c e (2 c d-b e) \left(24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right)-2 \left(8 c^2 d^2-4 b c e d-\frac{3 b^2 e^2}{2}+6 a c e^2\right) g \left(24 \left(3 e^2 f^2-3 d e g f+d^2 g^2\right) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right)\right) x\right) \sqrt{c x^2+b x+a}}{1536 c^4 e^6}",1,"(5120*(e*f - d*g)^3*(a + x*(b + c*x))^(3/2) + (1920*e*g*(e*f - d*g)^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3072*e^2*g^2*(e*f - d*g)*(a + x*(b + c*x))^(5/2))/c + (2560*e^3*g^2*(f + g*x)*(a + x*(b + c*x))^(5/2))/c + (360*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) - (60*e^2*g*(-2*c*f + b*g)*(e*f - d*g)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(7/2) + (e^3*g*(1792*g*(2*c*f - b*g)*(a + x*(b + c*x))^(5/2) + 5*(24*c^2*f^2 + 7*b^2*g^2 - 4*c*g*(6*b*f + a*g))*((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2))))/c^2 + (960*(e*f - d*g)^3*(-((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) - 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e + b*e*x)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])))/(c^(3/2)*e^3))/(15360*e^4)","A",1
863,1,536,662,1.2490183,"\int \frac{(f+g x)^2 \left(a+b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[((f + g*x)^2*(a + b*x + c*x^2)^(3/2))/(d + e*x),x]","\frac{\frac{90 e g \left(b^2-4 a c\right) (e f-d g) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right)}{c^{5/2}}+\frac{15 e^2 g (2 c f-b g) \left(\frac{3 \left(b^2-4 a c\right) \left(\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right)}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right)}{c}+\frac{240 (e f-d g)^2 \left(-\left((2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)-2 \sqrt{c} \left(e \sqrt{a+x (b+c x)} \left(-2 c e (4 a e-5 b d+b e x)-b^2 e^2+4 c^2 d (e x-2 d)\right)+8 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)\right)}{c^{3/2} e^3}+1280 (a+x (b+c x))^{3/2} (e f-d g)^2+\frac{480 e g (b+2 c x) (a+x (b+c x))^{3/2} (e f-d g)}{c}+\frac{768 e^2 g^2 (a+x (b+c x))^{5/2}}{c}}{3840 e^3}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(96 c^3 e^2 \left(-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right)+16 b c^2 e^3 \left(3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right)-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right)}{256 c^{7/2} e^6}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right)}{48 c^2 e^3}+\frac{\sqrt{a+b x+c x^2} \left(2 c e x \left(g \left(-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left(2 c e f^2-b d g^2\right)\right)-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)-32 c^3 e (5 b d-4 a e) (e f-d g)^2+8 b c^2 e^2 \left(3 a e g (2 e f-d g)+2 b (e f-d g)^2\right)+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right)}{128 c^3 e^5}+\frac{(e f-d g)^2 \left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^6}+\frac{g^2 \left(a+b x+c x^2\right)^{5/2}}{5 c e}",1,"(1280*(e*f - d*g)^2*(a + x*(b + c*x))^(3/2) + (480*e*g*(e*f - d*g)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (768*e^2*g^2*(a + x*(b + c*x))^(5/2))/c + (90*(b^2 - 4*a*c)*e*g*(e*f - d*g)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (15*e^2*g*(2*c*f - b*g)*((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2)))/c + (240*(e*f - d*g)^2*(-((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) - 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e + b*e*x)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])))/(c^(3/2)*e^3))/(3840*e^3)","A",1
864,1,420,441,1.1279332,"\int \frac{(f+g x) \left(a+b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[((f + g*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x),x]","\frac{\frac{3 \left(\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(48 c^2 e^2 \left(a^2 e^2 g+2 a b e (e f-d g)+b^2 d (d g-e f)\right)-8 b^2 c e^3 (3 a e g-b d g+b e f)-192 c^3 d e (b d-a e) (d g-e f)+3 b^4 e^4 g+128 c^4 d^3 (d g-e f)\right)+2 \sqrt{c} e \sqrt{a+x (b+c x)} \left(8 c^2 e (a e (-8 d g+8 e f+3 e g x)+2 b (e x-5 d) (e f-d g))+2 b c e^2 (6 a e g+b (-4 d g+4 e f-3 e g x))-3 b^3 e^3 g-32 c^3 d (e x-2 d) (e f-d g)\right)+128 c^{5/2} (d g-e f) \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)}{16 c^{3/2} e^3}+(a+x (b+c x))^{3/2} (3 b e g+c (-8 d g+8 e f+6 e g x))}{24 c e^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(48 c^2 e^2 \left(a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right)-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right)}{128 c^{5/2} e^5}-\frac{\sqrt{a+b x+c x^2} \left(2 c e x \left(-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right)}{64 c^2 e^4}+\frac{(e f-d g) \left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^5}+\frac{\left(a+b x+c x^2\right)^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}",1,"((a + x*(b + c*x))^(3/2)*(3*b*e*g + c*(8*e*f - 8*d*g + 6*e*g*x)) + (3*(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(-3*b^3*e^3*g - 32*c^3*d*(e*f - d*g)*(-2*d + e*x) + 2*b*c*e^2*(6*a*e*g + b*(4*e*f - 4*d*g - 3*e*g*x)) + 8*c^2*e*(2*b*(e*f - d*g)*(-5*d + e*x) + a*e*(8*e*f - 8*d*g + 3*e*g*x))) + (3*b^4*e^4*g + 128*c^4*d^3*(-(e*f) + d*g) - 192*c^3*d*e*(b*d - a*e)*(-(e*f) + d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e*g) + 48*c^2*e^2*(a^2*e^2*g + 2*a*b*e*(e*f - d*g) + b^2*d*(-(e*f) + d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 128*c^(5/2)*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(-(e*f) + d*g)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(16*c^(3/2)*e^3))/(24*c*e^2)","A",1
865,1,236,252,0.4119351,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{d+e x} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x),x]","\frac{2 \sqrt{c} \left(e \sqrt{a+x (b+c x)} \left(2 c e (16 a e-15 b d+7 b e x)+3 b^2 e^2+4 c^2 \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)-24 c \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)\right)-3 (2 c d-b e) \left(4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{48 c^{3/2} e^4}","\frac{\sqrt{a+b x+c x^2} \left(-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right)}{8 c e^3}-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} e^4}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^4}+\frac{\left(a+b x+c x^2\right)^{3/2}}{3 e}",1,"(-3*(2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(3*b^2*e^2 + 2*c*e*(-15*b*d + 16*a*e + 7*b*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 24*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(48*c^(3/2)*e^4)","A",1
866,1,323,491,1.0676449,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x) (f+g x)} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)),x]","\frac{\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-12 c e g (-a e g+b d g+b e f)+3 b^2 e^2 g^2+8 c^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)}{\sqrt{c}}+\frac{2 \left(-4 g^3 \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+e g \sqrt{a+x (b+c x)} (e f-d g) (5 b e g+c (-4 d g-4 e f+2 e g x))+4 e^3 \left(g (a g-b f)+c f^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)\right)}{e f-d g}}{8 e^3 g^3}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c g \left(3 b e f^2-a g (3 e f-d g)\right)+b g^2 (-4 a e g+b d g+3 b e f)+8 c^2 e f^3\right)}{8 \sqrt{c} e g^3 (e f-d g)}+\frac{\sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}{e^2 (e f-d g)}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^3 (e f-d g)}-\frac{(2 c d-b e) \left(a e^2-b d e+c d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{2 \sqrt{c} e^3 (e f-d g)}-\frac{\sqrt{a+b x+c x^2} \left(-g (-4 a e g-b d g+5 b e f)-2 c g x (e f-d g)+4 c e f^2\right)}{4 e g^2 (e f-d g)}-\frac{\left(a g^2-b f g+c f^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{g^3 (e f-d g)}",1,"(((3*b^2*e^2*g^2 - 12*c*e*g*(b*e*f + b*d*g - a*e*g) + 8*c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (2*(e*g*(e*f - d*g)*Sqrt[a + x*(b + c*x)]*(5*b*e*g + c*(-4*e*f - 4*d*g + 2*e*g*x)) - 4*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*g^3*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] + 4*e^3*(c*f^2 + g*(-(b*f) + a*g))^(3/2)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])]))/(e*f - d*g))/(8*e^3*g^3)","A",1
867,1,357,787,1.3991153,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x) (f+g x)^2} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)^2),x]","\frac{-2 g^3 (f+g x) \left(e (a e-b d)+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+e \left(2 g \sqrt{a+x (b+c x)} (d g-e f) (e g (b f-a g)+c d g (f+g x)-c e f (2 f+g x))-e (f+g x) \sqrt{g (a g-b f)+c f^2} (g (-2 a e g+3 b d g-b e f)+2 c f (2 e f-3 d g)) \tanh ^{-1}\left(\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)\right)-\sqrt{c} (f+g x) (e f-d g)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) (-3 b e g+2 c d g+4 c e f)}{2 e^2 g^3 (f+g x) (e f-d g)^2}","\frac{\sqrt{a+b x+c x^2} \left(-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right)}{8 c e (e f-d g)^2}-\frac{e \sqrt{a+b x+c x^2} \left(-2 c g (5 b f-4 a g)+b^2 g^2-2 c g x (2 c f-b g)+8 c^2 f^2\right)}{8 c g^2 (e f-d g)^2}-\frac{3 \left(-4 c g (2 b f-a g)+b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 \sqrt{c} g^3 (e f-d g)}-\frac{(2 c d-b e) \left(-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} e^2 (e f-d g)^2}+\frac{e (2 c f-b g) \left(-4 c g (2 b f-3 a g)-b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{16 c^{3/2} g^3 (e f-d g)^2}+\frac{\left(a e^2-b d e+c d^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^2 (e f-d g)^2}-\frac{e \left(a g^2-b f g+c f^2\right)^{3/2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{g^3 (e f-d g)^2}+\frac{3 (2 c f-b g) \sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 g^3 (e f-d g)}+\frac{3 \sqrt{a+b x+c x^2} (-3 b g+4 c f-2 c g x)}{4 g^2 (e f-d g)}+\frac{\left(a+b x+c x^2\right)^{3/2}}{(f+g x) (e f-d g)}",1,"(-(Sqrt[c]*(e*f - d*g)^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(f + g*x)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) - 2*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*g^3*(f + g*x)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] + e*(2*g*(-(e*f) + d*g)*Sqrt[a + x*(b + c*x)]*(e*g*(b*f - a*g) + c*d*g*(f + g*x) - c*e*f*(2*f + g*x)) - e*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*(2*c*f*(2*e*f - 3*d*g) + g*(-(b*e*f) + 3*b*d*g - 2*a*e*g))*(f + g*x)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])]))/(2*e^2*g^3*(e*f - d*g)^2*(f + g*x))","A",1
868,1,1036,1066,3.273186,"\int \frac{\left(a+b x+c x^2\right)^{3/2}}{(d+e x) (f+g x)^3} \, dx","Integrate[(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)^3),x]","\frac{1}{4} \left(-\frac{\left((2 c f-b g) \left(8 c^2 f^2-b^2 g^2+4 c g (3 a g-2 b f)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} \left(8 c \tanh ^{-1}\left(\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right) \left(c f^2+g (a g-b f)\right)^{3/2}+g \sqrt{a+x (b+c x)} \left(4 f (g x-2 f) c^2-2 g (-5 b f+4 a g+b g x) c-b^2 g^2\right)\right)\right) e^2}{4 c^{3/2} g^3 (d g-e f)^3}+\frac{4 (a+x (b+c x))^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac{3 \left(\left(8 c^2 f^2+b^2 g^2+4 c g (a g-2 b f)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+2 \sqrt{c} \left(g \sqrt{a+x (b+c x)} (-4 c f+3 b g+2 c g x)+2 (2 c f-b g) \sqrt{c f^2+g (a g-b f)} \tanh ^{-1}\left(\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)\right)\right) e}{2 \sqrt{c} g^3 (e f-d g)^2}+\frac{2 (a+x (b+c x))^{3/2}}{(e f-d g) (f+g x)^2}+\frac{3 \left(\frac{(b g-2 c f) (a+x (b+c x))^{3/2}}{f+g x}-\frac{\left(2 f (2 f-g x) c^2+g (-5 b f+2 a g+b g x) c+b^2 g^2\right) \sqrt{a+x (b+c x)}}{g^2}+\frac{4 \sqrt{c} (2 c f-b g) \left(c f^2+g (a g-b f)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+\left(8 c^2 f^2+b^2 g^2+4 c g (a g-2 b f)\right) \sqrt{c f^2+g (a g-b f)} \tanh ^{-1}\left(\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)}{2 g^3}\right)}{(e f-d g) \left(c f^2+g (a g-b f)\right)}+\frac{-\left((2 c d-b e) \left(8 c^2 d^2-b^2 e^2+4 c e (3 a e-2 b d)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)-2 \sqrt{c} \left(8 c \tanh ^{-1}\left(\frac{-b d-2 c x d+2 a e+b e x}{2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}}\right) \left(c d^2+e (a e-b d)\right)^{3/2}+e \sqrt{a+x (b+c x)} \left(4 d (e x-2 d) c^2-2 e (-5 b d+4 a e+b e x) c-b^2 e^2\right)\right)}{4 c^{3/2} (e f-d g)^3 e}\right)","\frac{(2 c f-b g) \left(8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) e^2}{16 c^{3/2} g^3 (e f-d g)^3}-\frac{\left(c f^2-b g f+a g^2\right)^{3/2} \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e^2}{g^3 (e f-d g)^3}-\frac{\left(8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right) \sqrt{c x^2+b x+a} e^2}{8 c g^2 (e f-d g)^3}+\frac{\left(c x^2+b x+a\right)^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac{3 \left(8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) e}{8 \sqrt{c} g^3 (e f-d g)^2}+\frac{3 (2 c f-b g) \sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e}{2 g^3 (e f-d g)^2}+\frac{3 (4 c f-3 b g-2 c g x) \sqrt{c x^2+b x+a} e}{4 g^2 (e f-d g)^2}+\frac{\left(c x^2+b x+a\right)^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{3 \sqrt{c} (2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{2 g^3 (e f-d g)}-\frac{3 \left(8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right) \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right)}{8 g^3 (e f-d g) \sqrt{c f^2-b g f+a g^2}}+\frac{\left(8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right) \sqrt{c x^2+b x+a}}{8 c (e f-d g)^3}-\frac{3 (4 c f-b g+2 c g x) \sqrt{c x^2+b x+a}}{4 g^2 (e f-d g) (f+g x)}-\frac{(2 c d-b e) \left(8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{16 c^{3/2} (e f-d g)^3 e}+\frac{\left(c d^2-b e d+a e^2\right)^{3/2} \tanh ^{-1}\left(\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right)}{(e f-d g)^3 e}",1,"((2*(a + x*(b + c*x))^(3/2))/((e*f - d*g)*(f + g*x)^2) + (4*e*(a + x*(b + c*x))^(3/2))/((e*f - d*g)^2*(f + g*x)) + (-((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) - 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e + b*e*x)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(4*c^(3/2)*e*(e*f - d*g)^3) - (3*e*((8*c^2*f^2 + b^2*g^2 + 4*c*g*(-2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(g*(-4*c*f + 3*b*g + 2*c*g*x)*Sqrt[a + x*(b + c*x)] + 2*(2*c*f - b*g)*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])))/(2*Sqrt[c]*g^3*(e*f - d*g)^2) + (3*(((-2*c*f + b*g)*(a + x*(b + c*x))^(3/2))/(f + g*x) - (Sqrt[a + x*(b + c*x)]*(b^2*g^2 + 2*c^2*f*(2*f - g*x) + c*g*(-5*b*f + 2*a*g + b*g*x)))/g^2 + (4*Sqrt[c]*(2*c*f - b*g)*(c*f^2 + g*(-(b*f) + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (8*c^2*f^2 + b^2*g^2 + 4*c*g*(-2*b*f + a*g))*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(2*g^3)))/((e*f - d*g)*(c*f^2 + g*(-(b*f) + a*g))) - (e^2*((2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 + 4*c*g*(-2*b*f + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(g*Sqrt[a + x*(b + c*x)]*(-(b^2*g^2) + 4*c^2*f*(-2*f + g*x) - 2*c*g*(-5*b*f + 4*a*g + b*g*x)) + 8*c*(c*f^2 + g*(-(b*f) + a*g))^(3/2)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])))/(4*c^(3/2)*g^3*(-(e*f) + d*g)^3))/4","A",1
869,1,647,886,2.5389231,"\int \frac{\left(a+b x+c x^2\right)^{5/2}}{(d+e x) (f+g x)} \, dx","Integrate[(a + b*x + c*x^2)^(5/2)/((d + e*x)*(f + g*x)),x]","\frac{3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(240 c^2 e^2 g^2 (e f-d g) \left(a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left(d^2 g^2+d e f g+e^2 f^2\right)\right)-40 b^2 c e^3 g^3 (e f-d g) (-3 a e g+b d g+b e f)+320 c^3 e g \left(-a d^3 e g^4+a e^4 f^3 g+b d^4 g^4-b e^4 f^4\right)+5 b^4 e^4 g^4 (d g-e f)+128 c^4 \left(e^5 f^5-d^5 g^5\right)\right)+2 \sqrt{c} \left(-e g \sqrt{a+x (b+c x)} (d g-e f) \left(8 c^2 e g \left(a e g (-56 d g-56 e f+27 e g x)+b \left(54 d^2 g^2+2 d e g (27 f-13 g x)+e^2 \left(54 f^2-26 f g x+17 g^2 x^2\right)\right)\right)+2 b c e^2 g^2 (278 a e g+b (-132 d g-132 e f+59 e g x))+15 b^3 e^3 g^3-16 c^3 \left(12 d^3 g^3-6 d^2 e g^2 (g x-2 f)+2 d e^2 g \left(6 f^2-3 f g x+2 g^2 x^2\right)+e^3 \left(12 f^3-6 f^2 g x+4 f g^2 x^2-3 g^3 x^3\right)\right)\right)-192 c g^5 \left(e (a e-b d)+c d^2\right)^{5/2} \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)+192 c e^5 \left(g (a g-b f)+c f^2\right)^{5/2} \tanh ^{-1}\left(\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)\right)}{384 c^{3/2} e^5 g^5 (e f-d g)}","\frac{\tanh ^{-1}\left(\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right) \left(c d^2-b e d+a e^2\right)^{5/2}}{e^5 (e f-d g)}+\frac{\left(c x^2+b x+a\right)^{3/2} \left(c d^2-b e d+a e^2\right)}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left(8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) \left(c d^2-b e d+a e^2\right)}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left(8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right) \sqrt{c x^2+b x+a} \left(c d^2-b e d+a e^2\right)}{8 c e^4 (e f-d g)}-\frac{\left(8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right) \left(c x^2+b x+a\right)^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left(128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left(5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right)-8 b c g^3 \left(5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left(c f^2-b g f+a g^2\right)^{5/2} \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right)}{g^5 (e f-d g)}-\frac{\left(64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left(22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right)-2 c g \left(16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left(6 b e f^2-a g (7 e f-3 d g)\right)\right) x\right) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)}",1,"(3*(5*b^4*e^4*g^4*(-(e*f) + d*g) - 40*b^2*c*e^3*g^3*(e*f - d*g)*(b*e*f + b*d*g - 3*a*e*g) + 320*c^3*e*g*(-(b*e^4*f^4) + a*e^4*f^3*g + b*d^4*g^4 - a*d^3*e*g^4) + 128*c^4*(e^5*f^5 - d^5*g^5) + 240*c^2*e^2*g^2*(e*f - d*g)*(a^2*e^2*g^2 - 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(-(e*g*(-(e*f) + d*g)*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 + 2*b*c*e^2*g^2*(278*a*e*g + b*(-132*e*f - 132*d*g + 59*e*g*x)) - 16*c^3*(12*d^3*g^3 - 6*d^2*e*g^2*(-2*f + g*x) + 2*d*e^2*g*(6*f^2 - 3*f*g*x + 2*g^2*x^2) + e^3*(12*f^3 - 6*f^2*g*x + 4*f*g^2*x^2 - 3*g^3*x^3)) + 8*c^2*e*g*(a*e*g*(-56*e*f - 56*d*g + 27*e*g*x) + b*(54*d^2*g^2 + 2*d*e*g*(27*f - 13*g*x) + e^2*(54*f^2 - 26*f*g*x + 17*g^2*x^2))))) - 192*c*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*g^5*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] + 192*c*e^5*(c*f^2 + g*(-(b*f) + a*g))^(5/2)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])]))/(384*c^(3/2)*e^5*g^5*(e*f - d*g))","A",1
