1,1,226,236,0.5063613,"\int (d+e x)^2 \left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2} \, dx","Integrate[(d + e*x)^2*(A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2],x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 \sin ^{-1}\left(\frac{e x}{d}\right) \left(2 d^3 e (5 A e+2 B d)+3 C d^5\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(2 e \left(5 A e \left(-16 d^3+9 d^2 e x+16 d e^2 x^2+6 e^3 x^3\right)+B \left(-56 d^4-30 d^3 e x+32 d^2 e^2 x^2+60 d e^3 x^3+24 e^4 x^4\right)\right)+C \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)\right)\right)}{240 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d^2 x \sqrt{d^2-e^2 x^2} \left(10 A e^2+4 B d e+3 C d^2\right)}{16 e^2}-\frac{x \left(d^2-e^2 x^2\right)^{3/2} \left(2 e (A e+2 B d)+3 C d^2\right)}{8 e^2}-\frac{d \left(d^2-e^2 x^2\right)^{3/2} \left(e (10 A e+7 B d)+4 C d^2\right)}{15 e^3}+\frac{d^4 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(10 A e^2+4 B d e+3 C d^2\right)}{16 e^3}-\frac{x^2 \left(d^2-e^2 x^2\right)^{3/2} (B e+2 C d)}{5 e}-\frac{1}{6} C x^3 \left(d^2-e^2 x^2\right)^{3/2}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(C*(-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) + 2*e*(5*A*e*(-16*d^3 + 9*d^2*e*x + 16*d*e^2*x^2 + 6*e^3*x^3) + B*(-56*d^4 - 30*d^3*e*x + 32*d^2*e^2*x^2 + 60*d*e^3*x^3 + 24*e^4*x^4))) + 15*(3*C*d^5 + 2*d^3*e*(2*B*d + 5*A*e))*ArcSin[(e*x)/d]))/(240*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
2,1,190,186,0.2990486,"\int (d+e x) \left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2} \, dx","Integrate[(d + e*x)*(A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2],x]","\frac{\sqrt{d^2-e^2 x^2} \left(15 \sin ^{-1}\left(\frac{e x}{d}\right) \left(d^2 e (4 A e+B d)+C d^4\right)+\sqrt{1-\frac{e^2 x^2}{d^2}} \left(5 e \left(4 A e \left(-2 d^2+3 d e x+2 e^2 x^2\right)+B \left(-8 d^3-3 d^2 e x+8 d e^2 x^2+6 e^3 x^3\right)\right)+C \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)\right)}{120 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{d x \sqrt{d^2-e^2 x^2} \left(e (4 A e+B d)+C d^2\right)}{8 e^2}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(5 e (A e+B d)+2 C d^2\right)}{15 e^3}+\frac{d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(e (4 A e+B d)+C d^2\right)}{8 e^3}-\frac{x \left(d^2-e^2 x^2\right)^{3/2} (B e+C d)}{4 e^2}-\frac{C x^2 \left(d^2-e^2 x^2\right)^{3/2}}{5 e}",1,"(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(C*(-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) + 5*e*(4*A*e*(-2*d^2 + 3*d*e*x + 2*e^2*x^2) + B*(-8*d^3 - 3*d^2*e*x + 8*d*e^2*x^2 + 6*e^3*x^3))) + 15*(C*d^4 + d^2*e*(B*d + 4*A*e))*ArcSin[(e*x)/d]))/(120*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
3,1,121,125,0.1332821,"\int \left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2} \, dx","Integrate[(A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2],x]","\frac{\sqrt{d^2-e^2 x^2} \left(e \sqrt{1-\frac{e^2 x^2}{d^2}} \left(12 A e^2 x-8 B d^2+8 B e^2 x^2-3 C d^2 x+6 C e^2 x^3\right)+3 \left(4 A d e^2+C d^3\right) \sin ^{-1}\left(\frac{e x}{d}\right)\right)}{24 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}","\frac{1}{8} x \sqrt{d^2-e^2 x^2} \left(4 A+\frac{C d^2}{e^2}\right)+\frac{d^2 \left(4 A e^2+C d^2\right) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{8 e^3}-\frac{B \left(d^2-e^2 x^2\right)^{3/2}}{3 e^2}-\frac{C x \left(d^2-e^2 x^2\right)^{3/2}}{4 e^2}",1,"(Sqrt[d^2 - e^2*x^2]*(e*Sqrt[1 - (e^2*x^2)/d^2]*(-8*B*d^2 - 3*C*d^2*x + 12*A*e^2*x + 8*B*e^2*x^2 + 6*C*e^2*x^3) + 3*(C*d^3 + 4*A*d*e^2)*ArcSin[(e*x)/d]))/(24*e^3*Sqrt[1 - (e^2*x^2)/d^2])","A",1