870,1,553,431,0.8880744,"\int \frac{(f+g x)^4}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)^4/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\frac{6 e^2 g (e f-d g) \left(\left(-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)+6 \sqrt{c} g \sqrt{a+x (b+c x)} (2 c f-b g)\right)}{c^{5/2}}+\frac{e^3 g \left(\frac{2 g \sqrt{a+x (b+c x)} \left(-2 c g (8 a g+27 b f+5 b g x)+15 b^2 g^2+4 c^2 f (16 f+5 g x)\right)}{c^2}+\frac{3 (2 c f-b g) \left(-4 c g (3 a g+2 b f)+5 b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{5/2}}\right)}{c}+\frac{24 e g (2 c f-b g) (e f-d g)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}+\frac{48 (e f-d g)^4 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{24 e^2 g^2 (f+g x) \sqrt{a+x (b+c x)} (e f-d g)}{c}+\frac{48 e g^2 \sqrt{a+x (b+c x)} (e f-d g)^2}{c}+\frac{48 g (e f-d g)^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+\frac{16 e^3 g^2 (f+g x)^2 \sqrt{a+x (b+c x)}}{c}}{48 e^4}","-\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(8 c^2 e g \left(a e g (4 e f-d g)+b \left(d^2 g^2-4 d e f g+6 e^2 f^2\right)\right)-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left(-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right)\right)}{16 c^{7/2} e^4}+\frac{g^2 \sqrt{a+b x+c x^2} \left(-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left(11 d^2 g^2-36 d e f g+36 e^2 f^2\right)\right)}{24 c^3 e^3}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac{(e f-d g)^4 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^4 \sqrt{a e^2-b d e+c d^2}}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}",1,"((48*e*g^2*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)])/c + (24*e^2*g^2*(e*f - d*g)*(f + g*x)*Sqrt[a + x*(b + c*x)])/c + (16*e^3*g^2*(f + g*x)^2*Sqrt[a + x*(b + c*x)])/c + (24*e*g*(2*c*f - b*g)*(e*f - d*g)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (48*g*(e*f - d*g)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (6*e^2*g*(e*f - d*g)*(6*Sqrt[c]*g*(2*c*f - b*g)*Sqrt[a + x*(b + c*x)] + (8*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (e^3*g*((2*g*Sqrt[a + x*(b + c*x)]*(15*b^2*g^2 + 4*c^2*f*(16*f + 5*g*x) - 2*c*g*(27*b*f + 8*a*g + 5*b*g*x)))/c^2 + (3*(2*c*f - b*g)*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)))/c + (48*(e*f - d*g)^4*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(48*e^4)","A",1
871,1,358,270,0.3800125,"\int \frac{(f+g x)^3}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)^3/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\frac{e^2 g \left(-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{5/2}}+\frac{4 e g (2 c f-b g) (e f-d g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{c^{3/2}}+\frac{6 e^2 g^2 \sqrt{a+x (b+c x)} (2 c f-b g)}{c^2}+\frac{8 (e f-d g)^3 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{8 e g^2 \sqrt{a+x (b+c x)} (e f-d g)}{c}+\frac{8 g (e f-d g)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}+\frac{4 e^2 g^2 (f+g x) \sqrt{a+x (b+c x)}}{c}}{8 e^3}","\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left(d^2 g^2-3 d e f g+3 e^2 f^2\right)\right)}{8 c^{5/2} e^3}+\frac{3 g^2 \sqrt{a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac{(e f-d g)^3 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^3 \sqrt{a e^2-b d e+c d^2}}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2}",1,"((6*e^2*g^2*(2*c*f - b*g)*Sqrt[a + x*(b + c*x)])/c^2 + (8*e*g^2*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/c + (4*e^2*g^2*(f + g*x)*Sqrt[a + x*(b + c*x)])/c + (4*e*g*(2*c*f - b*g)*(e*f - d*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (8*g*(e*f - d*g)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + (e^2*g*(8*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2) + (8*(e*f - d*g)^3*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(8*e^3)","A",1
872,1,170,176,0.4830269,"\int \frac{(f+g x)^2}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)^2/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{-\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) (b e g+2 c d g-4 c e f)}{c^{3/2}}+\frac{2 (e f-d g)^2 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 e g^2 \sqrt{a+x (b+c x)}}{c}}{2 e^2}","\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (-b e g-2 c d g+4 c e f)}{2 c^{3/2} e^2}+\frac{(e f-d g)^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^2 \sqrt{a e^2-b d e+c d^2}}+\frac{g^2 \sqrt{a+b x+c x^2}}{c e}",1,"((2*e*g^2*Sqrt[a + x*(b + c*x)])/c - (g*(-4*c*e*f + 2*c*d*g + b*e*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (2*(e*f - d*g)^2*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(2*e^2)","A",1
873,1,126,131,0.151053,"\int \frac{f+g x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\frac{(d g-e f) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}}{e}","\frac{(e f-d g) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e \sqrt{a e^2-b d e+c d^2}}+\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} e}",1,"((g*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + ((-(e*f) + d*g)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/e","A",1
874,1,78,79,0.0142552,"\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","-\frac{\tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}","\frac{\tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\sqrt{a e^2-b d e+c d^2}}",1,"-(ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]/Sqrt[c*d^2 + e*(-(b*d) + a*e)])","A",1
875,1,169,182,0.2491177,"\int \frac{1}{(d+e x) (f+g x) \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\frac{g \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\sqrt{g (a g-b f)+c f^2}}-\frac{e \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}}{d g-e f}","\frac{e \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g) \sqrt{a e^2-b d e+c d^2}}-\frac{g \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{(e f-d g) \sqrt{a g^2-b f g+c f^2}}",1,"(-((e*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)]) + (g*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*f^2 + g*(-(b*f) + a*g)])/(-(e*f) + d*g)","A",1
876,1,256,340,0.954901,"\int \frac{1}{(d+e x) (f+g x)^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^2*Sqrt[a + b*x + c*x^2]),x]","-\frac{-\frac{2 e^2 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 g^2 \sqrt{a+x (b+c x)} (d g-e f)}{(f+g x) \left(g (a g-b f)+c f^2\right)}+\frac{g (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g)) \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{3/2}}}{2 (e f-d g)^2}","\frac{e^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2}}+\frac{g^2 \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g) \left(a g^2-b f g+c f^2\right)}-\frac{e g \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{(e f-d g)^2 \sqrt{a g^2-b f g+c f^2}}-\frac{g (2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 (e f-d g) \left(a g^2-b f g+c f^2\right)^{3/2}}",1,"-1/2*((2*g^2*(-(e*f) + d*g)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)) - (2*e^2*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] + (g*(2*c*f*(2*e*f - d*g) + g*(-3*b*e*f + b*d*g + 2*a*e*g))*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2))/(e*f - d*g)^2","A",1
877,1,549,587,2.4288803,"\int \frac{1}{(d+e x) (f+g x)^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^3*Sqrt[a + b*x + c*x^2]),x]","\frac{g (e f-d g)^2 \left(\frac{6 g \sqrt{a+x (b+c x)} (2 c f-b g)}{(f+g x) \left(g (a g-b f)+c f^2\right)^2}-\frac{\left(-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{5/2}}\right)+\frac{8 e^3 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\sqrt{e (a e-b d)+c d^2}}+\frac{8 e g^2 \sqrt{a+x (b+c x)} (e f-d g)}{(f+g x) \left(g (a g-b f)+c f^2\right)}+\frac{4 g^2 \sqrt{a+x (b+c x)} (e f-d g)^2}{(f+g x)^2 \left(g (a g-b f)+c f^2\right)}+\frac{4 e g (b g-2 c f) (e f-d g) \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{3/2}}-\frac{8 e^2 g \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\sqrt{g (a g-b f)+c f^2}}}{8 (e f-d g)^3}","-\frac{g \left(-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{8 (e f-d g) \left(a g^2-b f g+c f^2\right)^{5/2}}+\frac{e^3 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g)^3 \sqrt{a e^2-b d e+c d^2}}-\frac{e^2 g \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{(e f-d g)^3 \sqrt{a g^2-b f g+c f^2}}+\frac{e g^2 \sqrt{a+b x+c x^2}}{(f+g x) (e f-d g)^2 \left(a g^2-b f g+c f^2\right)}+\frac{3 g^2 \sqrt{a+b x+c x^2} (2 c f-b g)}{4 (f+g x) (e f-d g) \left(a g^2-b f g+c f^2\right)^2}+\frac{g^2 \sqrt{a+b x+c x^2}}{2 (f+g x)^2 (e f-d g) \left(a g^2-b f g+c f^2\right)}-\frac{e g (2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 (e f-d g)^2 \left(a g^2-b f g+c f^2\right)^{3/2}}",1,"((4*g^2*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^2) + (8*e*g^2*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)) + (8*e^3*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] + (4*e*g*(-2*c*f + b*g)*(e*f - d*g)*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2) - (8*e^2*g*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*f^2 + g*(-(b*f) + a*g)] + g*(e*f - d*g)^2*((6*g*(2*c*f - b*g)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))^2*(f + g*x)) - ((8*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(2*b*f + a*g))*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(5/2)))/(8*(e*f - d*g)^3)","A",1
878,1,587,496,2.4570519,"\int \frac{(f+g x)^4}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{-\frac{2 e \left(b^2 \left(3 a^2 e^2 g^4+a c g^3 \left(d^2 g+4 d e (2 f+3 g x)+e^2 x (g x-8 f)\right)+c^2 \left(d^2 g^4 x^2-12 d e f^2 g^2 x+2 e^2 f^4\right)\right)-2 b c \left(a^2 e g^3 (-5 d g+4 e f+5 e g x)+2 a c g \left(d^2 g^3 x+d e g \left(3 f^2+6 f g x-g^2 x^2\right)+e^2 f^2 (2 f-3 g x)\right)+c^2 e f^3 (d (f-4 g x)-e f x)\right)-4 c \left(2 a^3 e^2 g^4+a^2 c g^2 \left(d^2 g^2+d e g (4 f+g x)+e^2 \left(-6 f^2-4 f g x+g^2 x^2\right)\right)+a c^2 \left(d^2 g^4 x^2-2 d e f^2 g (2 f+3 g x)+e^2 f^3 (f+4 g x)\right)+c^3 d e f^4 x\right)+b^3 g^3 (3 a e g (e x-d)+c d x (d g+8 e f-e g x))-3 b^4 d e g^4 x\right)}{c^2 \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{g^3 \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right) (-3 b e g-2 c d g+8 c e f)}{c^{5/2}}+\frac{2 (e f-d g)^4 \log (d+e x)}{\left(e (a e-b d)+c d^2\right)^{3/2}}-\frac{2 (e f-d g)^4 \log \left(2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}}{2 e^2}","-\frac{2 \left(-b^2 \left(a^2 e g^4+4 a c d f g^3+c^2 e f^4\right)+x \left(2 c^2 g^2 \left(a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right)-b c g^3 \left(-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right)+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right)+b c \left(a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right)+2 a c \left(a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right)+a b^3 d g^4\right)}{c^2 \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{g^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e^2 \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"((-2*e*(-3*b^4*d*e*g^4*x + b^3*g^3*(3*a*e*g*(-d + e*x) + c*d*x*(8*e*f + d*g - e*g*x)) + b^2*(3*a^2*e^2*g^4 + c^2*(2*e^2*f^4 - 12*d*e*f^2*g^2*x + d^2*g^4*x^2) + a*c*g^3*(d^2*g + e^2*x*(-8*f + g*x) + 4*d*e*(2*f + 3*g*x))) - 2*b*c*(a^2*e*g^3*(4*e*f - 5*d*g + 5*e*g*x) + c^2*e*f^3*(-(e*f*x) + d*(f - 4*g*x)) + 2*a*c*g*(d^2*g^3*x + e^2*f^2*(2*f - 3*g*x) + d*e*g*(3*f^2 + 6*f*g*x - g^2*x^2))) - 4*c*(2*a^3*e^2*g^4 + c^3*d*e*f^4*x + a*c^2*(d^2*g^4*x^2 - 2*d*e*f^2*g*(2*f + 3*g*x) + e^2*f^3*(f + 4*g*x)) + a^2*c*g^2*(d^2*g^2 + d*e*g*(4*f + g*x) + e^2*(-6*f^2 - 4*f*g*x + g^2*x^2)))))/(c^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (2*(e*f - d*g)^4*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + (g^3*(8*c*e*f - 2*c*d*g - 3*b*e*g)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) - (2*(e*f - d*g)^4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(2*e^2)","A",1
879,1,373,357,1.0444466,"\int \frac{(f+g x)^3}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(f + g*x)^3/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 \left(b \left(a^2 e g^3+3 a c g (d g (f+g x)+e f (f-g x))+c^2 f^2 (d (f-3 g x)-e f x)\right)+2 c \left(a^2 g^2 (d g-e (3 f+g x))+a c f (e f (f+3 g x)-3 d g (f+g x))+c^2 d f^3 x\right)+b^2 \left(a g^3 (e x-d)+c \left(3 d f g^2 x-e f^3\right)\right)-b^3 d g^3 x\right)}{c \left(4 a c-b^2\right) \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}+\frac{g^3 \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)}{c^{3/2} e}+\frac{(e f-d g)^3 \log (d+e x)}{e \left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{(d g-e f)^3 \log \left(2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right)}{e \left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{2 \left(-x \left(c g^2 \left(-2 a^2 e g+3 a b d g-3 a b e f+3 b^2 d f\right)-b^2 g^3 (b d-a e)+c^2 f (6 a g (e f-d g)-b f (3 d g+e f))+2 c^3 d f^3\right)-b \left(a^2 e g^3+3 a c f g (d g+e f)+c^2 d f^3\right)+b^2 \left(a d g^3+c e f^3\right)-2 a c \left(c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{g^3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2} e}+\frac{(e f-d g)^3 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{e \left(a e^2-b d e+c d^2\right)^{3/2}}",1,"(2*(-(b^3*d*g^3*x) + b^2*(a*g^3*(-d + e*x) + c*(-(e*f^3) + 3*d*f*g^2*x)) + b*(a^2*e*g^3 + c^2*f^2*(-(e*f*x) + d*(f - 3*g*x)) + 3*a*c*g*(e*f*(f - g*x) + d*g*(f + g*x))) + 2*c*(c^2*d*f^3*x + a^2*g^2*(d*g - e*(3*f + g*x)) + a*c*f*(-3*d*g*(f + g*x) + e*f*(f + 3*g*x)))))/(c*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]) + ((e*f - d*g)^3*Log[d + e*x])/(e*(c*d^2 + e*(-(b*d) + a*e))^(3/2)) + (g^3*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(c^(3/2)*e) + ((-(e*f) + d*g)^3*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(e*(c*d^2 + e*(-(b*d) + a*e))^(3/2))","A",1
880,1,265,240,0.6326043,"\int \frac{(f+g x)^2}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 \left(-2 a^2 e g^2+a b g (d g+2 e f-e g x)-2 a c d g (2 f+g x)+2 a c e f (f+2 g x)+b^2 \left(d g^2 x-e f^2\right)+b c f (d (f-2 g x)-e f x)+2 c^2 d f^2 x\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{(e f-d g)^2 \log (d+e x)}{\left(e (a e-b d)+c d^2\right)^{3/2}}-\frac{(e f-d g)^2 \log \left(2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{2 \left(-x \left(c (2 a g (2 e f-d g)-b f (2 d g+e f))+b g^2 (b d-a e)+2 c^2 d f^2\right)-b \left(a g (d g+2 e f)+c d f^2\right)+2 a \left(a e g^2-c f (e f-2 d g)\right)+b^2 e f^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}+\frac{(e f-d g)^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\left(a e^2-b d e+c d^2\right)^{3/2}}",1,"(2*(-2*a^2*e*g^2 + 2*c^2*d*f^2*x - 2*a*c*d*g*(2*f + g*x) + 2*a*c*e*f*(f + 2*g*x) + a*b*g*(2*e*f + d*g - e*g*x) + b^2*(-(e*f^2) + d*g^2*x) + b*c*f*(-(e*f*x) + d*(f - 2*g*x))))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + ((e*f - d*g)^2*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) - ((e*f - d*g)^2*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)","A",1
881,1,183,187,0.1608903,"\int \frac{f+g x}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(f + g*x)/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{2 b (a e g+c d (f-g x)-c e f x)+4 c (-a d g+a e (f+g x)+c d f x)-2 b^2 e f}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{e (d g-e f) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}","\frac{e (e f-d g) \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{2 \left(c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"(-2*b^2*e*f + 2*b*(a*e*g - c*e*f*x + c*d*(f - g*x)) + 4*c*(-(a*d*g) + c*d*f*x + a*e*(f + g*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (e*(-(e*f) + d*g)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)","A",1
882,1,162,155,0.2555672,"\int \frac{1}{(d+e x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{\frac{e^2 \left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{\sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}}{4 a c-b^2}","\frac{e^2 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{\left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}",1,"((-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/((c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]) + ((b^2 - 4*a*c)*e^2*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(-b^2 + 4*a*c)","A",1