4,1,103,148,0.220709,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{d+e x} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x),x]","\frac{\sqrt{d^2-e^2 x^2} \left(3 e (2 A e-2 B d+B e x)+C \left(4 d^2-3 d e x+2 e^2 x^2\right)\right)+3 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(e (2 A e-B d)+C d^2\right)}{6 e^3}","\frac{\sqrt{d^2-e^2 x^2} \left(C d^2-e (B d-2 A e)\right)}{2 e^3}+\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(C d^2-e (B d-2 A e)\right)}{2 e^3}+\frac{\left(d^2-e^2 x^2\right)^{3/2} (C d-B e)}{2 e^3 (d+e x)}-\frac{C \left(d^2-e^2 x^2\right)^{3/2}}{3 e^3}",1,"(Sqrt[d^2 - e^2*x^2]*(3*e*(-2*B*d + 2*A*e + B*e*x) + C*(4*d^2 - 3*d*e*x + 2*e^2*x^2)) + 3*d*(C*d^2 + e*(-(B*d) + 2*A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^3)","A",1
5,1,109,170,0.2272308,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{(d+e x)^2} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x)^2,x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(2 e (-2 A e+3 B d+B e x)+C \left(-8 d^2-3 d e x+e^2 x^2\right)\right)}{d+e x}-\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(2 e (A e-2 B d)+5 C d^2\right)}{2 e^3}","-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{d e^3 (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2} \left(5 C d^2-2 e (2 B d-A e)\right)}{2 d e^3}-\frac{\tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(5 C d^2-2 e (2 B d-A e)\right)}{2 e^3}-\frac{C \left(d^2-e^2 x^2\right)^{3/2}}{2 e^3 (d+e x)}",1,"((Sqrt[d^2 - e^2*x^2]*(2*e*(3*B*d - 2*A*e + B*e*x) + C*(-8*d^2 - 3*d*e*x + e^2*x^2)))/(d + e*x) - (5*C*d^2 + 2*e*(-2*B*d + A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^3)","A",1
6,1,114,149,0.2207082,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x)^3,x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(e (A e (e x-d)-B d (5 d+7 e x))+C d \left(14 d^2+19 d e x+3 e^2 x^2\right)\right)}{d (d+e x)^2}+3 (3 C d-B e) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{3 e^3}","-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{3 d e^3 (d+e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2} (3 C d-B e)}{e^3 (d+e x)}+\frac{(3 C d-B e) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}-\frac{C \left(d^2-e^2 x^2\right)^{3/2}}{e^3 (d+e x)^2}",1,"((Sqrt[d^2 - e^2*x^2]*(C*d*(14*d^2 + 19*d*e*x + 3*e^2*x^2) + e*(A*e*(-d + e*x) - B*d*(5*d + 7*e*x))))/(d*(d + e*x)^2) + 3*(3*C*d - B*e)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e^3)","A",1