883,1,317,352,1.2239743,"\int \frac{1}{(d+e x) (f+g x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{-\frac{2 e \left(-2 c (a e+c d x)+b^2 e+b c (e x-d)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(e (b d-a e)-c d^2\right)}+\frac{2 g \left(-2 c (a g+c f x)+b^2 g+b c (g x-f)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(g (b f-a g)-c f^2\right)}+\frac{e^3 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{\left(e (a e-b d)+c d^2\right)^{3/2}}-\frac{g^3 \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{3/2}}}{e f-d g}","-\frac{2 e \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (e f-d g) \left(a e^2-b d e+c d^2\right)}+\frac{2 g \left(2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (e f-d g) \left(a g^2-b f g+c f^2\right)}+\frac{e^3 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g) \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{g^3 \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{(e f-d g) \left(a g^2-b f g+c f^2\right)^{3/2}}",1,"((-2*e*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(c*f^2) + g*(b*f - a*g))*Sqrt[a + x*(b + c*x)]) + (e^3*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) - (g^3*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2))/(e*f - d*g)","A",1
884,1,623,642,5.0843543,"\int \frac{1}{(d+e x) (f+g x)^2 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^2*(a + b*x + c*x^2)^(3/2)),x]","\frac{g^2 \left(\frac{3 g (b g-2 c f) \tanh ^{-1}\left(\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{\left(g (a g-b f)+c f^2\right)^{5/2}}-\frac{2 \sqrt{a+x (b+c x)} \left(-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right)}{\left(b^2-4 a c\right) (f+g x) \left(g (a g-b f)+c f^2\right)^2}\right)}{2 (d g-e f)}-\frac{2 e^2 \left(-2 c (a e+c d x)+b^2 e+b c (e x-d)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} (e f-d g)^2 \left(e (b d-a e)-c d^2\right)}+\frac{2 e g \left(-2 c (a g+c f x)+b^2 g+b c (g x-f)\right)}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} (e f-d g)^2 \left(g (b f-a g)-c f^2\right)}-\frac{2 g \left(-2 c (a g+c f x)+b^2 g+b c (g x-f)\right)}{\left(b^2-4 a c\right) (f+g x) \sqrt{a+x (b+c x)} (d g-e f) \left(g (b f-a g)-c f^2\right)}+\frac{e^4 \tanh ^{-1}\left(\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right)}{(e f-d g)^2 \left(e (a e-b d)+c d^2\right)^{3/2}}-\frac{e g^3 \tanh ^{-1}\left(\frac{-2 a g+b (f-g x)+2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right)}{(e f-d g)^2 \left(g (a g-b f)+c f^2\right)^{3/2}}","\frac{g^2 \sqrt{a+b x+c x^2} \left(-4 c g (2 a g+b f)+3 b^2 g^2+4 c^2 f^2\right)}{\left(b^2-4 a c\right) (f+g x) (e f-d g) \left(a g^2-b f g+c f^2\right)^2}-\frac{2 e^2 \left(2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (e f-d g)^2 \left(a e^2-b d e+c d^2\right)}+\frac{2 e g \left(2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (e f-d g)^2 \left(a g^2-b f g+c f^2\right)}+\frac{2 g \left(2 a c g+b^2 (-g)+c x (2 c f-b g)+b c f\right)}{\left(b^2-4 a c\right) (f+g x) \sqrt{a+b x+c x^2} (e f-d g) \left(a g^2-b f g+c f^2\right)}+\frac{e^4 \tanh ^{-1}\left(\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right)}{(e f-d g)^2 \left(a e^2-b d e+c d^2\right)^{3/2}}-\frac{e g^3 \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{(e f-d g)^2 \left(a g^2-b f g+c f^2\right)^{3/2}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left(\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right)}{2 (e f-d g) \left(a g^2-b f g+c f^2\right)^{5/2}}",1,"(-2*e^2*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)]) + (2*e*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(e*f - d*g)^2*(-(c*f^2) + g*(b*f - a*g))*Sqrt[a + x*(b + c*x)]) - (2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)*(-(c*f^2) + g*(b*f - a*g))*(f + g*x)*Sqrt[a + x*(b + c*x)]) + (g^2*((-2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*(c*f^2 + g*(-(b*f) + a*g))^2*(f + g*x)) + (3*g*(-2*c*f + b*g)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(5/2)))/(2*(-(e*f) + d*g)) + (e^4*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/((c*d^2 + e*(-(b*d) + a*e))^(3/2)*(e*f - d*g)^2) - (e*g^3*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/((e*f - d*g)^2*(c*f^2 + g*(-(b*f) + a*g))^(3/2))","A",1
885,1,1013,1064,5.7384913,"\int \frac{1}{(d+e x) (f+g x)^3 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)),x]","-\frac{\tanh ^{-1}\left(\frac{-2 a e+2 c d x+b (d-e x)}{2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}}\right) e^5}{\left(c d^2+e (a e-b d)\right)^{3/2} (d g-e f)^3}-\frac{2 \left(e b^2+c (e x-d) b-2 c (a e+c d x)\right) e^3}{\left(b^2-4 a c\right) \left(e (b d-a e)-c d^2\right) (e f-d g)^3 \sqrt{a+x (b+c x)}}-\frac{2 g \left(g b^2+c (g x-f) b-2 c (a g+c f x)\right) e^2}{\left(b^2-4 a c\right) (d g-e f)^3 \left(g (b f-a g)-c f^2\right) \sqrt{a+x (b+c x)}}-\frac{g^3 \tanh ^{-1}\left(\frac{-2 a g+2 c f x+b (f-g x)}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right) e^2}{(e f-d g)^3 \left(c f^2+g (a g-b f)\right)^{3/2}}+\frac{2 g \left(g b^2+c (g x-f) b-2 c (a g+c f x)\right) e}{\left(b^2-4 a c\right) (e f-d g)^2 \left(g (b f-a g)-c f^2\right) (f+g x) \sqrt{a+x (b+c x)}}+\frac{g^2 \left(\frac{2 \sqrt{a+x (b+c x)} \left(4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right)}{\left(b^2-4 a c\right) \left(c f^2+g (a g-b f)\right)^2 (f+g x)}+\frac{3 g (2 c f-b g) \tanh ^{-1}\left(\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)}{\left(c f^2+g (a g-b f)\right)^{5/2}}\right) e}{2 (e f-d g)^2}-\frac{2 g \left(g b^2+c (g x-f) b-2 c (a g+c f x)\right)}{\left(b^2-4 a c\right) (d g-e f) \left(g (b f-a g)-c f^2\right) (f+g x)^2 \sqrt{a+x (b+c x)}}-\frac{g^2 \left(\frac{4 \sqrt{a+x (b+c x)} \left(8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right)}{(f+g x)^2}+\frac{3 \left(b^2-4 a c\right) g \left(16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right) \tanh ^{-1}\left(\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right)}{\left(c f^2+g (a g-b f)\right)^{3/2}}+\frac{2 (2 c f-b g) \left(8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right) \sqrt{a+x (b+c x)}}{\left(c f^2+g (a g-b f)\right) (f+g x)}\right)}{8 \left(b^2-4 a c\right) (d g-e f) \left(c f^2+g (a g-b f)\right)^2}","\frac{\tanh ^{-1}\left(\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right) e^5}{\left(c d^2-b e d+a e^2\right)^{3/2} (e f-d g)^3}-\frac{2 \left(-e b^2+c d b+2 a c e+c (2 c d-b e) x\right) e^3}{\left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) (e f-d g)^3 \sqrt{c x^2+b x+a}}-\frac{g^3 \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e^2}{(e f-d g)^3 \left(c f^2-b g f+a g^2\right)^{3/2}}+\frac{2 g \left(-g b^2+c f b+2 a c g+c (2 c f-b g) x\right) e^2}{\left(b^2-4 a c\right) (e f-d g)^3 \left(c f^2-b g f+a g^2\right) \sqrt{c x^2+b x+a}}-\frac{3 g^3 (2 c f-b g) \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right) e}{2 (e f-d g)^2 \left(c f^2-b g f+a g^2\right)^{5/2}}+\frac{g^2 \left(4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right) \sqrt{c x^2+b x+a} e}{\left(b^2-4 a c\right) (e f-d g)^2 \left(c f^2-b g f+a g^2\right)^2 (f+g x)}+\frac{2 g \left(-g b^2+c f b+2 a c g+c (2 c f-b g) x\right) e}{\left(b^2-4 a c\right) (e f-d g)^2 \left(c f^2-b g f+a g^2\right) (f+g x) \sqrt{c x^2+b x+a}}-\frac{3 g^3 \left(16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right) \tanh ^{-1}\left(\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right)}{8 (e f-d g) \left(c f^2-b g f+a g^2\right)^{7/2}}+\frac{g^2 (2 c f-b g) \left(8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right) \sqrt{c x^2+b x+a}}{4 \left(b^2-4 a c\right) (e f-d g) \left(c f^2-b g f+a g^2\right)^3 (f+g x)}+\frac{g^2 \left(8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right) \sqrt{c x^2+b x+a}}{2 \left(b^2-4 a c\right) (e f-d g) \left(c f^2-b g f+a g^2\right)^2 (f+g x)^2}+\frac{2 g \left(-g b^2+c f b+2 a c g+c (2 c f-b g) x\right)}{\left(b^2-4 a c\right) (e f-d g) \left(c f^2-b g f+a g^2\right) (f+g x)^2 \sqrt{c x^2+b x+a}}",1,"(-2*e^3*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(e*f - d*g)^3*Sqrt[a + x*(b + c*x)]) - (2*e^2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)^3*(-(c*f^2) + g*(b*f - a*g))*Sqrt[a + x*(b + c*x)]) - (2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)*(-(c*f^2) + g*(b*f - a*g))*(f + g*x)^2*Sqrt[a + x*(b + c*x)]) + (2*e*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(e*f - d*g)^2*(-(c*f^2) + g*(b*f - a*g))*(f + g*x)*Sqrt[a + x*(b + c*x)]) + (e*g^2*((2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*(c*f^2 + g*(-(b*f) + a*g))^2*(f + g*x)) + (3*g*(2*c*f - b*g)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(5/2)))/(2*(e*f - d*g)^2) - (g^2*((4*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*Sqrt[a + x*(b + c*x)])/(f + g*x)^2 + (2*(2*c*f - b*g)*(8*c^2*f^2 + 15*b^2*g^2 - 4*c*g*(2*b*f + 13*a*g))*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)) + (3*(b^2 - 4*a*c)*g*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2)))/(8*(b^2 - 4*a*c)*(-(e*f) + d*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - (e^5*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/((c*d^2 + e*(-(b*d) + a*e))^(3/2)*(-(e*f) + d*g)^3) - (e^2*g^3*ArcTanh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/((e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g))^(3/2))","A",1
886,1,26600,1551,17.7498455,"\int (d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","\frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^4}{11 e}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(2 f^2 \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right) c^5-g \left(b f \left(56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right)-18 a g \left(6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right)\right) c^4-g^2 \left(\left(37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right) b^2-9 a e g \left(15 e^2 f^2-110 d e g f-319 d^2 g^2\right) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right) c^3+b e g^3 \left(-\left(\left(37 e^2 f^2-264 d e g f-792 d^2 g^2\right) b^2\right)+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3465 c^5 g^5 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c f^2-b g f+a g^2\right) \left(-2 f \left(64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right) c^4-g \left(6 a e g \left(2 e^2 f^2-33 d e g f+165 d^2 g^2\right)+b \left(8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right)\right) c^3+3 e g^2 \left(3 \left(e^2 f^2-11 d e g f+44 d^2 g^2\right) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3465 c^5 g^5 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt{c x^2+b x+a}}{99 c g^4}-\frac{2 e \left(\left(29 e^2 f^2-96 d e g f+81 d^2 g^2\right) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{693 c^2 g^4}+\frac{2 \left(\left(233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right) c^3-e g \left(2 a e g (74 e f-231 d g)-3 b \left(24 e^2 f^2-88 d e g f+99 d^2 g^2\right)\right) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{3465 c^3 g^4}-\frac{2 \left(\left(187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right) c^4-e g \left(6 a e g \left(2 e^2 f^2-33 d e g f+165 d^2 g^2\right)+b \left(8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right)\right) c^3+3 e^2 g^2 \left(3 \left(e^2 f^2-11 d e g f+44 d^2 g^2\right) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{3465 c^4 e g^4}",1,"Result too large to show","C",0
887,1,15781,1015,15.5106163,"\int (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","\frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(f^2 \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right) c^4+g \left(3 a g \left(3 e^2 f^2-16 d e g f-21 d^2 g^2\right)-b f \left(4 e^2 f^2-15 d e g f+21 d^2 g^2\right)\right) c^3+3 g^2 \left(-\left(\left(e^2 f^2-5 d e g f-7 d^2 g^2\right) b^2\right)+a e g (5 e f+29 d g) b+7 a^2 e^2 g^2\right) c^2-4 b^2 e g^3 (b e f+6 b d g+9 a e g) c+8 b^4 e^2 g^4\right) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{315 c^4 g^4 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c f^2-b g f+a g^2\right) \left(-2 f \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right) c^3-3 g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^2 g^3\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{315 c^4 g^4 \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{2 e (c e f-3 c d g+b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^3}-\frac{4 \left(\left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right) c^2+e g (4 b e f-9 b d g-7 a e g) c+3 b^2 e^2 g^2\right) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^3}+\frac{2 \left(\left(19 e^3 f^3-57 d e^2 g f^2+63 d^2 e g^2 f-35 d^3 g^3\right) c^3-3 e g^2 (2 a e (e f-10 d g)+b d (2 e f-7 d g)) c^2+3 b e^2 g^2 (b e f-8 b d g-9 a e g) c+8 b^3 e^3 g^3\right) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 e g^3}",1,"Result too large to show","C",0
888,1,8432,652,13.9490348,"\int (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left(-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right)}{105 c^2 g^2}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^3 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(\left(-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^3 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e \sqrt{f+g x} \left(a+b x+c x^2\right)^{3/2}}{7 c}",1,"Result too large to show","C",1
889,1,697,513,10.6856453,"\int \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx","Integrate[Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]","\frac{-\frac{4 g^2 (a+x (b+c x)) \left(-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right)}{\sqrt{f+g x}}+\frac{i (f+g x) \sqrt{1-\frac{2 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right)}} \sqrt{\frac{4 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f\right)}+2} \left(\left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right) \left(-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)+\left(c \left(a g^2 \left(3 \sqrt{g^2 \left(b^2-4 a c\right)}+8 c f\right)-c f^2 \sqrt{g^2 \left(b^2-4 a c\right)}\right)-b^2 g^2 \left(\sqrt{g^2 \left(b^2-4 a c\right)}+2 c f\right)+b c g \left(f \sqrt{g^2 \left(b^2-4 a c\right)}-4 a g^2\right)+b^3 g^3\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{\sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f}}}+2 c g^2 \sqrt{f+g x} (a+x (b+c x)) (b g+c (f+3 g x))}{15 c^2 g^3 \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c f-b g) \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^2 g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^2 g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{5 g}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (2 c f-b g)}{15 c g}",1,"((-4*g^2*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*(a + x*(b + c*x)))/Sqrt[f + g*x] + 2*c*g^2*Sqrt[f + g*x]*(a + x*(b + c*x))*(b*g + c*(f + 3*g*x)) + (I*(f + g*x)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[2 + (4*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (b^3*g^3 - b^2*g^2*(2*c*f + Sqrt[(b^2 - 4*a*c)*g^2]) + b*c*g*(-4*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2]) + c*(-(c*f^2*Sqrt[(b^2 - 4*a*c)*g^2]) + a*g^2*(8*c*f + 3*Sqrt[(b^2 - 4*a*c)*g^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/(15*c^2*g^3*Sqrt[a + x*(b + c*x)])","C",1
890,1,35245,764,15.1703164,"\int \frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{d+e x} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(d + e*x),x]","\text{Result too large to show}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(e g (2 a e g-3 b d g+b e f)+c \left(3 d^2 g^2-e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{3 c e^3 g \sqrt{f+g x} \sqrt{a+x (b+c x)}}-\frac{\sqrt{2} \left(a e^2-b d e+c d^2\right) \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} e^3 \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (b e g-3 c d g+c e f) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c e^2 g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 e}",1,"Result too large to show","C",0
891,1,16573,743,13.8441337,"\int \frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(d + e*x)^2,x]","\text{Result too large to show}","\frac{\sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f)) \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{2} \sqrt{c} e^3 \sqrt{a+b x+c x^2} (e f-d g)}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} (2 b e g-c (3 d g+e f)) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{c e^3 \sqrt{f+g x} \sqrt{a+x (b+c x)}}+\frac{3 \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\sqrt{2} e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{e (d+e x)}",1,"Result too large to show","C",0
892,1,33765,1034,16.5253401,"\int \frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx","Integrate[(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(d + e*x)^3,x]","\text{Result too large to show}","-\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 \sqrt{2} e^2 \left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 e \left(c d^2-b e d+a e^2\right) (e f-d g) (d+e x)}-\frac{\sqrt{b^2-4 a c} (e (b e f+4 b d g-5 a e g)-c d (2 e f+3 d g)) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{2 \sqrt{2} e^3 \left(c d^2+e (a e-b d)\right) \sqrt{f+g x} \sqrt{a+x (b+c x)}}+\frac{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g} \left(b^2 f^2 e^4+a^2 g^2 e^4-2 a c \left(2 e^2 f^2-6 d e g f+3 d^2 g^2\right) e^2-2 b g \left(a f e^3+c d^2 (3 e f-2 d g)\right) e+c^2 d^3 g (4 e f-3 d g)\right) \sqrt{\frac{g \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}} \sqrt{\frac{g \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \Pi \left(\frac{2 c e f-b e g+\sqrt{b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g}}\right)|\frac{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{4 \sqrt{2} \sqrt{c} e^3 \left(c d^2+e (a e-b d)\right) (e f-d g)^2 \sqrt{a+x (b+c x)}}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a}}{2 e (d+e x)^2}",1,"Result too large to show","C",0
893,1,17771,1098,15.8828366,"\int \frac{(d+e x)^3 \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^3*Sqrt[a + b*x + c*x^2])/Sqrt[f + g*x],x]","\text{Result too large to show}","\frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} (d+e x)^3}{9 g}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(-2 f \left(64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right) c^4-g \left(6 a e g \left(10 e^2 f^2-39 d e g f+63 d^2 g^2\right)-b \left(40 e^3 f^3-144 d e^2 g f^2+189 d^2 e g^2 f-105 d^3 g^3\right)\right) c^3+3 e g^2 \left(\left(7 e^2 f^2-27 d e g f+42 d^2 g^2\right) b^2-a e g (19 e f-87 d g) b+14 a^2 e^2 g^2\right) c^2+8 b^2 e^2 g^3 (2 b e f-9 b d g-9 a e g) c+16 b^4 e^3 g^4\right) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{315 c^4 g^5 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c f^2-b g f+a g^2\right) \left(2 \left(64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right) c^3-3 e g \left(6 a e g (2 e f-5 d g)-b \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right)\right) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{315 c^4 g^5 \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{2 e^2 (8 c e f-6 c d g-b e g) (f+g x)^{5/2} \sqrt{c x^2+b x+a}}{63 c g^4}-\frac{2 e \left(-2 \left(64 e^2 f^2-111 d e g f+42 d^2 g^2\right) c^2+e g (17 b e f-27 b d g-14 a e g) c+6 b^2 e^2 g^2\right) (f+g x)^{3/2} \sqrt{c x^2+b x+a}}{315 c^2 g^4}+\frac{2 \left(-\left(\left(152 e^3 f^3-408 d e^2 g f^2+336 d^2 e g^2 f-70 d^3 g^3\right) c^3\right)-3 e g \left(6 a e g (2 e f-5 d g)-b \left(8 e^2 f^2-24 d e g f+21 d^2 g^2\right)\right) c^2+3 b e^2 g^2 (5 b e f-12 b d g-9 a e g) c+8 b^3 e^3 g^3\right) \sqrt{f+g x} \sqrt{c x^2+b x+a}}{315 c^3 g^4}",1,"Result too large to show","C",0
894,1,10030,755,14.1403215,"\int \frac{(d+e x)^2 \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)^2*Sqrt[a + b*x + c*x^2])/Sqrt[f + g*x],x]","\text{Result too large to show}","-\frac{4 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left(c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2-\left(c^2 \left(10 d^2 g^2-34 d e f g+21 e^2 f^2\right)\right)\right)}{105 c^2 g^3}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(c e g (-5 a e g-7 b d g+4 b e f)+2 b^2 e^2 g^2+c^2 \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^3 g^4 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c^2 g \left(2 a e g (13 e f-42 d g)-b \left(35 d^2 g^2-42 d e f g+16 e^2 f^2\right)\right)+b c e g^2 (-29 a e g-28 b d g+9 b e f)+8 b^3 e^2 g^3-2 c^3 f \left(35 d^2 g^2-56 d e f g+24 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^3 g^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 e (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-b e g-4 c d g+6 c e f)}{35 c g^3}+\frac{2 (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 g}",1,"Result too large to show","C",1
895,1,911,519,10.5131936,"\int \frac{(d+e x) \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[((d + e*x)*Sqrt[a + b*x + c*x^2])/Sqrt[f + g*x],x]","\frac{2 \sqrt{a+x (b+c x)} \left(\left(2 f (4 e f-5 d g) c^2+g (-3 b e f+5 b d g+6 a e g) c-2 b^2 e g^2\right) \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)+\frac{i \sqrt{1-\frac{2 \left(c f^2+g (a g-b f)\right)}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 \left(c f^2+g (a g-b f)\right)}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}+1} \left(\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \left(2 f (5 d g-4 e f) c^2+g (3 b e f-5 b d g-6 a e g) c+2 b^2 e g^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)+\left(2 b^3 e g^3-b^2 \left(-c f e+2 \sqrt{\left(b^2-4 a c\right) g^2} e+5 c d g\right) g^2+b c \left(\sqrt{\left(b^2-4 a c\right) g^2} (5 d g-3 e f)-8 a e g^2\right) g+2 c \left(a \left(-2 c f e+3 \sqrt{\left(b^2-4 a c\right) g^2} e+10 c d g\right) g^2+c f \sqrt{\left(b^2-4 a c\right) g^2} (4 e f-5 d g)\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{f+g x}}\right) (f+g x)^{3/2}}{15 c^2 g^4 \sqrt{c x^2+b x+a} \sqrt{\frac{(f+g x)^2 \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)}{g^2}}}+\left(\frac{2 (-4 c e f+5 c d g+b e g)}{15 c g^2}+\frac{2 e x}{5 g}\right) \sqrt{a+x (b+c x)} \sqrt{f+g x}","-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} (b e g-10 c d g+8 c e f) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^2 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2}",1,"((2*(-4*c*e*f + 5*c*d*g + b*e*g))/(15*c*g^2) + (2*e*x)/(5*g))*Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)] + (2*(f + g*x)^(3/2)*Sqrt[a + x*(b + c*x)]*((-2*b^2*e*g^2 + 2*c^2*f*(4*e*f - 5*d*g) + c*g*(-3*b*e*f + 5*b*d*g + 6*a*e*g))*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)) + ((I/2)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(2*b^2*e*g^2 + 2*c^2*f*(-4*e*f + 5*d*g) + c*g*(3*b*e*f - 5*b*d*g - 6*a*e*g))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (2*b^3*e*g^3 - b^2*g^2*(-(c*e*f) + 5*c*d*g + 2*e*Sqrt[(b^2 - 4*a*c)*g^2]) + b*c*g*(-8*a*e*g^2 + Sqrt[(b^2 - 4*a*c)*g^2]*(-3*e*f + 5*d*g)) + 2*c*(c*f*Sqrt[(b^2 - 4*a*c)*g^2]*(4*e*f - 5*d*g) + a*g^2*(-2*c*e*f + 10*c*d*g + 3*e*Sqrt[(b^2 - 4*a*c)*g^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/(Sqrt[2]*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(15*c^2*g^4*Sqrt[a + b*x + c*x^2]*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2])","C",1
896,1,936,444,8.3104289,"\int \frac{\sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/Sqrt[f + g*x],x]","\frac{\sqrt{f+g x} \left(4 (a+x (b+c x)) g^2+\frac{(f+g x) \left(\frac{4 (b g-2 c f) \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} (a+x (b+c x)) g^2}{(f+g x)^2}+\frac{i \sqrt{2} (2 c f-b g) \left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \sqrt{\frac{-2 a g^2+2 c f x g+\sqrt{\left(b^2-4 a c\right) g^2} x g+b (f-g x) g+f \sqrt{\left(b^2-4 a c\right) g^2}}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+\sqrt{\left(b^2-4 a c\right) g^2} x g+b (g x-f) g+f \sqrt{\left(b^2-4 a c\right) g^2}}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}-\frac{i \sqrt{2} \left(b^2 g^2-4 a c g^2-b \sqrt{\left(b^2-4 a c\right) g^2} g+2 c f \sqrt{\left(b^2-4 a c\right) g^2}\right) \sqrt{\frac{-2 a g^2+2 c f x g+\sqrt{\left(b^2-4 a c\right) g^2} x g+b (f-g x) g+f \sqrt{\left(b^2-4 a c\right) g^2}}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+\sqrt{\left(b^2-4 a c\right) g^2} x g+b (g x-f) g+f \sqrt{\left(b^2-4 a c\right) g^2}}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}\right)}{c \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}\right)}{6 g^3 \sqrt{a+x (b+c x)}}","\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 c f-b g) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 g}",1,"(Sqrt[f + g*x]*(4*g^2*(a + x*(b + c*x)) + ((f + g*x)*((4*g^2*(-2*c*f + b*g)*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*(a + x*(b + c*x)))/(f + g*x)^2 + (I*Sqrt[2]*(2*c*f - b*g)*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*Sqrt[(-2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] + 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(f - g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[(2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] - 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(-f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x] - (I*Sqrt[2]*(b^2*g^2 - 4*a*c*g^2 + 2*c*f*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*Sqrt[(b^2 - 4*a*c)*g^2])*Sqrt[(-2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] + 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(f - g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[(2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] - 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(-f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x]))/(c*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])))/(6*g^3*Sqrt[a + x*(b + c*x)])","C",1
897,1,16471,700,13.9182838,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x) \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)*Sqrt[f + g*x]),x]","\text{Result too large to show}","-\frac{\sqrt{2} \left(a e^2-b d e+c d^2\right) \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} e^2 \sqrt{a+b x+c x^2} (e f-d g)}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} (-b e g+c d g+c e f) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c e^2 g \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{e g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"Result too large to show","C",1
898,1,6911,736,13.3888306,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^2 \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)^2*Sqrt[f + g*x]),x]","\text{Result too large to show}","-\frac{\sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(e^2 (b f-a g)-c d (2 e f-d g)\right) \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{2} \sqrt{c} e^2 \sqrt{a+b x+c x^2} (e f-d g)^2}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\sqrt{2} e \sqrt{a+b x+c x^2} (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{e^2 \sqrt{f+g x} \sqrt{a+x (b+c x)}}-\frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(d+e x) (e f-d g)}",1,"Result too large to show","C",1
899,1,36616,1049,16.6779235,"\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3 \sqrt{f+g x}} \, dx","Integrate[Sqrt[a + b*x + c*x^2]/((d + e*x)^3*Sqrt[f + g*x]),x]","\text{Result too large to show}","-\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \sqrt{2} e \left(c d^2-b e d+a e^2\right) (e f-d g)^2 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \left(c d^2-b e d+a e^2\right) (e f-d g)^2 (d+e x)}-\frac{\sqrt{b^2-4 a c} \left((b f-a g) e^2+c d (d g-2 e f)\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{2 \sqrt{2} e^2 \left(c d^2+e (a e-b d)\right) (e f-d g) \sqrt{f+g x} \sqrt{a+x (b+c x)}}-\frac{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g} \left(3 a^2 g^2 e^4+b^2 f (4 d g-e f) e^3+2 a c \left(2 e^2 f^2-2 d e g f+3 d^2 g^2\right) e^2-2 b g \left(3 c f d^2+a e (e f+2 d g)\right) e^2+c^2 d^3 g (4 e f-d g)\right) \sqrt{\frac{g \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}} \sqrt{\frac{g \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \Pi \left(\frac{2 c e f-b e g+\sqrt{b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g}}\right)|\frac{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{4 \sqrt{2} \sqrt{c} e^2 \left(c d^2+e (a e-b d)\right) (e f-d g)^3 \sqrt{a+x (b+c x)}}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a}}{2 (e f-d g) (d+e x)^2}",1,"Result too large to show","C",0
900,1,10649,774,14.6741052,"\int \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)^3*Sqrt[f + g*x])/Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left(105 d^2 g^2-42 d e f g+8 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left(c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2-\left(c^2 \left(-90 d^2 g^2+12 d e f g+7 e^2 f^2\right)\right)\right)}{105 c^3 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(c^2 e g \left(a e g (189 d g+19 e f)-b \left(-210 d^2 g^2-63 d e f g+9 e^2 f^2\right)\right)-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3-\left(c^3 \left(105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right)\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}",1,"Result too large to show","C",0
901,1,1002,567,11.4381375,"\int \frac{(d+e x)^2 \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)^2*Sqrt[f + g*x])/Sqrt[a + b*x + c*x^2],x]","\frac{\left(\frac{2 e^2 x}{5 c}-\frac{2 e (-c e f-10 c d g+4 b e g)}{15 c^2 g}\right) \sqrt{f+g x} \left(c x^2+b x+a\right)}{\sqrt{a+x (b+c x)}}-\frac{2 (f+g x)^{3/2} \sqrt{c x^2+b x+a} \left(\left(\left(2 e^2 f^2-10 d e g f-15 d^2 g^2\right) c^2+e g (3 b e f+20 b d g+9 a e g) c-8 b^2 e^2 g^2\right) \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)+\frac{i \sqrt{1-\frac{2 \left(c f^2+g (a g-b f)\right)}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 \left(c f^2+g (a g-b f)\right)}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}+1} \left(\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \left(\left(-2 e^2 f^2+10 d e g f+15 d^2 g^2\right) c^2-e g (3 b e f+20 b d g+9 a e g) c+8 b^2 e^2 g^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)+\left(-30 d^2 f g^2 c^3-\left(-15 b d (2 e f+d g) g^2-2 a e (7 e f+10 d g) g^2+\sqrt{\left(b^2-4 a c\right) g^2} \left(-2 e^2 f^2+10 d e g f+15 d^2 g^2\right)\right) c^2+e g \left(-g (11 e f+20 d g) b^2-17 a e g^2 b+\sqrt{\left(b^2-4 a c\right) g^2} (3 e f+20 d g) b+9 a e g \sqrt{\left(b^2-4 a c\right) g^2}\right) c+8 b^2 e^2 g^2 \left(b g-\sqrt{\left(b^2-4 a c\right) g^2}\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{f+g x}}\right)}{15 c^3 g^3 \sqrt{a+x (b+c x)} \sqrt{\frac{(f+g x)^2 \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)}{g^2}}}","\frac{4 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} (2 b e g-5 c d g+c e f) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^3 g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e g (9 a e g+20 b d g+3 b e f)+8 b^2 e^2 g^2-\left(c^2 \left(-15 d^2 g^2-10 d e f g+2 e^2 f^2\right)\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^3 g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} (-4 b e g+7 c d g+c e f)}{15 c^2 g}+\frac{2 e (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{5 c}",1,"(((-2*e*(-(c*e*f) - 10*c*d*g + 4*b*e*g))/(15*c^2*g) + (2*e^2*x)/(5*c))*Sqrt[f + g*x]*(a + b*x + c*x^2))/Sqrt[a + x*(b + c*x)] - (2*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((-8*b^2*e^2*g^2 + c*e*g*(3*b*e*f + 20*b*d*g + 9*a*e*g) + c^2*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)) + ((I/2)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(8*b^2*e^2*g^2 - c*e*g*(3*b*e*f + 20*b*d*g + 9*a*e*g) + c^2*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*g^2))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (-30*c^3*d^2*f*g^2 + 8*b^2*e^2*g^2*(b*g - Sqrt[(b^2 - 4*a*c)*g^2]) + c*e*g*(-17*a*b*e*g^2 + 9*a*e*g*Sqrt[(b^2 - 4*a*c)*g^2] + b*Sqrt[(b^2 - 4*a*c)*g^2]*(3*e*f + 20*d*g) - b^2*g*(11*e*f + 20*d*g)) - c^2*(-15*b*d*g^2*(2*e*f + d*g) - 2*a*e*g^2*(7*e*f + 10*d*g) + Sqrt[(b^2 - 4*a*c)*g^2]*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*g^2)))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/(Sqrt[2]*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(15*c^3*g^3*Sqrt[a + x*(b + c*x)]*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2])","C",1
902,1,638,452,7.1425406,"\int \frac{(d+e x) \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)*Sqrt[f + g*x])/Sqrt[a + b*x + c*x^2],x]","\frac{2 \sqrt{f+g x} \left(c e (a+x (b+c x))+\frac{(f+g x) \left(\frac{g^2 (a+x (b+c x)) (-2 b e g+3 c d g+c e f)}{(f+g x)^2}+\frac{i \sqrt{1-\frac{2 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right)}} \sqrt{\frac{2 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f\right)}+1} \left(\left(c \left(\sqrt{g^2 \left(b^2-4 a c\right)} (3 d g+e f)-2 a e g^2-3 b g (d g+e f)\right)+2 b e g \left(b g-\sqrt{g^2 \left(b^2-4 a c\right)}\right)+6 c^2 d f g\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)+\left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right) (2 b e g-c (3 d g+e f)) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{2 \sqrt{2} \sqrt{f+g x} \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f}}}\right)}{g^2}\right)}{3 c^2 \sqrt{a+x (b+c x)}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (-2 b e g+3 c d g+c e f) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c^2 g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c^2 g \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 c}",1,"(2*Sqrt[f + g*x]*(c*e*(a + x*(b + c*x)) + ((f + g*x)*((g^2*(c*e*f + 3*c*d*g - 2*b*e*g)*(a + x*(b + c*x)))/(f + g*x)^2 + ((I/2)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(2*b*e*g - c*(e*f + 3*d*g))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (6*c^2*d*f*g + 2*b*e*g*(b*g - Sqrt[(b^2 - 4*a*c)*g^2]) + c*(-2*a*e*g^2 - 3*b*g*(e*f + d*g) + Sqrt[(b^2 - 4*a*c)*g^2]*(e*f + 3*d*g)))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/(Sqrt[2]*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/g^2))/(3*c^2*Sqrt[a + x*(b + c*x)])","C",1
903,1,365,188,0.7774644,"\int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2],x]","\frac{i \left(g \left(\sqrt{b^2-4 a c}-b\right)+2 c f\right) \sqrt{\frac{g \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}} \sqrt{1-\frac{2 c (f+g x)}{g \left(\sqrt{b^2-4 a c}-b\right)+2 c f}} \left(E\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \sqrt{f+g x}\right)|\frac{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}\right)-F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \sqrt{f+g x}\right)|\frac{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}\right)\right)}{\sqrt{2} c g \sqrt{a+x (b+c x)} \sqrt{\frac{c}{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(I*(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]))/(Sqrt[2]*c*g*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + x*(b + c*x)])","C",1
904,1,379,467,1.623275,"\int \frac{\sqrt{f+g x}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","-\frac{i \sqrt{2} \sqrt{\frac{g \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}} \sqrt{1-\frac{2 c (f+g x)}{g \left(\sqrt{b^2-4 a c}-b\right)+2 c f}} \left(F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \sqrt{f+g x}\right)|\frac{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}\right)-\Pi \left(\frac{e \left(2 c f-\left(b+\sqrt{b^2-4 a c}\right) g\right)}{2 c (e f-d g)};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \sqrt{f+g x}\right)|\frac{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}\right)\right)}{e \sqrt{a+x (b+c x)} \sqrt{\frac{c}{g \left(\sqrt{b^2-4 a c}+b\right)-2 c f}}}","\frac{2 \sqrt{2} g \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c e \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} e \sqrt{a+b x+c x^2}}",1,"((-I)*Sqrt[2]*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)] - EllipticPi[(e*(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g))/(2*c*(e*f - d*g)), I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]))/(e*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + x*(b + c*x)])","C",1