7,1,112,196,0.2984586,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]","-\frac{\frac{\sqrt{d^2-e^2 x^2} \left(e (d-e x) (A e (4 d+e x)+B d (d+4 e x))+3 C d^2 \left(8 d^2+19 d e x+13 e^2 x^2\right)\right)}{d^2 (d+e x)^3}+15 C \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{15 e^3}","-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{15 d^2 e^3 (d+e x)^3}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{5 d e^3 (d+e x)^4}+\frac{\left(d^2-e^2 x^2\right)^{3/2} (2 C d-B e)}{3 d e^3 (d+e x)^3}-\frac{2 C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{C \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}",1,"-1/15*((Sqrt[d^2 - e^2*x^2]*(3*C*d^2*(8*d^2 + 19*d*e*x + 13*e^2*x^2) + e*(d - e*x)*(A*e*(4*d + e*x) + B*d*(d + 4*e*x))))/(d^2*(d + e*x)^3) + 15*C*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^3","A",1
8,1,109,180,0.2033117,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{(d+e x)^5} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x)^5,x]","-\frac{(d-e x) \sqrt{d^2-e^2 x^2} \left(e \left(A e \left(23 d^2+10 d e x+2 e^2 x^2\right)+5 B d \left(d^2+5 d e x+e^2 x^2\right)\right)+C d^2 \left(2 d^2+10 d e x+23 e^2 x^2\right)\right)}{105 d^3 e^3 (d+e x)^4}","-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(e (2 A e+5 B d)+23 C d^2\right)}{35 d^2 e^3 (d+e x)^4}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{7 d e^3 (d+e x)^5}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(e (2 A e+5 B d)+23 C d^2\right)}{105 d^3 e^3 (d+e x)^3}+\frac{C \left(d^2-e^2 x^2\right)^{3/2}}{e^3 (d+e x)^4}",1,"-1/105*((d - e*x)*Sqrt[d^2 - e^2*x^2]*(C*d^2*(2*d^2 + 10*d*e*x + 23*e^2*x^2) + e*(5*B*d*(d^2 + 5*d*e*x + e^2*x^2) + A*e*(23*d^2 + 10*d*e*x + 2*e^2*x^2))))/(d^3*e^3*(d + e*x)^4)","A",1
9,1,144,234,0.2248387,"\int \frac{\left(A+B x+C x^2\right) \sqrt{d^2-e^2 x^2}}{(d+e x)^6} \, dx","Integrate[((A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2])/(d + e*x)^6,x]","-\frac{(d-e x) \sqrt{d^2-e^2 x^2} \left(e \left(A e \left(58 d^3+33 d^2 e x+12 d e^2 x^2+2 e^3 x^3\right)+B d \left(11 d^3+66 d^2 e x+24 d e^2 x^2+4 e^3 x^3\right)\right)+C d^2 \left(4 d^3+24 d^2 e x+66 d e^2 x^2+11 e^3 x^3\right)\right)}{315 d^4 e^3 (d+e x)^5}","-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(2 e (A e+2 B d)+11 C d^2\right)}{42 d^2 e^3 (d+e x)^5}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(A e^2-B d e+C d^2\right)}{9 d e^3 (d+e x)^6}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(2 e (A e+2 B d)+11 C d^2\right)}{315 d^4 e^3 (d+e x)^3}-\frac{\left(d^2-e^2 x^2\right)^{3/2} \left(2 e (A e+2 B d)+11 C d^2\right)}{105 d^3 e^3 (d+e x)^4}+\frac{C \left(d^2-e^2 x^2\right)^{3/2}}{2 e^3 (d+e x)^5}",1,"-1/315*((d - e*x)*Sqrt[d^2 - e^2*x^2]*(C*d^2*(4*d^3 + 24*d^2*e*x + 66*d*e^2*x^2 + 11*e^3*x^3) + e*(A*e*(58*d^3 + 33*d^2*e*x + 12*d*e^2*x^2 + 2*e^3*x^3) + B*d*(11*d^3 + 66*d^2*e*x + 24*d*e^2*x^2 + 4*e^3*x^3))))/(d^4*e^3*(d + e*x)^5)","A",1
10,1,174,236,0.4528175,"\int \frac{(d+e x)^3 \left(A+B x+C x^2\right)}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[((d + e*x)^3*(A + B*x + C*x^2))/Sqrt[d^2 - e^2*x^2],x]","\frac{15 d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(5 e (4 A e+3 B d)+13 C d^2\right)-\sqrt{d^2-e^2 x^2} \left(5 e \left(4 A e \left(22 d^2+9 d e x+2 e^2 x^2\right)+3 B \left(24 d^3+15 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right)\right)+C \left(304 d^4+195 d^3 e x+152 d^2 e^2 x^2+90 d e^3 x^3+24 e^4 x^4\right)\right)}{120 e^3}","-\frac{x^2 \sqrt{d^2-e^2 x^2} \left(5 e (A e+3 B d)+19 C d^2\right)}{15 e}-\frac{d x \sqrt{d^2-e^2 