905,1,18563,994,13.733761,"\int \frac{\sqrt{f+g x}}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^2*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e}{\left(c d^2-b e d+a e^2\right) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\sqrt{2} \left(c d^2-b e d+a e^2\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\left(c d^2-b e d+a e^2\right) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} d g \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\left(c d^2-b e d+a e^2\right) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}+\frac{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} \left(e^2 (b f-a g)-c d (2 e f-d g)\right) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{c x^2+b x+a} e}",1,"Result too large to show","C",0
906,1,36634,1786,17.6646176,"\int \frac{\sqrt{f+g x}}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[f + g*x]/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}{4 \left(c d^2-b e d+a e^2\right)^2 (e f-d g) (d+e x)}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e}{2 \left(c d^2-b e d+a e^2\right) (d+e x)^2}+\frac{\sqrt{b^2-4 a c} (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{4 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}-\frac{\sqrt{b^2-4 a c} f (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{2 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{\sqrt{b^2-4 a c} g \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\sqrt{2} \left(c d^2-b e d+a e^2\right) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}+\frac{\sqrt{b^2-4 a c} d g (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{2 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a} e}+\frac{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} (c e f-3 c d g+b e g) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{c x^2+b x+a} e}-\frac{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{4 \sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right)^2 (e f-d g)^2 \sqrt{c x^2+b x+a} e}",1,"Result too large to show","C",0
907,1,1385,675,4.5496481,"\int \frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)^(3/2)/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\frac{\sqrt{2} \sqrt{\frac{c (f+g x)}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}} \left(-\frac{4 \sqrt{b^2-4 a c} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \Pi \left(\frac{2 \sqrt{b^2-4 a c} e}{2 c d-b e+\sqrt{b^2-4 a c} e};\sin ^{-1}\left(\frac{\sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{2 c f-b g+\sqrt{b^2-4 a c} g}\right) f^2}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}+\frac{2 g \left(b+2 c x-\sqrt{b^2-4 a c}\right) \sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{2 c f-b g+\sqrt{b^2-4 a c} g}\right) f}{c e \sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}+\frac{8 \sqrt{b^2-4 a c} d g \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \Pi \left(\frac{2 \sqrt{b^2-4 a c} e}{2 c d-b e+\sqrt{b^2-4 a c} e};\sin ^{-1}\left(\frac{\sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{2 c f-b g+\sqrt{b^2-4 a c} g}\right) f}{e \left(2 c d+\left(\sqrt{b^2-4 a c}-b\right) e\right)}-\frac{d g^2 \left(b+2 c x-\sqrt{b^2-4 a c}\right) \sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{2 c f-b g+\sqrt{b^2-4 a c} g}\right)}{c e^2 \sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}+\frac{g \left(-b-2 c x+\sqrt{b^2-4 a c}\right) \sqrt{\frac{g \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \left(\left(\left(b+\sqrt{b^2-4 a c}\right) g-2 c f\right) E\left(\sin ^{-1}\left(\sqrt{2} \sqrt{\frac{c (f+g x)}{2 c f-b g+\sqrt{b^2-4 a c} g}}\right)|\frac{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)-\left(b+\sqrt{b^2-4 a c}\right) g F\left(\sin ^{-1}\left(\sqrt{2} \sqrt{\frac{c (f+g x)}{2 c f-b g+\sqrt{b^2-4 a c} g}}\right)|\frac{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)\right)}{2 c^2 e \sqrt{\frac{g \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}}}-\frac{4 \sqrt{b^2-4 a c} d^2 g^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \Pi \left(\frac{2 \sqrt{b^2-4 a c} e}{2 c d-b e+\sqrt{b^2-4 a c} e};\sin ^{-1}\left(\frac{\sqrt{\frac{-b-2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{2 c f-b g+\sqrt{b^2-4 a c} g}\right)}{e^2 \left(2 c d+\left(\sqrt{b^2-4 a c}-b\right) e\right)}\right)}{\sqrt{f+g x} \sqrt{a+x (b+c x)}}","\frac{2 \sqrt{2} g \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} (e f-d g) \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} e^2 \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} g \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c e \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"(Sqrt[2]*Sqrt[(c*(f + g*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]*((2*f*g*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]*EllipticF[ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(c*e*Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]) - (d*g^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]*EllipticF[ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(c*e^2*Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]) + (g*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*((-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)*EllipticE[ArcSin[Sqrt[2]*Sqrt[(c*(f + g*x))/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)]], (2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)] - (b + Sqrt[b^2 - 4*a*c])*g*EllipticF[ArcSin[Sqrt[2]*Sqrt[(c*(f + g*x))/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)]], (2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]))/(2*c^2*e*Sqrt[(g*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]) - (4*Sqrt[b^2 - 4*a*c]*f^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e) + (8*Sqrt[b^2 - 4*a*c]*d*f*g*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)) - (4*Sqrt[b^2 - 4*a*c]*d^2*g^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticPi[(2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e), ArcSin[Sqrt[(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g)])/(e^2*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))))/(Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)])","B",0
908,1,37137,1138,16.2536997,"\int \frac{(f+g x)^{5/2}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(f + g*x)^(5/2)/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","\frac{2 \sqrt{f+g x} \sqrt{c x^2+b x+a} g^2}{3 c e}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c f-b g) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) g}{3 c^2 e \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} (e f-d g) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) g}{c e^2 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (e f-d g)^2 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) g}{c e^3 \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c f^2-b g f+a g^2\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) g}{3 c^2 e \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} (e f-d g)^2 \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} e^3 \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
909,1,12746,631,13.6920133,"\int \frac{(d+e x)^3}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^3/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","\frac{\sqrt{2} e \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left(45 d^2 g^2-30 d e f g+8 e^2 f^2\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^3 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left(-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{15 c^3 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{8 e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (b e g-3 c d g+c e f)}{15 c^2 g^2}+\frac{2 e^2 (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{5 c g}",1,"Result too large to show","C",1
910,1,1080,479,12.7886326,"\int \frac{(d+e x)^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^2/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{f+g x} \left(c x^2+b x+a\right) e^2}{3 c g \sqrt{a+x (b+c x)}}+\frac{(f+g x)^{3/2} \sqrt{c x^2+b x+a} \left(-4 e (c e f-3 c d g+b e g) \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)+\frac{i \sqrt{2} e (c e f-3 c d g+b e g) \left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \sqrt{\frac{-\frac{2 a g^2}{f+g x}+b \left(\frac{2 f}{f+g x}-1\right) g-2 c f \left(\frac{f}{f+g x}-1\right)+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{\frac{\frac{2 a g^2}{f+g x}+2 c f \left(\frac{f}{f+g x}-1\right)+b \left(g-\frac{2 f g}{f+g x}\right)+\sqrt{\left(b^2-4 a c\right) g^2}}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}-\frac{i \sqrt{2} \left(b g \left(\sqrt{\left(b^2-4 a c\right) g^2}-b g\right) e^2+c \left(3 b d g^2+a e g^2+\sqrt{\left(b^2-4 a c\right) g^2} (e f-3 d g)\right) e-3 c^2 d^2 g^2\right) \sqrt{\frac{-\frac{2 a g^2}{f+g x}+b \left(\frac{2 f}{f+g x}-1\right) g-2 c f \left(\frac{f}{f+g x}-1\right)+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{\frac{\frac{2 a g^2}{f+g x}+2 c f \left(\frac{f}{f+g x}-1\right)+b \left(g-\frac{2 f g}{f+g x}\right)+\sqrt{\left(b^2-4 a c\right) g^2}}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}\right)}{3 c^2 g^3 \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{a+x (b+c x)} \sqrt{\frac{(f+g x)^2 \left(c \left(\frac{f}{f+g x}-1\right)^2+\frac{g \left(-\frac{f b}{f+g x}+b+\frac{a g}{f+g x}\right)}{f+g x}\right)}{g^2}}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \left(e^2 g (b f-a g)+c \left(3 d^2 g^2-6 d e f g+2 e^2 f^2\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c^2 g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (b e g-3 c d g+c e f) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 c^2 g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 e^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 c g}",1,"(2*e^2*Sqrt[f + g*x]*(a + b*x + c*x^2))/(3*c*g*Sqrt[a + x*(b + c*x)]) + ((f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2]*(-4*e*(c*e*f - 3*c*d*g + b*e*g)*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)) + (I*Sqrt[2]*e*(c*e*f - 3*c*d*g + b*e*g)*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*g^2] - (2*a*g^2)/(f + g*x) - 2*c*f*(-1 + f/(f + g*x)) + b*g*(-1 + (2*f)/(f + g*x)))/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*g^2] + (2*a*g^2)/(f + g*x) + 2*c*f*(-1 + f/(f + g*x)) + b*(g - (2*f*g)/(f + g*x)))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x] - (I*Sqrt[2]*(-3*c^2*d^2*g^2 + b*e^2*g*(-(b*g) + Sqrt[(b^2 - 4*a*c)*g^2]) + c*e*(3*b*d*g^2 + a*e*g^2 + Sqrt[(b^2 - 4*a*c)*g^2]*(e*f - 3*d*g)))*Sqrt[(Sqrt[(b^2 - 4*a*c)*g^2] - (2*a*g^2)/(f + g*x) - 2*c*f*(-1 + f/(f + g*x)) + b*g*(-1 + (2*f)/(f + g*x)))/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*g^2] + (2*a*g^2)/(f + g*x) + 2*c*f*(-1 + f/(f + g*x)) + b*(g - (2*f*g)/(f + g*x)))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x]))/(3*c^2*g^3*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[a + x*(b + c*x)]*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2])","C",1
911,1,814,393,5.878469,"\int \frac{d+e x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","-\frac{(f+g x)^{3/2} \left(-\frac{4 e \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} (a+x (b+c x)) g^2}{(f+g x)^2}+\frac{i \sqrt{2} e \left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}-\frac{i \sqrt{2} \left(2 c d g+e \left(\sqrt{\left(b^2-4 a c\right) g^2}-b g\right)\right) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\sqrt{f+g x}}\right)}{2 c g^2 \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{a+x (b+c x)}}","\frac{\sqrt{2} e \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c g \sqrt{f+g x} \sqrt{a+b x+c x^2}}",1,"-1/2*((f + g*x)^(3/2)*((-4*e*g^2*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*(a + x*(b + c*x)))/(f + g*x)^2 + (I*Sqrt[2]*e*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[(2*a*g^2 - 2*c*f*g*x + b*g*(-f + g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x] - (I*Sqrt[2]*(2*c*d*g + e*(-(b*g) + Sqrt[(b^2 - 4*a*c)*g^2]))*Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[(2*a*g^2 - 2*c*f*g*x + b*g*(-f + g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x]))/(c*g^2*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[a + x*(b + c*x)])","C",1
912,1,308,189,0.6352244,"\int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{i (f+g x) \sqrt{2-\frac{4 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right)}} \sqrt{\frac{2 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f\right)}+1} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{g \sqrt{a+x (b+c x)} \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f}}}","\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{c \sqrt{f+g x} \sqrt{a+b x+c x^2}}",1,"(I*(f + g*x)*Sqrt[2 - (4*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/(g*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[a + x*(b + c*x)])","C",1
913,1,499,280,1.7496515,"\int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{i (f+g x) \sqrt{2-\frac{4 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}-b g+2 c f\right)}} \sqrt{\frac{2 \left(g (a g-b f)+c f^2\right)}{(f+g x) \left(\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f\right)}+1} \left(F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)-\Pi \left(\frac{(e f-d g) \left(2 c f-b g-\sqrt{\left(b^2-4 a c\right) g^2}\right)}{2 e \left(c f^2+g (a g-b f)\right)};i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{\sqrt{a+x (b+c x)} (d g-e f) \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left(b^2-4 a c\right)}+b g-2 c f}}}","-\frac{\sqrt{2} \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} \sqrt{a+b x+c x^2} (e f-d g)}",1,"(I*(f + g*x)*Sqrt[2 - (4*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*(EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] - EllipticPi[((e*f - d*g)*(2*c*f - b*g - Sqrt[(b^2 - 4*a*c)*g^2]))/(2*e*(c*f^2 + g*(-(b*f) + a*g))), I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/((-(e*f) + d*g)*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[a + x*(b + c*x)])","C",1
914,1,10881,1037,13.9483443,"\int \frac{1}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e^2}{\left(c d^2-b e d+a e^2\right) (e f-d g) (d+e x)}+\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) e}{\sqrt{2} \left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} f \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) e}{\left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} d g \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\left(c d^2-b e d+a e^2\right) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+b x+a}}-\frac{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{2} \sqrt{c} \left(c d^2-b e d+a e^2\right) (e f-d g)^2 \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
915,1,40396,1114,17.8821358,"\int \frac{1}{(d+e x)^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{3 (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt{f+g x} \sqrt{c x^2+b x+a} e^2}{4 \left(c d^2-b e d+a e^2\right)^2 (e f-d g)^2 (d+e x)}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} e^2}{2 \left(c d^2-b e d+a e^2\right) (e f-d g) (d+e x)^2}+\frac{3 \sqrt{b^2-4 a c} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) e}{4 \sqrt{2} \left(c d^2-b e d+a e^2\right)^2 (e f-d g)^2 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{b^2-4 a c} (c d (7 d g-6 e f)+e (3 b e f-4 b d g+a e g)) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-4 a c} g}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}\right)}{2 \sqrt{2} \left(c d^2+e (a e-b d)\right)^2 (e f-d g) \sqrt{f+g x} \sqrt{a+x (b+c x)}}+\frac{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g} \left(c^2 \left(8 e^2 f^2-20 d e g f+15 d^2 g^2\right) d^2+2 c e \left(b d \left(-4 e^2 f^2+11 d e g f-10 d^2 g^2\right)+a e \left(-2 e^2 f^2+2 d e g f+3 d^2 g^2\right)\right)+e^2 \left(\left(3 e^2 f^2-8 d e g f+8 d^2 g^2\right) b^2+2 a e g (e f-4 d g) b+3 a^2 e^2 g^2\right)\right) \sqrt{\frac{g \left(-b-2 c x+\sqrt{b^2-4 a c}\right)}{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}} \sqrt{\frac{g \left(b+2 c x+\sqrt{b^2-4 a c}\right)}{\left(b+\sqrt{b^2-4 a c}\right) g-2 c f}} \Pi \left(\frac{2 c e f-b e g+\sqrt{b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g}}\right)|\frac{2 c f+\left(\sqrt{b^2-4 a c}-b\right) g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{4 \sqrt{2} \sqrt{c} \left(c d^2+e (a e-b d)\right)^2 (d g-e f)^3 \sqrt{a+x (b+c x)}}",1,"Result too large to show","C",0