x^2} \left(12 A e^2+15 B d e+13 C d^2\right)}{8 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2} \left(55 A e^2+45 B d e+38 C d^2\right)}{15 e^3}+\frac{d^3 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(20 A e^2+15 B d e+13 C d^2\right)}{8 e^3}-\frac{1}{4} x^3 \sqrt{d^2-e^2 x^2} (B e+3 C d)-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}",1,"(-(Sqrt[d^2 - e^2*x^2]*(C*(304*d^4 + 195*d^3*e*x + 152*d^2*e^2*x^2 + 90*d*e^3*x^3 + 24*e^4*x^4) + 5*e*(4*A*e*(22*d^2 + 9*d*e*x + 2*e^2*x^2) + 3*B*(24*d^3 + 15*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3)))) + 15*d^3*(13*C*d^2 + 5*e*(3*B*d + 4*A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(120*e^3)","A",1
11,1,139,191,0.1774015,"\int \frac{(d+e x)^2 \left(A+B x+C x^2\right)}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[((d + e*x)^2*(A + B*x + C*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) \left(4 e (3 A e+2 B d)+7 C d^2\right)-\sqrt{d^2-e^2 x^2} \left(4 e \left(3 A e (4 d+e x)+2 B \left(5 d^2+3 d e x+e^2 x^2\right)\right)+C \left(32 d^3+21 d^2 e x+16 d e^2 x^2+6 e^3 x^3\right)\right)}{24 e^3}","-\frac{x \sqrt{d^2-e^2 x^2} \left(4 e (A e+2 B d)+7 C d^2\right)}{8 e^2}-\frac{d \sqrt{d^2-e^2 x^2} \left(e (6 A e+5 B d)+4 C d^2\right)}{3 e^3}+\frac{d^2 \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(12 A e^2+8 B d e+7 C d^2\right)}{8 e^3}-\frac{x^2 \sqrt{d^2-e^2 x^2} (B e+2 C d)}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}",1,"(-(Sqrt[d^2 - e^2*x^2]*(C*(32*d^3 + 21*d^2*e*x + 16*d*e^2*x^2 + 6*e^3*x^3) + 4*e*(3*A*e*(4*d + e*x) + 2*B*(5*d^2 + 3*d*e*x + e^2*x^2)))) + 3*d^2*(7*C*d^2 + 4*e*(2*B*d + 3*A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e^3)","A",1
12,1,103,143,0.1072733,"\int \frac{(d+e x) \left(A+B x+C x^2\right)}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[((d + e*x)*(A + B*x + C*x^2))/Sqrt[d^2 - e^2*x^2],x]","\frac{3 d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(e (2 A e+B d)+C d^2\right)-\sqrt{d^2-e^2 x^2} \left(3 e (2 A e+2 B d+B e x)+C \left(4 d^2+3 d e x+2 e^2 x^2\right)\right)}{6 e^3}","-\frac{\sqrt{d^2-e^2 x^2} \left(3 e (A e+B d)+2 C d^2\right)}{3 e^3}+\frac{d \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right) \left(e (2 A e+B d)+C d^2\right)}{2 e^3}-\frac{x \sqrt{d^2-e^2 x^2} (B e+C d)}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}",1,"(-(Sqrt[d^2 - e^2*x^2]*(3*e*(2*B*d + 2*A*e + B*e*x) + C*(4*d^2 + 3*d*e*x + 2*e^2*x^2))) + 3*d*(C*d^2 + e*(B*d + 2*A*e))*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(6*e^3)","A",1
13,1,67,87,0.039824,"\int \frac{A+B x+C x^2}{\sqrt{d^2-e^2 x^2}} \, dx","Integrate[(A + B*x + C*x^2)/Sqrt[d^2 - e^2*x^2],x]","\frac{\left(2 A e^2+C d^2\right) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-e (2 B+C x) \sqrt{d^2-e^2 x^2}}{2 e^3}","\frac{\left(2 A e^2+C d^2\right) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{2 e^3}-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2}",1,"(-(e*(2*B + C*x)*Sqrt[d^2 - e^2*x^2]) + (C*d^2 + 2*A*e^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^3)","A",1