916,1,950,553,6.1167215,"\int \frac{1}{(d+e x) (f+g x)^{3/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \left(\frac{(a+x (b+c x)) g^2}{e f-d g}+\frac{(f+g x)^2 \left(\frac{c f^2}{(f+g x)^2}-\frac{2 c f}{f+g x}-\frac{b g f}{(f+g x)^2}+c-\frac{i \sqrt{1-\frac{2 \left(c f^2+g (a g-b f)\right)}{\left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} \sqrt{\frac{4 \left(c f^2+g (a g-b f)\right)}{\left(-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}+2} \left((e f-d g) \left(2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}\right) \left(E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)-F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)-2 e \left(c f^2+g (a g-b f)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)+2 e \left(c f^2+g (a g-b f)\right) \Pi \left(\frac{(e f-d g) \left(2 c f-b g-\sqrt{\left(b^2-4 a c\right) g^2}\right)}{2 e \left(c f^2+g (a g-b f)\right)};i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}}}{\sqrt{f+g x}}\right)|-\frac{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)\right)}{4 (e f-d g) \sqrt{\frac{c f^2+g (a g-b f)}{-2 c f+b g+\sqrt{\left(b^2-4 a c\right) g^2}}} \sqrt{f+g x}}+\frac{b g}{f+g x}+\frac{a g^2}{(f+g x)^2}\right)}{d g-e f}\right)}{\left(c f^2+g (a g-b f)\right) \sqrt{f+g x} \sqrt{a+x (b+c x)}}","-\frac{\sqrt{2} g \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{\sqrt{a+b x+c x^2} (e f-d g) \left(a g^2-b f g+c f^2\right) \sqrt{\frac{c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{\sqrt{2} e \sqrt{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right)}{\sqrt{c} \sqrt{a+b x+c x^2} (e f-d g)^2}+\frac{2 g^2 \sqrt{a+b x+c x^2}}{\sqrt{f+g x} (e f-d g) \left(a g^2-b f g+c f^2\right)}",1,"(2*((g^2*(a + x*(b + c*x)))/(e*f - d*g) + ((f + g*x)^2*(c + (c*f^2)/(f + g*x)^2 - (b*f*g)/(f + g*x)^2 + (a*g^2)/(f + g*x)^2 - (2*c*f)/(f + g*x) + (b*g)/(f + g*x) - ((I/4)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[2 + (4*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*((e*f - d*g)*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]) - 2*e*(c*f^2 + g*(-(b*f) + a*g))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + 2*e*(c*f^2 + g*(-(b*f) + a*g))*EllipticPi[((e*f - d*g)*(2*c*f - b*g - Sqrt[(b^2 - 4*a*c)*g^2]))/(2*e*(c*f^2 + g*(-(b*f) + a*g))), I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/((e*f - d*g)*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(-(e*f) + d*g)))/((c*f^2 + g*(-(b*f) + a*g))*Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)])","C",1
917,1,14759,1125,15.5229592,"\int \frac{1}{(d+e x) (f+g x)^{5/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/((d + e*x)*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{\sqrt{2} \sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \Pi \left(\frac{e \left(2 c f-b g+\sqrt{b^2-4 a c} g\right)}{2 c (e f-d g)};\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left(b-\sqrt{b^2-4 a c}\right) g}}\right)|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right) e^2}{\sqrt{c} (e f-d g)^3 \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} g \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right) e}{(e f-d g)^2 \left(c f^2-b g f+a g^2\right) \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{2 g^2 \sqrt{c x^2+b x+a} e}{(e f-d g)^2 \left(c f^2-b g f+a g^2\right) \sqrt{f+g x}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g (2 c f-b g) \sqrt{f+g x} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 (e f-d g) \left(c f^2-b g f+a g^2\right)^2 \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g \sqrt{\frac{c (f+g x)}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}} \sqrt{-\frac{c \left(c x^2+b x+a\right)}{b^2-4 a c}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g}\right)}{3 (e f-d g) \left(c f^2-b g f+a g^2\right) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{4 g^2 (2 c f-b g) \sqrt{c x^2+b x+a}}{3 (e f-d g) \left(c f^2-b g f+a g^2\right)^2 \sqrt{f+g x}}+\frac{2 g^2 \sqrt{c x^2+b x+a}}{3 (e f-d g) \left(c f^2-b g f+a g^2\right) (f+g x)^{3/2}}",1,"Result too large to show","C",0
918,1,1118,475,9.4706603,"\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","-\frac{\sqrt{2} \sqrt{-\frac{g \left(c f^2+g (a g-b f)\right) (d+e x)}{\left(-2 a e g^2-2 c d f g+b (e f+d g) g-d \sqrt{\left(b^2-4 a c\right) g^2} g+e f \sqrt{\left(b^2-4 a c\right) g^2}\right) (f+g x)}} (f+g x)^{3/2} \left(\frac{2 e f \sqrt{\left(b^2-4 a c\right) g^2} \sqrt{-\frac{\left(c f^2+g (a g-b f)\right) (a+x (b+c x))}{\left(b^2-4 a c\right) (f+g x)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}}}{\sqrt{2}}\right)|\frac{2 \sqrt{\left(b^2-4 a c\right) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left(b^2-4 a c\right) g^2} g-e f \sqrt{\left(b^2-4 a c\right) g^2}}\right)}{c f^2+g (a g-b f)}+\frac{d g \left(2 a g^2-2 c f x g-\sqrt{\left(b^2-4 a c\right) g^2} x g+b (g x-f) g-f \sqrt{\left(b^2-4 a c\right) g^2}\right) \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}}}{\sqrt{2}}\right)|\frac{2 \sqrt{\left(b^2-4 a c\right) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left(b^2-4 a c\right) g^2} g-e f \sqrt{\left(b^2-4 a c\right) g^2}}\right)}{\left(c f^2+g (a g-b f)\right) (f+g x) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}}}-\frac{4 e \sqrt{\left(b^2-4 a c\right) g^2} \sqrt{-\frac{\left(c f^2+g (a g-b f)\right) (a+x (b+c x))}{\left(b^2-4 a c\right) (f+g x)^2}} \Pi \left(\frac{2 \sqrt{\left(b^2-4 a c\right) g^2}}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}};\sin ^{-1}\left(\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}{\sqrt{\left(b^2-4 a c\right) g^2} (f+g x)}}}{\sqrt{2}}\right)|\frac{2 \sqrt{\left(b^2-4 a c\right) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left(b^2-4 a c\right) g^2} g-e f \sqrt{\left(b^2-4 a c\right) g^2}}\right)}{2 c f-b g+\sqrt{\left(b^2-4 a c\right) g^2}}\right)}{g^2 \sqrt{d+e x} \sqrt{a+x (b+c x)}}","\frac{\sqrt{2} (d+e x) \sqrt{-\sqrt{b^2-4 a c}+b+2 c x} \sqrt{2 c f-g \left(\sqrt{b^2-4 a c}+b\right)} \sqrt{\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) (e f-d g)}{(d+e x) \left(2 c f-g \left(\sqrt{b^2-4 a c}+b\right)\right)}} \sqrt{\frac{\left(x \left(\sqrt{b^2-4 a c}+b\right)+2 a\right) (e f-d g)}{(d+e x) \left(f \sqrt{b^2-4 a c}-2 a g+b f\right)}} \Pi \left(\frac{e \left(2 c f-\left(b+\sqrt{b^2-4 a c}\right) g\right)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) g};\sin ^{-1}\left(\frac{\sqrt{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e} \sqrt{f+g x}}{\sqrt{2 c f-\left(b+\sqrt{b^2-4 a c}\right) g} \sqrt{d+e x}}\right)|\frac{\left(b d+\sqrt{b^2-4 a c} d-2 a e\right) \left(2 c f-\left(b+\sqrt{b^2-4 a c}\right) g\right)}{\left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) \left(b f+\sqrt{b^2-4 a c} f-2 a g\right)}\right)}{g \sqrt{\frac{2 a c}{\sqrt{b^2-4 a c}+b}+c x} \sqrt{a+b x+c x^2} \sqrt{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}",1,"-((Sqrt[2]*Sqrt[-((g*(c*f^2 + g*(-(b*f) + a*g))*(d + e*x))/((-2*c*d*f*g - 2*a*e*g^2 + e*f*Sqrt[(b^2 - 4*a*c)*g^2] - d*g*Sqrt[(b^2 - 4*a*c)*g^2] + b*g*(e*f + d*g))*(f + g*x)))]*(f + g*x)^(3/2)*((2*e*f*Sqrt[(b^2 - 4*a*c)*g^2]*Sqrt[-(((c*f^2 + g*(-(b*f) + a*g))*(a + x*(b + c*x)))/((b^2 - 4*a*c)*(f + g*x)^2))]*EllipticF[ArcSin[Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] + d*g*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*(e*f + d*g))])/(c*f^2 + g*(-(b*f) + a*g)) + (d*g*(2*a*g^2 - f*Sqrt[(b^2 - 4*a*c)*g^2] - 2*c*f*g*x - g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(-f + g*x))*Sqrt[(2*a*g^2 - 2*c*f*g*x + b*g*(-f + g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]*EllipticF[ArcSin[Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] + d*g*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*(e*f + d*g))])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)*Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]) - (4*e*Sqrt[(b^2 - 4*a*c)*g^2]*Sqrt[-(((c*f^2 + g*(-(b*f) + a*g))*(a + x*(b + c*x)))/((b^2 - 4*a*c)*(f + g*x)^2))]*EllipticPi[(2*Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]), ArcSin[Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] + d*g*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*(e*f + d*g))])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])))/(g^2*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]))","B",0
919,1,375,588,3.5206351,"\int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx","Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{2} e \sqrt{a+x (b+c x)} \sqrt{-\frac{e (f+g x) \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(-d g \sqrt{e^2 \left(b^2-4 a c\right)}+e f \sqrt{e^2 \left(b^2-4 a c\right)}-2 a e^2 g+b e (d g+e f)-2 c d e f\right)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{2 a e^2-2 c d x e+b (e x-d) e+\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}{\sqrt{\left(b^2-4 a c\right) e^2} (d+e x)}}}{\sqrt{2}}\right)|\frac{2 \sqrt{\left(b^2-4 a c\right) e^2} (e f-d g)}{-2 a g e^2-2 c d f e+\sqrt{\left(b^2-4 a c\right) e^2} f e+b (e f+d g) e-d \sqrt{\left(b^2-4 a c\right) e^2} g}\right)}{\sqrt{d+e x} \sqrt{f+g x} \sqrt{e^2 \left(b^2-4 a c\right)} \sqrt{-\frac{(a+x (b+c x)) \left(e (a e-b d)+c d^2\right)}{\left(b^2-4 a c\right) (d+e x)^2}}}","-\frac{(d+e x) \sqrt[4]{c f^2-g (b f-a g)} \sqrt{\frac{\left(a+b x+c x^2\right) (e f-d g)^2}{(d+e x)^2 \left(a g^2-b f g+c f^2\right)}} \left(\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right) \sqrt{\frac{\frac{(f+g x)^2 \left(a e^2-b d e+c d^2\right)}{(d+e x)^2 \left(c f^2-g (b f-a g)\right)}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left(a g^2-b f g+c f^2\right)}+1}{\left(\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c d^2-b e d+a e^2} \sqrt{f+g x}}{\sqrt[4]{c f^2-b g f+a g^2} \sqrt{d+e x}}\right)|\frac{1}{4} \left(\frac{2 c d f+2 a e g-b (e f+d g)}{\sqrt{c d^2-e (b d-a e)} \sqrt{c f^2-g (b f-a g)}}+2\right)\right)}{\sqrt{a+b x+c x^2} (e f-d g) \sqrt[4]{a e^2-b d e+c d^2} \sqrt{\frac{(f+g x)^2 \left(a e^2-b d e+c d^2\right)}{(d+e x)^2 \left(c f^2-g (b f-a g)\right)}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left(a g^2-b f g+c f^2\right)}+1}}",1,"(2*Sqrt[2]*e*Sqrt[-((e*(c*d^2 + e*(-(b*d) + a*e))*(f + g*x))/((-2*c*d*e*f + e*Sqrt[(b^2 - 4*a*c)*e^2]*f - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a*c)*e^2]*g + b*e*(e*f + d*g))*(d + e*x)))]*Sqrt[a + x*(b + c*x)]*EllipticF[ArcSin[Sqrt[(2*a*e^2 - 2*c*d*e*x + b*e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/(Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*e^2]*(e*f - d*g))/(-2*c*d*e*f + e*Sqrt[(b^2 - 4*a*c)*e^2]*f - 2*a*e^2*g - d*Sqrt[(b^2 - 4*a*c)*e^2]*g + b*e*(e*f + d*g))])/(Sqrt[(b^2 - 4*a*c)*e^2]*Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[-(((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x)))/((b^2 - 4*a*c)*(d + e*x)^2))])","A",1
920,1,198,220,0.2725772,"\int (d+e x)^m (f+g x)^2 \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2),x]","\frac{(d+e x)^{m+1} \left(\frac{(d+e x)^2 \left(e g (a e g-3 b d g+2 b e f)+c \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)\right)}{m+3}+\frac{(e f-d g)^2 \left(e (a e-b d)+c d^2\right)}{m+1}+\frac{(d+e x) (e f-d g) (e (2 a e g-3 b d g+b e f)+2 c d (2 d g-e f))}{m+2}+\frac{g (d+e x)^3 (b e g-4 c d g+2 c e f)}{m+4}+\frac{c g^2 (d+e x)^4}{m+5}\right)}{e^5}","\frac{(d+e x)^{m+3} \left(e g (a e g-3 b d g+2 b e f)+c \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)\right)}{e^5 (m+3)}+\frac{(e f-d g)^2 (d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)}{e^5 (m+1)}-\frac{(e f-d g) (d+e x)^{m+2} (2 c d (e f-2 d g)-e (2 a e g-3 b d g+b e f))}{e^5 (m+2)}+\frac{g (d+e x)^{m+4} (b e g-4 c d g+2 c e f)}{e^5 (m+4)}+\frac{c g^2 (d+e x)^{m+5}}{e^5 (m+5)}",1,"((d + e*x)^(1 + m)*(((c*d^2 + e*(-(b*d) + a*e))*(e*f - d*g)^2)/(1 + m) + ((e*f - d*g)*(2*c*d*(-(e*f) + 2*d*g) + e*(b*e*f - 3*b*d*g + 2*a*e*g))*(d + e*x))/(2 + m) + ((e*g*(2*b*e*f - 3*b*d*g + a*e*g) + c*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2))*(d + e*x)^2)/(3 + m) + (g*(2*c*e*f - 4*c*d*g + b*e*g)*(d + e*x)^3)/(4 + m) + (c*g^2*(d + e*x)^4)/(5 + m)))/e^5","A",1
921,1,180,144,0.3309729,"\int (d+e x)^m (f+g x) \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2),x]","\frac{(d+e x)^{m+1} \left(\frac{(d+e x) \left(c e (2 a e g (m+3)+b d g (m-2)+b e f (m+4))-b^2 e^2 g (m+2)+2 c^2 d (3 d g-e f (m+4))\right)}{e^2 (m+2)}-\frac{\left(e (a e-b d)+c d^2\right) (b e g (m+1)+6 c d g-2 c e f (m+4))}{e^2 (m+1)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (m+4)+e g (m+3) x))\right)}{c e^2 (m+3) (m+4)}","\frac{(e f-d g) (d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)}{e^4 (m+1)}-\frac{(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac{(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac{c g (d+e x)^{m+4}}{e^4 (m+4)}",1,"((d + e*x)^(1 + m)*(-(((c*d^2 + e*(-(b*d) + a*e))*(6*c*d*g + b*e*g*(1 + m) - 2*c*e*f*(4 + m)))/(e^2*(1 + m))) + ((-(b^2*e^2*g*(2 + m)) + 2*c^2*d*(3*d*g - e*f*(4 + m)) + c*e*(b*d*g*(-2 + m) + 2*a*e*g*(3 + m) + b*e*f*(4 + m)))*(d + e*x))/(e^2*(2 + m)) + (a + x*(b + c*x))*(b*e*g + c*(-3*d*g + e*f*(4 + m) + e*g*(3 + m)*x))))/(c*e^2*(3 + m)*(4 + m))","A",1
922,1,111,129,0.1399242,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)}{f+g x} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x),x]","\frac{(d+e x)^{m+1} \left(\frac{\left(g (a g-b f)+c f^2\right) \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)}+\frac{b e g-c (d g+e f)}{e^2 (m+1)}+\frac{c g (d+e x)}{e^2 (m+2)}\right)}{g^2}","\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right) \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right)}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)}",1,"((d + e*x)^(1 + m)*((b*e*g - c*(e*f + d*g))/(e^2*(1 + m)) + (c*g*(d + e*x))/(e^2*(2 + m)) + ((c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)*(1 + m))))/g^2","A",1
923,1,134,157,0.1459638,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)}{(f+g x)^2} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x)^2,x]","\frac{(d+e x)^{m+1} \left(e^2 \left(g (a g-b f)+c f^2\right) \, _2F_1\left(2,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)-e (2 c f-b g) (e f-d g) \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)+c (e f-d g)^2\right)}{e g^2 (m+1) (e f-d g)^2}","\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right) (c f (2 d g-e f (m+2))-g (a e g m+b (d g-e f (m+1))))}{g^2 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left(a+\frac{f (c f-b g)}{g^2}\right)}{(f+g x) (e f-d g)}+\frac{c (d+e x)^{m+1}}{e g^2 (m+1)}",1,"((d + e*x)^(1 + m)*(c*(e*f - d*g)^2 - e*(2*c*f - b*g)*(e*f - d*g)*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e^2*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[2, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)]))/(e*g^2*(e*f - d*g)^2*(1 + m))","A",1
924,1,157,245,0.1597455,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)}{(f+g x)^3} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x)^3,x]","-\frac{(d+e x)^{m+1} \left(e \left(e \left(g (a g-b f)+c f^2\right) \, _2F_1\left(3,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)-(2 c f-b g) (e f-d g) \, _2F_1\left(2,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)\right)+c (e f-d g)^2 \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)\right)}{g^2 (m+1) (d g-e f)^3}","\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right) \left(c \left(2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left(m^2+3 m+2\right)\right)-e g m (a e g (1-m)-b (2 d g-e f (m+1)))\right)}{2 g^2 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} (g (a e g (1-m)-b (2 d g-e f (m+1)))+c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left(a+\frac{f (c f-b g)}{g^2}\right)}{2 (f+g x)^2 (e f-d g)}",1,"-(((d + e*x)^(1 + m)*(c*(e*f - d*g)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e*(-((2*c*f - b*g)*(e*f - d*g)*Hypergeometric2F1[2, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)]) + e*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[3, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])))/(g^2*(-(e*f) + d*g)^3*(1 + m)))","A",1
925,1,492,525,0.7729224,"\int (d+e x)^m (f+g x)^2 \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{(d+e x)^2 \left(e^2 \left(a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)\right)+2 c e \left(a e \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)+b d \left(-10 d^2 g^2+12 d e f g-3 e^2 f^2\right)\right)+c^2 d^2 \left(15 d^2 g^2-20 d e f g+6 e^2 f^2\right)\right)}{m+3}+\frac{(d+e x)^4 \left(2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left(15 d^2 g^2-10 d e f g+e^2 f^2\right)\right)}{m+5}+\frac{2 (d+e x)^3 \left(c e \left(2 a e g (e f-2 d g)+b \left(10 d^2 g^2-8 d e f g+e^2 f^2\right)\right)+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left(5 d^2 g^2-5 d e f g+e^2 f^2\right)\right)}{m+4}-\frac{2 (d+e x) (d g-e f) \left(e (a e-b d)+c d^2\right) (e (a e g-2 b d g+b e f)+c d (3 d g-2 e f))}{m+2}+\frac{(e f-d g)^2 \left(e (a e-b d)+c d^2\right)^2}{m+1}+\frac{2 c g (d+e x)^5 (b e g-3 c d g+c e f)}{m+6}+\frac{c^2 g^2 (d+e x)^6}{m+7}\right)}{e^7}","\frac{(d+e x)^{m+3} \left(e^2 \left(a^2 e^2 g^2+2 a b e g (2 e f-3 d g)+b^2 \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)\right)+2 c e \left(a e \left(6 d^2 g^2-6 d e f g+e^2 f^2\right)-b d \left(10 d^2 g^2-12 d e f g+3 e^2 f^2\right)\right)+c^2 d^2 \left(15 d^2 g^2-20 d e f g+6 e^2 f^2\right)\right)}{e^7 (m+3)}+\frac{(d+e x)^{m+5} \left(2 c e g (a e g-5 b d g+2 b e f)+b^2 e^2 g^2+c^2 \left(15 d^2 g^2-10 d e f g+e^2 f^2\right)\right)}{e^7 (m+5)}+\frac{2 (d+e x)^{m+4} \left(c e \left(2 a e g (e f-2 d g)+b \left(10 d^2 g^2-8 d e f g+e^2 f^2\right)\right)+b e^2 g (a e g-2 b d g+b e f)-2 c^2 d \left(5 d^2 g^2-5 d e f g+e^2 f^2\right)\right)}{e^7 (m+4)}+\frac{(e f-d g)^2 (d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)^2}{e^7 (m+1)}-\frac{2 (e f-d g) (d+e x)^{m+2} \left(a e^2-b d e+c d^2\right) (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^7 (m+2)}+\frac{2 c g (d+e x)^{m+6} (b e g-3 c d g+c e f)}{e^7 (m+6)}+\frac{c^2 g^2 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*(((c*d^2 + e*(-(b*d) + a*e))^2*(e*f - d*g)^2)/(1 + m) - (2*(c*d^2 + e*(-(b*d) + a*e))*(-(e*f) + d*g)*(c*d*(-2*e*f + 3*d*g) + e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x))/(2 + m) + ((c^2*d^2*(6*e^2*f^2 - 20*d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2)) + 2*c*e*(b*d*(-3*e^2*f^2 + 12*d*e*f*g - 10*d^2*g^2) + a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2)))*(d + e*x)^2)/(3 + m) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) + c*e*(2*a*e*g*(e*f - 2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^3)/(4 + m) + ((b^2*e^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^4)/(5 + m) + (2*c*g*(c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^5)/(6 + m) + (c^2*g^2*(d + e*x)^6)/(7 + m)))/e^7","A",1