14,1,83,103,0.1575,"\int \frac{A+B x+C x^2}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]","\frac{(B e-C d) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)-\frac{\sqrt{d^2-e^2 x^2} (e (A e-B d)+C d (2 d+e x))}{d (d+e x)}}{e^3}","-\frac{\sqrt{d^2-e^2 x^2} \left(A e^2-B d e+C d^2\right)}{d e^3 (d+e x)}-\frac{(C d-B e) \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}-\frac{C \sqrt{d^2-e^2 x^2}}{e^3}",1,"(-((Sqrt[d^2 - e^2*x^2]*(e*(-(B*d) + A*e) + C*d*(2*d + e*x)))/(d*(d + e*x))) + (-(C*d) + B*e)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^3","A",1
15,1,95,163,0.2180256,"\int \frac{A+B x+C x^2}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^2*Sqrt[d^2 - e^2*x^2]),x]","\frac{\frac{\sqrt{d^2-e^2 x^2} \left(C d^2 (4 d+5 e x)-e (A e (2 d+e x)+B d (d+2 e x))\right)}{d^2 (d+e x)^2}+3 C \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{3 e^3}","-\frac{\sqrt{d^2-e^2 x^2} \left(A e^2-B d e+C d^2\right)}{3 d^2 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left(A e^2-B d e+C d^2\right)}{3 d e^3 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2} (2 C d-B e)}{d e^3 (d+e x)}+\frac{C \tan ^{-1}\left(\frac{e x}{\sqrt{d^2-e^2 x^2}}\right)}{e^3}",1,"((Sqrt[d^2 - e^2*x^2]*(C*d^2*(4*d + 5*e*x) - e*(A*e*(2*d + e*x) + B*d*(d + 2*e*x))))/(d^2*(d + e*x)^2) + 3*C*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(3*e^3)","A",1
16,1,103,180,0.1970583,"\int \frac{A+B x+C x^2}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(e \left(A e \left(7 d^2+6 d e x+2 e^2 x^2\right)+3 B d \left(d^2+3 d e x+e^2 x^2\right)\right)+C d^2 \left(2 d^2+6 d e x+7 e^2 x^2\right)\right)}{15 d^3 e^3 (d+e x)^3}","-\frac{\sqrt{d^2-e^2 x^2} \left(e (2 A e+3 B d)+7 C d^2\right)}{15 d^2 e^3 (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2} \left(A e^2-B d e+C d^2\right)}{5 d e^3 (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2} \left(e (2 A e+3 B d)+7 C d^2\right)}{15 d^3 e^3 (d+e x)}+\frac{C \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)^2}",1,"-1/15*(Sqrt[d^2 - e^2*x^2]*(C*d^2*(2*d^2 + 6*d*e*x + 7*e^2*x^2) + e*(3*B*d*(d^2 + 3*d*e*x + e^2*x^2) + A*e*(7*d^2 + 6*d*e*x + 2*e^2*x^2))))/(d^3*e^3*(d + e*x)^3)","A",1
17,1,139,234,0.2212124,"\int \frac{A+B x+C x^2}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]","-\frac{\sqrt{d^2-e^2 x^2} \left(e \left(3 A e \left(12 d^3+13 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right)+B d \left(13 d^3+52 d^2 e x+32 d e^2 x^2+8 e^3 x^3\right)\right)+C d^2 \left(8 d^3+32 d^2 e x+52 d e^2 x^2+13 e^3 x^3\right)\right)}{105 d^4 e^3 (d+e x)^4}","-\frac{\sqrt{d^2-e^2 x^2} \left(6 A e^2+8 B d e+13 C d^2\right)}{70 d^2 e^3 (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2} \left(A e^2-B d e+C d^2\right)}{7 d e^3 (d+e x)^4}-\frac{\sqrt{d^2-e^2 x^2} \left(6 A e^2+8 B d e+13 C d^2\right)}{105 d^4 e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2} \left(6 A e^2+8 B d e+13 C d^2\right)}{105 d^3 e^3 (d+e x)^2}+\frac{C \sqrt{d^2-e^2 x^2}}{2 e^3 (d+e x)^3}",1,"-1/105*(Sqrt[d^2 - e^2*x^2]*(C*d^2*(8*d^3 + 32*d^2*e*x + 52*d*e^2*x^2 + 13*e^3*x^3) + e*(3*A*e*(12*d^3 + 13*d^2*e*x + 8*d*e^2*x^2 + 2*e^3*x^3) + B*d*(13*d^3 + 52*d^2*e*x + 32*d*e^2*x^2 + 8*e^3*x^3))))/(d^4*e^3*(d + e*x)^4)","A",1