926,1,655,311,1.5228554,"\int (d+e x)^m (f+g x) \left(a+b x+c x^2\right)^2 \, dx","Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{2 \left(\frac{(d+e x) \left(c^2 e^2 \left(4 a^2 e^2 g \left(m^2+8 m+15\right)+2 a b e \left(d g \left(4 m^2+11 m-18\right)+e f \left(2 m^2+19 m+42\right)\right)+b^2 d \left(d g \left(m^2-13 m+6\right)+2 e f \left(m^2+5 m-6\right)\right)\right)-b^2 c e^3 (m+2) (a e g (5 m+21)+b d g (2 m-3)+b e f (m+6))+2 c^3 d e \left(3 b d (d g (m-14)+3 e f (m+6))-2 a e \left(d g \left(m^2-4 m-30\right)+e f \left(2 m^2+19 m+42\right)\right)\right)+b^4 e^4 g \left(m^2+5 m+6\right)+12 c^4 d^3 (5 d g-e f (m+6))\right)}{e^2 (m+2)}+\frac{\left(e (a e-b d)+c d^2\right) \left(2 c^2 e \left(2 a e \left(d g \left(m^2+m-15\right)+e f \left(m^2+10 m+24\right)\right)-3 b d (d g (m-9)+2 e f (m+6))\right)-b c e^2 (m+1) (2 a e g (2 m+9)+b d g (m-6)+b e f (m+6))+b^3 e^3 g \left(m^2+4 m+3\right)+12 c^3 d^2 (e f (m+6)-5 d g)\right)}{e^2 (m+1)}-(a+x (b+c x)) \left(c e (m+3) x \left(-c e (2 a e g (m+5)+b d g (m-4)+b e f (m+6))+b^2 e^2 g (m+3)+2 c^2 d (e f (m+6)-5 d g)\right)-(3 c d-b e) \left(-c e (2 a e g (m+5)+b d g (m-4)+b e f (m+6))+b^2 e^2 g (m+3)+2 c^2 d (e f (m+6)-5 d g)\right)+c e (m+4) \left(c e f (m+6) (b d-2 a e)+a b e^2 g (m+1)-2 a c d e g m+b d g (2 b e-5 c d)\right)\right)\right)}{c e^2 (m+3) (m+4)}+(a+x (b+c x))^2 (2 b e g+c (-5 d g+e f (m+6)+e g (m+5) x))\right)}{c e^2 (m+5) (m+6)}","\frac{(d+e x)^{m+4} \left(2 c e (a e g-4 b d g+b e f)+b^2 e^2 g-2 c^2 d (2 e f-5 d g)\right)}{e^6 (m+4)}+\frac{(d+e x)^{m+3} \left(2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))+b e^2 (2 a e g-3 b d g+b e f)+2 c^2 d^2 (3 e f-5 d g)\right)}{e^6 (m+3)}+\frac{(e f-d g) (d+e x)^{m+1} \left(a e^2-b d e+c d^2\right)^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left(a e^2-b d e+c d^2\right) (c d (4 e f-5 d g)-e (a e g-3 b d g+2 b e f))}{e^6 (m+2)}+\frac{c (d+e x)^{m+5} (2 b e g-5 c d g+c e f)}{e^6 (m+5)}+\frac{c^2 g (d+e x)^{m+6}}{e^6 (m+6)}",1,"((d + e*x)^(1 + m)*((a + x*(b + c*x))^2*(2*b*e*g + c*(-5*d*g + e*f*(6 + m) + e*g*(5 + m)*x)) + (2*(((c*d^2 + e*(-(b*d) + a*e))*(b^3*e^3*g*(3 + 4*m + m^2) + 12*c^3*d^2*(-5*d*g + e*f*(6 + m)) - b*c*e^2*(1 + m)*(b*d*g*(-6 + m) + b*e*f*(6 + m) + 2*a*e*g*(9 + 2*m)) + 2*c^2*e*(-3*b*d*(d*g*(-9 + m) + 2*e*f*(6 + m)) + 2*a*e*(d*g*(-15 + m + m^2) + e*f*(24 + 10*m + m^2)))))/(e^2*(1 + m)) + ((b^4*e^4*g*(6 + 5*m + m^2) + 12*c^4*d^3*(5*d*g - e*f*(6 + m)) - b^2*c*e^3*(2 + m)*(b*e*f*(6 + m) + b*d*g*(-3 + 2*m) + a*e*g*(21 + 5*m)) + 2*c^3*d*e*(3*b*d*(d*g*(-14 + m) + 3*e*f*(6 + m)) - 2*a*e*(d*g*(-30 - 4*m + m^2) + e*f*(42 + 19*m + 2*m^2))) + c^2*e^2*(4*a^2*e^2*g*(15 + 8*m + m^2) + b^2*d*(d*g*(6 - 13*m + m^2) + 2*e*f*(-6 + 5*m + m^2)) + 2*a*b*e*(e*f*(42 + 19*m + 2*m^2) + d*g*(-18 + 11*m + 4*m^2))))*(d + e*x))/(e^2*(2 + m)) - (c*e*(4 + m)*(b*d*(-5*c*d + 2*b*e)*g - 2*a*c*d*e*g*m + a*b*e^2*g*(1 + m) + c*e*(b*d - 2*a*e)*f*(6 + m)) - (3*c*d - b*e)*(b^2*e^2*g*(3 + m) + 2*c^2*d*(-5*d*g + e*f*(6 + m)) - c*e*(b*d*g*(-4 + m) + 2*a*e*g*(5 + m) + b*e*f*(6 + m))) + c*e*(3 + m)*(b^2*e^2*g*(3 + m) + 2*c^2*d*(-5*d*g + e*f*(6 + m)) - c*e*(b*d*g*(-4 + m) + 2*a*e*g*(5 + m) + b*e*f*(6 + m)))*x)*(a + x*(b + c*x))))/(c*e^2*(3 + m)*(4 + m))))/(c*e^2*(5 + m)*(6 + m))","B",1
927,1,265,287,0.414715,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)^2}{f+g x} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2)^2)/(f + g*x),x]","\frac{(d+e x)^{m+1} \left(\frac{g (d+e x) \left(2 c e g (a e g-b (2 d g+e f))+b^2 e^2 g^2+c^2 \left(3 d^2 g^2+2 d e f g+e^2 f^2\right)\right)}{e^4 (m+2)}-\frac{(-b e g+c d g+c e f) \left(e g (2 a e g-b (d g+e f))+c \left(d^2 g^2+e^2 f^2\right)\right)}{e^4 (m+1)}+\frac{\left(g (a g-b f)+c f^2\right)^2 \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)}-\frac{c g^2 (d+e x)^2 (-2 b e g+3 c d g+c e f)}{e^4 (m+3)}+\frac{c^2 g^3 (d+e x)^3}{e^4 (m+4)}\right)}{g^4}","\frac{(d+e x)^{m+2} \left(2 c e g (a e g-b (2 d g+e f))+b^2 e^2 g^2+c^2 \left(3 d^2 g^2+2 d e f g+e^2 f^2\right)\right)}{e^4 g^3 (m+2)}+\frac{(d+e x)^{m+1} (b e g-c (d g+e f)) \left(e g (2 a e g-b (d g+e f))+c \left(d^2 g^2+e^2 f^2\right)\right)}{e^4 g^4 (m+1)}+\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right)^2 \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right)}{g^4 (m+1) (e f-d g)}-\frac{c (d+e x)^{m+3} (-2 b e g+3 c d g+c e f)}{e^4 g^2 (m+3)}+\frac{c^2 (d+e x)^{m+4}}{e^4 g (m+4)}",1,"((d + e*x)^(1 + m)*(-(((c*e*f + c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g))))/(e^4*(1 + m))) + (g*(b^2*e^2*g^2 + c^2*(e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 2*c*e*g*(a*e*g - b*(e*f + 2*d*g)))*(d + e*x))/(e^4*(2 + m)) - (c*g^2*(c*e*f + 3*c*d*g - 2*b*e*g)*(d + e*x)^2)/(e^4*(3 + m)) + (c^2*g^3*(d + e*x)^3)/(e^4*(4 + m)) + ((c*f^2 + g*(-(b*f) + a*g))^2*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)*(1 + m))))/g^4","A",1
928,1,261,298,0.3910948,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)^2}{(f+g x)^2} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2)^2)/(f + g*x)^2,x]","\frac{(d+e x)^{m+1} \left(\frac{2 c e g (a e g-b (d g+2 e f))+b^2 e^2 g^2+c^2 \left(d^2 g^2+2 d e f g+3 e^2 f^2\right)}{e^3 (m+1)}+\frac{e \left(g (a g-b f)+c f^2\right)^2 \, _2F_1\left(2,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)^2}-\frac{2 (2 c f-b g) \left(g (a g-b f)+c f^2\right) \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)}-\frac{2 c g (d+e x) (-b e g+c d g+c e f)}{e^3 (m+2)}+\frac{c^2 g^2 (d+e x)^2}{e^3 (m+3)}\right)}{g^4}","\frac{(d+e x)^{m+1} \left(2 c e g (a e g-b (d g+2 e f))+b^2 e^2 g^2+c^2 \left(d^2 g^2+2 d e f g+3 e^2 f^2\right)\right)}{e^3 g^4 (m+1)}+\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right) \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right) (c f (4 d g-e f (m+4))-g (a e g m+b (2 d g-e f (m+2))))}{g^4 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right)^2}{g^4 (f+g x) (e f-d g)}-\frac{2 c (d+e x)^{m+2} (-b e g+c d g+c e f)}{e^3 g^3 (m+2)}+\frac{c^2 (d+e x)^{m+3}}{e^3 g^2 (m+3)}",1,"((d + e*x)^(1 + m)*((b^2*e^2*g^2 + c^2*(3*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*c*e*g*(a*e*g - b*(2*e*f + d*g)))/(e^3*(1 + m)) - (2*c*g*(c*e*f + c*d*g - b*e*g)*(d + e*x))/(e^3*(2 + m)) + (c^2*g^2*(d + e*x)^2)/(e^3*(3 + m)) - (2*(2*c*f - b*g)*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)*(1 + m)) + (e*(c*f^2 + g*(-(b*f) + a*g))^2*Hypergeometric2F1[2, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)^2*(1 + m))))/g^4","A",1
929,1,257,461,0.3447176,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)^2}{(f+g x)^3} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2)^2)/(f + g*x)^3,x]","\frac{(d+e x)^{m+1} \left(\frac{\left(2 c g (a g-3 b f)+b^2 g^2+6 c^2 f^2\right) \, _2F_1\left(1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)}+\frac{e^2 \left(g (a g-b f)+c f^2\right)^2 \, _2F_1\left(3,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)^3}-\frac{2 e (2 c f-b g) \left(g (a g-b f)+c f^2\right) \, _2F_1\left(2,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right)}{(m+1) (e f-d g)^2}-\frac{c (-2 b e g+c d g+3 c e f)}{e^2 (m+1)}+\frac{c^2 g (d+e x)}{e^2 (m+2)}\right)}{g^4}","\frac{(d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right) \left(-g^2 \left(a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))-\left(b^2 \left(2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left(m^2+3 m+2\right)\right)\right)\right)+2 c g \left(a g \left(2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left(m^2+3 m+2\right)\right)-b f \left(6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left(m^2+5 m+6\right)\right)\right)+c^2 f^2 \left(12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left(m^2+7 m+12\right)\right)\right)}{2 g^4 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right) (g (a e g (1-m)-b (4 d g-e f (m+3)))+c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left(a g^2-b f g+c f^2\right)^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)}",1,"((d + e*x)^(1 + m)*(-((c*(3*c*e*f + c*d*g - 2*b*e*g))/(e^2*(1 + m))) + (c^2*g*(d + e*x))/(e^2*(2 + m)) + ((6*c^2*f^2 + b^2*g^2 + 2*c*g*(-3*b*f + a*g))*Hypergeometric2F1[1, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)*(1 + m)) - (2*e*(2*c*f - b*g)*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[2, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)^2*(1 + m)) + (e^2*(c*f^2 + g*(-(b*f) + a*g))^2*Hypergeometric2F1[3, 1 + m, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])/((e*f - d*g)^3*(1 + m))))/g^4","A",1
930,1,151,183,0.3297706,"\int \frac{(2+3 x)^4 (1+4 x)^m}{1-5 x+3 x^2} \, dx","Integrate[((2 + 3*x)^4*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]","\frac{3}{832} (4 x+1)^{m+1} \left(-\frac{32 \left(1631 \sqrt{13}-5499\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}-\frac{32 \left(5499+1631 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{117 (4 x+1)^2}{m+3}+\frac{1794 (4 x+1)}{m+2}+\frac{15977}{m+1}\right)","-\frac{3 \left(5499-1631 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{26 \left(13-2 \sqrt{13}\right) (m+1)}-\frac{3 \left(5499+1631 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{26 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{3687 (4 x+1)^{m+1}}{64 (m+1)}+\frac{207 (4 x+1)^{m+2}}{32 (m+2)}+\frac{27 (4 x+1)^{m+3}}{64 (m+3)}",1,"(3*(1 + 4*x)^(1 + m)*(15977/(1 + m) + (1794*(1 + 4*x))/(2 + m) + (117*(1 + 4*x)^2)/(3 + m) - (32*(-5499 + 1631*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) - (32*(5499 + 1631*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/832","A",1
931,1,117,165,0.1384176,"\int \frac{(2+3 x)^3 (1+4 x)^m}{1-5 x+3 x^2} \, dx","Integrate[((2 + 3*x)^3*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]","\frac{(4 x+1)^{m+1} \left(16 \left(71 \sqrt{13}-146\right) (m+2) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)-16 \left(146+71 \sqrt{13}\right) (m+2) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)+117 (4 m (3 x+11)+12 x+85)\right)}{624 \left(m^2+3 m+2\right)}","-\frac{3 \left(416-135 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{13 \left(13-2 \sqrt{13}\right) (m+1)}-\frac{3 \left(416+135 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{13 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{123 (4 x+1)^{m+1}}{16 (m+1)}+\frac{9 (4 x+1)^{m+2}}{16 (m+2)}",1,"((1 + 4*x)^(1 + m)*(117*(85 + 12*x + 4*m*(11 + 3*x)) + 16*(-146 + 71*Sqrt[13])*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] - 16*(146 + 71*Sqrt[13])*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(624*(2 + 3*m + m^2))","A",1
932,1,91,147,0.0990098,"\int \frac{(2+3 x)^2 (1+4 x)^m}{1-5 x+3 x^2} \, dx","Integrate[((2 + 3*x)^2*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]","\frac{(4 x+1)^{m+1} \left(\left(58 \sqrt{13}-46\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)-2 \left(23+29 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)+117\right)}{156 (m+1)}","-\frac{3 \left(117-47 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{26 \left(13-2 \sqrt{13}\right) (m+1)}-\frac{3 \left(117+47 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{26 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{3 (4 x+1)^{m+1}}{4 (m+1)}",1,"((1 + 4*x)^(1 + m)*(117 + (-46 + 58*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] - 2*(23 + 29*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(156*(1 + m))","A",1
933,1,89,129,0.0861605,"\int \frac{(2+3 x) (1+4 x)^m}{1-5 x+3 x^2} \, dx","Integrate[((2 + 3*x)*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]","\frac{(4 x+1)^{m+1} \left(\left(5+7 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)+\left(5-7 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)\right)}{78 (m+1)}","-\frac{3 \left(13-9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{26 \left(13-2 \sqrt{13}\right) (m+1)}-\frac{3 \left(13+9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{26 \left(13+2 \sqrt{13}\right) (m+1)}",1,"((1 + 4*x)^(1 + m)*((5 + 7*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] + (5 - 7*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(78*(1 + m))","A",1
934,1,94,117,0.1155397,"\int \frac{(1+4 x)^m}{1-5 x+3 x^2} \, dx","Integrate[(1 + 4*x)^m/(1 - 5*x + 3*x^2),x]","\frac{(4 x+1)^{m+1} \left(\left(13+2 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)+\left(2 \sqrt{13}-13\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)\right)}{39 \sqrt{13} (m+1)}","\frac{3 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{\sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}-\frac{3 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{\sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}",1,"((1 + 4*x)^(1 + m)*((13 + 2*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] + (-13 + 2*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(39*Sqrt[13]*(1 + m))","A",1
935,1,110,164,0.1941052,"\int \frac{(1+4 x)^m}{(2+3 x) \left(1-5 x+3 x^2\right)} \, dx","Integrate[(1 + 4*x)^m/((2 + 3*x)*(1 - 5*x + 3*x^2)),x]","\frac{(4 x+1)^{m+1} \left(234 \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)+5 \left(31+11 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)+5 \left(31-11 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)\right)}{6630 (m+1)}","\frac{3 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{85 (m+1)}+\frac{3 \left(13+9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{442 \left(13-2 \sqrt{13}\right) (m+1)}+\frac{3 \left(13-9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{442 \left(13+2 \sqrt{13}\right) (m+1)}",1,"((1 + 4*x)^(1 + m)*(234*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5] + 5*(31 + 11*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] + 5*(31 - 11*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(6630*(1 + m))","A",1
936,1,152,199,0.1589706,"\int \frac{(1+4 x)^m}{(2+3 x)^2 \left(1-5 x+3 x^2\right)} \, dx","Integrate[(1 + 4*x)^m/((2 + 3*x)^2*(1 - 5*x + 3*x^2)),x]","\frac{(4 x+1)^{m+1} \left(10530 \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)+25 \left(211+65 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)-1625 \sqrt{13} \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)+5275 \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)+15912 \, _2F_1\left(2,m+1;m+2;-\frac{3}{5} (4 x+1)\right)\right)}{563550 (m+1)}","\frac{27 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{1445 (m+1)}+\frac{3 \left(117+47 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{7514 \left(13-2 \sqrt{13}\right) (m+1)}+\frac{3 \left(117-47 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{7514 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{12 (4 x+1)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{425 (m+1)}",1,"((1 + 4*x)^(1 + m)*(10530*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5] + 25*(211 + 65*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] + 5275*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])] - 1625*Sqrt[13]*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])] + 15912*Hypergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5]))/(563550*(1 + m))","A",1
937,1,251,202,0.4438155,"\int \frac{(2+3 x)^4 (1+4 x)^m}{\left(1-5 x+3 x^2\right)^2} \, dx","Integrate[((2 + 3*x)^4*(1 + 4*x)^m)/(1 - 5*x + 3*x^2)^2,x]","\frac{(4 x+1)^{m+1} \left(-\frac{1053 \left(128 \sqrt{13}-117\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}-\frac{1053 \left(117+128 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}-\frac{\left(2 \left(5731+667 \sqrt{13}\right) m-14679 \left(\sqrt{13}-2\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)-\left(2 \left(667 \sqrt{13}-5731\right) m-14679 \left(2+\sqrt{13}\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{m+1}+\frac{13689}{4 m+4}+\frac{39 (844-2355 x)}{3 x^2-5 x+1}\right)}{1521}","-\frac{\left(13689-\sqrt{13} \left(-1570 \sqrt{13} m+4474 m+297\right)\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{169 \left(13-2 \sqrt{13}\right) (m+1)}-\frac{\left(\sqrt{13} \left(1570 \sqrt{13} m+4474 m+297\right)+13689\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{169 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(844-2355 x) (4 x+1)^{m+1}}{39 \left(3 x^2-5 x+1\right)}+\frac{9 (4 x+1)^{m+1}}{4 (m+1)}",1,"((1 + 4*x)^(1 + m)*(13689/(4 + 4*m) + (39*(844 - 2355*x))/(1 - 5*x + 3*x^2) - (1053*(-117 + 128*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) - (1053*(117 + 128*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m)) - (-((-14679*(2 + Sqrt[13]) + 2*(-5731 + 667*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])]) + (-14679*(-2 + Sqrt[13]) + 2*(5731 + 667*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/(1 + m)))/1521","A",1
938,1,252,181,0.3768486,"\int \frac{(2+3 x)^3 (1+4 x)^m}{\left(1-5 x+3 x^2\right)^2} \, dx","Integrate[((2 + 3*x)^3*(1 + 4*x)^m)/(1 - 5*x + 3*x^2)^2,x]","\frac{(4 x+1)^{m+1} \left(-\frac{12 \left(\sqrt{13} (1215-292 m)+1846 m\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(13-2 \sqrt{13}\right) (m+1)}-\frac{351 \left(27 \sqrt{13}-13\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}+\frac{12 \left(\sqrt{13} (1215-292 m)-1846 m\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}-\frac{351 \left(13+27 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{5434-11076 x}{3 x^2-5 x+1}\right)}{1014}","-\frac{\left(\sqrt{13} \left(568 \sqrt{13} m-1168 m+1701\right)+1521\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{338 \left(13-2 \sqrt{13}\right) (m+1)}+\frac{\left(\sqrt{13} (1701-1168 m)-13 (568 m+117)\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{338 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(209-426 x) (4 x+1)^{m+1}}{39 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((5434 - 11076*x)/(1 - 5*x + 3*x^2) - (351*(-13 + 27*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) - (12*(Sqrt[13]*(1215 - 292*m) + 1846*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((13 - 2*Sqrt[13])*(1 + m)) - (351*(13 + 27*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m)) + (12*(Sqrt[13]*(1215 - 292*m) - 1846*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/1014","A",1
939,1,156,179,0.3842535,"\int \frac{(2+3 x)^2 (1+4 x)^m}{\left(1-5 x+3 x^2\right)^2} \, dx","Integrate[((2 + 3*x)^2*(1 + 4*x)^m)/(1 - 5*x + 3*x^2)^2,x]","\frac{1}{507} (4 x+1)^{m+1} \left(-\frac{6 \left(\left(23 \sqrt{13}-377\right) m-153 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}-\frac{6 \left(\left(377+23 \sqrt{13}\right) m-153 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{793-1131 x}{3 x^2-5 x+1}\right)","-\frac{2 \left(153-\left(23-29 \sqrt{13}\right) m\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}+\frac{2 \left(153-\left(23+29 \sqrt{13}\right) m\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(61-87 x) (4 x+1)^{m+1}}{39 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((793 - 1131*x)/(1 - 5*x + 3*x^2) - (6*(-153*Sqrt[13] + (-377 + 23*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) - (6*(-153*Sqrt[13] + (377 + 23*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/507","A",1
940,1,149,179,0.2430605,"\int \frac{(2+3 x) (1+4 x)^m}{\left(1-5 x+3 x^2\right)^2} \, dx","Integrate[((2 + 3*x)*(1 + 4*x)^m)/(1 - 5*x + 3*x^2)^2,x]","\frac{1}{507} (4 x+1)^{m+1} \left(\frac{3 \left(182 m+\sqrt{13} (10 m+81)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}-\frac{3 \left(182 m-\sqrt{13} (10 m+81)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{260-273 x}{3 x^2-5 x+1}\right)","-\frac{\left(2 \left(5+7 \sqrt{13}\right) m+81\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}+\frac{\left(2 \left(5-7 \sqrt{13}\right) m+81\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(20-21 x) (4 x+1)^{m+1}}{39 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((260 - 273*x)/(1 - 5*x + 3*x^2) + (3*(182*m + Sqrt[13]*(81 + 10*m))*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) - (3*(182*m - Sqrt[13]*(81 + 10*m))*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/507","A",1
941,1,150,177,0.2034821,"\int \frac{(1+4 x)^m}{\left(1-5 x+3 x^2\right)^2} \, dx","Integrate[(1 + 4*x)^m/(1 - 5*x + 3*x^2)^2,x]","\frac{1}{507} (4 x+1)^{m+1} \left(\frac{6 \left(26 m+\sqrt{13} (4 m+9)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(2 \sqrt{13}-13\right) (m+1)}+\frac{6 \sqrt{13} \left(9-2 \left(\sqrt{13}-2\right) m\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{91-78 x}{3 x^2-5 x+1}\right)","-\frac{2 \left(2 \left(2+\sqrt{13}\right) m+9\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}+\frac{2 \left(2 \left(2-\sqrt{13}\right) m+9\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{13 \sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(7-6 x) (4 x+1)^{m+1}}{39 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((91 - 78*x)/(1 - 5*x + 3*x^2) + (6*(26*m + Sqrt[13]*(9 + 4*m))*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) + (6*Sqrt[13]*(9 - 2*(-2 + Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/507","A",1
942,1,274,340,0.7734824,"\int \frac{(1+4 x)^m}{(2+3 x) \left(1-5 x+3 x^2\right)^2} \, dx","Integrate[(1 + 4*x)^m/((2 + 3*x)*(1 - 5*x + 3*x^2)^2),x]","\frac{(4 x+1)^{m+1} \left(\frac{9126 \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{m+1}+\frac{1755 \left(13+9 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(13-2 \sqrt{13}\right) (m+1)}+\frac{1755 \left(13-9 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}+\frac{510 \sqrt{13} \left(\frac{\left(\left(62+22 \sqrt{13}\right) m+81\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{2 \sqrt{13}-13}+\frac{\left(\left(62-22 \sqrt{13}\right) m+81\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{13+2 \sqrt{13}}\right)}{m+1}+\frac{2210 (43-33 x)}{3 x^2-5 x+1}\right)}{1465230}","\frac{9 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{1445 (m+1)}-\frac{\left(\left(62+22 \sqrt{13}\right) m+81\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{221 \sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}+\frac{9 \left(13+9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{7514 \left(13-2 \sqrt{13}\right) (m+1)}+\frac{\left(\left(62-22 \sqrt{13}\right) m+81\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{221 \sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}+\frac{9 \left(13-9 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{7514 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{(43-33 x) (4 x+1)^{m+1}}{663 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((2210*(43 - 33*x))/(1 - 5*x + 3*x^2) + (9126*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(1 + m) + (1755*(13 + 9*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((13 - 2*Sqrt[13])*(1 + m)) + (1755*(13 - 9*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m)) + (510*Sqrt[13]*(((81 + (62 + 22*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/(-13 + 2*Sqrt[13]) + ((81 + (62 - 22*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/(13 + 2*Sqrt[13])))/(1 + m)))/1465230","A",1
943,1,287,376,0.6358265,"\int \frac{(1+4 x)^m}{(2+3 x)^2 \left(1-5 x+3 x^2\right)^2} \, dx","Integrate[(1 + 4*x)^m/((2 + 3*x)^2*(1 - 5*x + 3*x^2)^2),x]","\frac{(4 x+1)^{m+1} \left(\frac{1232010 \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{m+1}+\frac{26325 \left(117+64 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)}{\left(13-2 \sqrt{13}\right) (m+1)}+\frac{26325 \left(117-64 \sqrt{13}\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)}{\left(13+2 \sqrt{13}\right) (m+1)}-\frac{425 \left(\left(\left(2534+682 \sqrt{13}\right) m+423 \left(2+\sqrt{13}\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right)+\left(\left(2534-682 \sqrt{13}\right) m-423 \left(\sqrt{13}-2\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right)\right)}{m+1}+\frac{930852 \, _2F_1\left(2,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{m+1}+\frac{16575 (268-195 x)}{3 x^2-5 x+1}\right)}{186816825}","\frac{162 (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{24565 (m+1)}-\frac{\left(2 \left(211+65 \sqrt{13}\right) m+423\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{3757 \sqrt{13} \left(13-2 \sqrt{13}\right) (m+1)}+\frac{9 \left(117+64 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right)}{63869 \left(13-2 \sqrt{13}\right) (m+1)}+\frac{\left(\left(422-130 \sqrt{13}\right) m+423\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{3757 \sqrt{13} \left(13+2 \sqrt{13}\right) (m+1)}+\frac{9 \left(117-64 \sqrt{13}\right) (4 x+1)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right)}{63869 \left(13+2 \sqrt{13}\right) (m+1)}+\frac{36 (4 x+1)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{3}{5} (4 x+1)\right)}{7225 (m+1)}+\frac{(268-195 x) (4 x+1)^{m+1}}{11271 \left(3 x^2-5 x+1\right)}",1,"((1 + 4*x)^(1 + m)*((16575*(268 - 195*x))/(1 - 5*x + 3*x^2) + (1232010*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(1 + m) + (26325*(117 + 64*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((13 - 2*Sqrt[13])*(1 + m)) + (26325*(117 - 64*Sqrt[13])*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m)) - (425*((423*(2 + Sqrt[13]) + (2534 + 682*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])] + (-423*(-2 + Sqrt[13]) + (2534 - 682*Sqrt[13])*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])]))/(1 + m) + (930852*Hypergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(1 + m)))/186816825","A",1
944,1,171,237,0.2854895,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)}{(e+f x)^{3/2}} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(e + f*x)^(3/2),x]","\frac{2 (d+e x)^m \left(\frac{f (d+e x)}{d f-e^2}\right)^{-m} \left(-3 \left(f (a f-b e)+c e^2\right) \, _2F_1\left(-\frac{1}{2},-m;\frac{1}{2};\frac{e (e+f x)}{e^2-d f}\right)-(e+f x) \left((6 c e-3 b f) \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};\frac{e (e+f x)}{e^2-d f}\right)-c (e+f x) \, _2F_1\left(\frac{3}{2},-m;\frac{5}{2};\frac{e (e+f x)}{e^2-d f}\right)\right)\right)}{3 f^3 \sqrt{e+f x}}","\frac{2 \sqrt{e+f x} (d+e x)^m \left(-\frac{f (d+e x)}{e^2-d f}\right)^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};\frac{e (e+f x)}{e^2-d f}\right) \left(c \left(d^2 f^2+4 d e^2 f (m+1)-4 e^4 \left(m^2+3 m+2\right)\right)-e f (2 m+3) \left(a e f (2 m+1)+b \left(d f-2 e^2 (m+1)\right)\right)\right)}{e f^3 (2 m+3) \left(e^2-d f\right)}+\frac{2 (d+e x)^{m+1} \left(a+\frac{e (c e-b f)}{f^2}\right)}{\left(e^2-d f\right) \sqrt{e+f x}}+\frac{2 c \sqrt{e+f x} (d+e x)^{m+1}}{e f^2 (2 m+3)}",1,"(2*(d + e*x)^m*(-3*(c*e^2 + f*(-(b*e) + a*f))*Hypergeometric2F1[-1/2, -m, 1/2, (e*(e + f*x))/(e^2 - d*f)] - (e + f*x)*((6*c*e - 3*b*f)*Hypergeometric2F1[1/2, -m, 3/2, (e*(e + f*x))/(e^2 - d*f)] - c*(e + f*x)*Hypergeometric2F1[3/2, -m, 5/2, (e*(e + f*x))/(e^2 - d*f)])))/(3*f^3*((f*(d + e*x))/(-e^2 + d*f))^m*Sqrt[e + f*x])","A",1
945,0,0,509,1.3915973,"\int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^m*(f + g*x)^2*Sqrt[a + b*x + c*x^2],x]","\int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx","\frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) \left(e g^2 (m+1) (b d-a e)+c \left(3 d^2 g^2-2 d e f g (m+4)+e^2 f^2 (m+4)\right)\right)}{c e^3 (m+1) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} (b e g (2 m+5)+2 c (3 d g-2 e f (m+4))) F_1\left(m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c e^3 (m+2) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{g^2 \left(a+b x+c x^2\right)^{3/2} (d+e x)^{m+1}}{c e (m+4)}",1,"Integrate[(d + e*x)^m*(f + g*x)^2*Sqrt[a + b*x + c*x^2], x]","F",-1
946,0,0,388,0.7454816,"\int (d+e x)^m (f+g x) \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^m*(f + g*x)*Sqrt[a + b*x + c*x^2],x]","\int (d+e x)^m (f+g x) \sqrt{a+b x+c x^2} \, dx","\frac{\sqrt{a+b x+c x^2} (e f-d g) (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+1) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} F_1\left(m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+2) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"Integrate[(d + e*x)^m*(f + g*x)*Sqrt[a + b*x + c*x^2], x]","F",-1
947,1,207,189,0.0195612,"\int (d+e x)^m \sqrt{a+b x+c x^2} \, dx","Integrate[(d + e*x)^m*Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1) \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}}}","\frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}",1,"((d + e*x)^(1 + m)*Sqrt[a + x*(b + c*x)]*AppellF1[1 + m, -1/2, -1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)])","A",0
948,0,0,32,0.114084,"\int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx","Integrate[((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x),x]","\int \frac{(d+e x)^m \sqrt{a+b x+c x^2}}{f+g x} \, dx","\text{Int}\left(\frac{\sqrt{a+b x+c x^2} (d+e x)^m}{f+g x},x\right)",0,"Integrate[((d + e*x)^m*Sqrt[a + b*x + c*x^2])/(f + g*x), x]","A",-1
949,0,0,502,1.3023711,"\int \frac{(d+e x)^m (f+g x)^2}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)^m*(f + g*x)^2)/Sqrt[a + b*x + c*x^2],x]","\int \frac{(d+e x)^m (f+g x)^2}{\sqrt{a+b x+c x^2}} \, dx","\frac{(d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) \left(e g^2 (m+1) (b d-a e)+c \left(d^2 g^2-2 d e f g (m+2)+e^2 f^2 (m+2)\right)\right)}{c e^3 (m+1) (m+2) \sqrt{a+b x+c x^2}}-\frac{g (d+e x)^{m+2} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} (b e g (2 m+3)+c (2 d g-4 e f (m+2))) F_1\left(m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{2 c e^3 (m+2)^2 \sqrt{a+b x+c x^2}}+\frac{g^2 \sqrt{a+b x+c x^2} (d+e x)^{m+1}}{c e (m+2)}",1,"Integrate[((d + e*x)^m*(f + g*x)^2)/Sqrt[a + b*x + c*x^2], x]","F",-1
950,0,0,388,0.7759378,"\int \frac{(d+e x)^m (f+g x)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)^m*(f + g*x))/Sqrt[a + b*x + c*x^2],x]","\int \frac{(d+e x)^m (f+g x)}{\sqrt{a+b x+c x^2}} \, dx","\frac{(e f-d g) (d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+1) \sqrt{a+b x+c x^2}}+\frac{g (d+e x)^{m+2} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F_1\left(m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+2) \sqrt{a+b x+c x^2}}",1,"Integrate[((d + e*x)^m*(f + g*x))/Sqrt[a + b*x + c*x^2], x]","F",-1
951,1,207,189,0.0251858,"\int \frac{(d+e x)^m}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^m/Sqrt[a + b*x + c*x^2],x]","\frac{(d+e x)^{m+1} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}} \sqrt{\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1) \sqrt{a+x (b+c x)}}","\frac{(d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1) \sqrt{a+b x+c x^2}}",1,"(Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*Sqrt[a + x*(b + c*x)])","A",0
952,0,0,32,0.120874,"\int \frac{(d+e x)^m}{(f+g x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x)^m/((f + g*x)*Sqrt[a + b*x + c*x^2]),x]","\int \frac{(d+e x)^m}{(f+g x) \sqrt{a+b x+c x^2}} \, dx","\text{Int}\left(\frac{(d+e x)^m}{(f+g x) \sqrt{a+b x+c x^2}},x\right)",0,"Integrate[(d + e*x)^m/((f + g*x)*Sqrt[a + b*x + c*x^2]), x]","A",-1
953,1,187,265,0.2158848,"\int (d+e x)^m (f+g x)^n \left(a+b x+c x^2\right) \, dx","Integrate[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2),x]","\frac{(d+e x)^{m+1} (f+g x)^n \left(\frac{e (f+g x)}{e f-d g}\right)^{-n} \left(e \left(e \left(g (a g-b f)+c f^2\right) \, _2F_1\left(m+1,-n;m+2;\frac{g (d+e x)}{d g-e f}\right)-(2 c f-b g) (e f-d g) \, _2F_1\left(m+1,-n-1;m+2;\frac{g (d+e x)}{d g-e f}\right)\right)+c (e f-d g)^2 \, _2F_1\left(m+1,-n-2;m+2;\frac{g (d+e x)}{d g-e f}\right)\right)}{e^3 g^2 (m+1)}","\frac{(d+e x)^{m+1} (f+g x)^n \left(\frac{e (f+g x)}{e f-d g}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right) \left(g (m+n+2) \left(a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right)-(d g (n+1)+e f (m+1)) (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))\right)}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}+\frac{(d+e x)^{m+1} (f+g x)^{n+1} (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)}",1,"((d + e*x)^(1 + m)*(f + g*x)^n*(c*(e*f - d*g)^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e*(-((2*c*f - b*g)*(e*f - d*g)*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)]) + e*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[1 + m, -n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])))/(e^3*g^2*(1 + m)*((e*(f + g*x))/(e*f - d*g))^n)","A",1
954,0,0,525,2.2447084,"\int (d+e x)^m (f+g x)^2 \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2)^p,x]","\int (d+e x)^m (f+g x)^2 \left(a+b x+c x^2\right)^p \, dx","\frac{(d+e x)^{m+1} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right) \left(e g^2 (m+1) (b d-a e)+c \left(2 d^2 g^2 (p+1)-2 d e f g (m+2 p+3)+e^2 f^2 (m+2 p+3)\right)\right)}{c e^3 (m+1) (m+2 p+3)}-\frac{g (d+e x)^{m+2} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} (b e g (m+p+2)+2 c (d g (p+1)-e f (m+2 p+3))) F_1\left(m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c e^3 (m+2) (m+2 p+3)}+\frac{g^2 (d+e x)^{m+1} \left(a+b x+c x^2\right)^{p+1}}{c e (m+2 p+3)}",1,"Integrate[(d + e*x)^m*(f + g*x)^2*(a + b*x + c*x^2)^p, x]","F",-1
955,0,0,384,1.321249,"\int (d+e x)^m (f+g x) \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p,x]","\int (d+e x)^m (f+g x) \left(a+b x+c x^2\right)^p \, dx","\frac{(e f-d g) (d+e x)^{m+1} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+1)}+\frac{g (d+e x)^{m+2} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e^2 (m+2)}",1,"Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x]","F",-1
956,1,205,187,0.1005055,"\int (d+e x)^m \left(a+b x+c x^2\right)^p \, dx","Integrate[(d + e*x)^m*(a + b*x + c*x^2)^p,x]","\frac{(d+e x)^{m+1} (a+x (b+c x))^p \left(\frac{e \left(\sqrt{b^2-4 a c}-b-2 c x\right)}{e \left(\sqrt{b^2-4 a c}-b\right)+2 c d}\right)^{-p} \left(\frac{e \left(\sqrt{b^2-4 a c}+b+2 c x\right)}{e \left(\sqrt{b^2-4 a c}+b\right)-2 c d}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d+\left(\sqrt{b^2-4 a c}-b\right) e}\right)}{e (m+1)}","\frac{(d+e x)^{m+1} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (d+e x)}{2 c d-e \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left(b-\sqrt{b^2-4 a c}\right) e},\frac{2 c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{e (m+1)}",1,"((d + e*x)^(1 + m)*(a + x*(b + c*x))^p*AppellF1[1 + m, -p, -p, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*((e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e))^p)","A",0
957,0,0,30,0.1697727,"\int \frac{(d+e x)^m \left(a+b x+c x^2\right)^p}{f+g x} \, dx","Integrate[((d + e*x)^m*(a + b*x + c*x^2)^p)/(f + g*x),x]","\int \frac{(d+e x)^m \left(a+b x+c x^2\right)^p}{f+g x} \, dx","\text{Int}\left(\frac{(d+e x)^m \left(a+b x+c x^2\right)^p}{f+g x},x\right)",0,"Integrate[((d + e*x)^m*(a + b*x + c*x^2)^p)/(f + g*x), x]","A",-1
958,1,188,89,0.6359745,"\int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx","Integrate[1/(Sqrt[1 - 1/(c^2*x^2)]*x^2*Sqrt[d + e*x]),x]","-\frac{2 i (d+e x) \sqrt{\frac{e (c x-1)}{c (d+e x)}} \sqrt{\frac{c e x+e}{c d+c e x}} \left(F\left(i \sinh ^{-1}\left(\frac{\sqrt{-\frac{c d+e}{c}}}{\sqrt{d+e x}}\right)|\frac{c d-e}{c d+e}\right)-\Pi \left(\frac{c d}{c d+e};i \sinh ^{-1}\left(\frac{\sqrt{-\frac{c d+e}{c}}}{\sqrt{d+e x}}\right)|\frac{c d-e}{c d+e}\right)\right)}{d x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c d+e}{c}}}","-\frac{2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"((-2*I)*Sqrt[(e*(-1 + c*x))/(c*(d + e*x))]*(d + e*x)*Sqrt[(e + c*e*x)/(c*d + c*e*x)]*(EllipticF[I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x]], (c*d - e)/(c*d + e)] - EllipticPi[(c*d)/(c*d + e), I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x]], (c*d - e)/(c*d + e)]))/(d*Sqrt[-((c*d + e)/c)]*Sqrt[1 - 1/(c^2*x^2)]*x)","C",1