18,1,208,175,0.0870373,"\int (d+e x)^3 \left(a+c x^2\right) \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^3*(a + c*x^2)*(A + B*x + C*x^2),x]","\frac{1}{5} x^5 \left(a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right)+\frac{1}{6} e x^6 \left(a C e^2+c e (A e+3 B d)+3 c C d^2\right)+\frac{1}{3} d x^3 \left(A \left(3 a e^2+c d^2\right)+a d (3 B e+C d)\right)+\frac{1}{4} x^4 \left(a A e^3+3 a B d e^2+3 a C d^2 e+3 A c d^2 e+B c d^3\right)+\frac{1}{2} a d^2 x^2 (3 A e+B d)+a A d^3 x+\frac{1}{7} c e^2 x^7 (B e+3 C d)+\frac{1}{8} c C e^3 x^8","\frac{(d+e x)^6 \left(a C e^2+c \left(6 C d^2-e (3 B d-A e)\right)\right)}{6 e^5}-\frac{(d+e x)^5 \left(a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)\right)}{5 e^5}+\frac{(d+e x)^4 \left(a e^2+c d^2\right) \left(A e^2-B d e+C d^2\right)}{4 e^5}-\frac{c (d+e x)^7 (4 C d-B e)}{7 e^5}+\frac{c C (d+e x)^8}{8 e^5}",1,"a*A*d^3*x + (a*d^2*(B*d + 3*A*e)*x^2)/2 + (d*(a*d*(C*d + 3*B*e) + A*(c*d^2 + 3*a*e^2))*x^3)/3 + ((B*c*d^3 + 3*A*c*d^2*e + 3*a*C*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + ((c*C*d^3 + 3*c*d*e*(B*d + A*e) + a*e^2*(3*C*d + B*e))*x^5)/5 + (e*(3*c*C*d^2 + a*C*e^2 + c*e*(3*B*d + A*e))*x^6)/6 + (c*e^2*(3*C*d + B*e)*x^7)/7 + (c*C*e^3*x^8)/8","A",1
19,1,150,175,0.0602131,"\int (d+e x)^2 \left(a+c x^2\right) \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^2*(a + c*x^2)*(A + B*x + C*x^2),x]","\frac{1}{5} x^5 \left(a C e^2+A c e^2+2 B c d e+c C d^2\right)+\frac{1}{4} x^4 \left(a B e^2+2 a C d e+2 A c d e+B c d^2\right)+\frac{1}{3} x^3 \left(a A e^2+2 a B d e+a C d^2+A c d^2\right)+\frac{1}{2} a d x^2 (2 A e+B d)+a A d^2 x+\frac{1}{6} c e x^6 (B e+2 C d)+\frac{1}{7} c C e^2 x^7","\frac{(d+e x)^5 \left(a C e^2+c \left(6 C d^2-e (3 B d-A e)\right)\right)}{5 e^5}-\frac{(d+e x)^4 \left(a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)\right)}{4 e^5}+\frac{(d+e x)^3 \left(a e^2+c d^2\right) \left(A e^2-B d e+C d^2\right)}{3 e^5}-\frac{c (d+e x)^6 (4 C d-B e)}{6 e^5}+\frac{c C (d+e x)^7}{7 e^5}",1,"a*A*d^2*x + (a*d*(B*d + 2*A*e)*x^2)/2 + ((A*c*d^2 + a*C*d^2 + 2*a*B*d*e + a*A*e^2)*x^3)/3 + ((B*c*d^2 + 2*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x^4)/4 + ((c*C*d^2 + 2*B*c*d*e + A*c*e^2 + a*C*e^2)*x^5)/5 + (c*e*(2*C*d + B*e)*x^6)/6 + (c*C*e^2*x^7)/7","A",1
20,1,86,86,0.0291793,"\int (d+e x) \left(a+c x^2\right) \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)*(a + c*x^2)*(A + B*x + C*x^2),x]","\frac{1}{4} x^4 (a C e+A c e+B c d)+\frac{1}{3} x^3 (a B e+a C d+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{5} c x^5 (B e+C d)+\frac{1}{6} c C e x^6","\frac{1}{4} x^4 (a C e+A c e+B c d)+\frac{1}{3} x^3 (a B e+a C d+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{5} c x^5 (B e+C d)+\frac{1}{6} c C e x^6",1,"a*A*d*x + (a*(B*d + A*e)*x^2)/2 + ((A*c*d + a*C*d + a*B*e)*x^3)/3 + ((B*c*d + A*c*e + a*C*e)*x^4)/4 + (c*(C*d + B*e)*x^5)/5 + (c*C*e*x^6)/6","A",1
21,1,46,46,0.0130096,"\int \left(a+c x^2\right) \left(A+B x+C x^2\right) \, dx","Integrate[(a + c*x^2)*(A + B*x + C*x^2),x]","\frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} a B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5","\frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} a B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5",1,"a*A*x + (a*B*x^2)/2 + ((A*c + a*C)*x^3)/3 + (B*c*x^4)/4 + (c*C*x^5)/5","A",1
22,1,136,145,0.0736305,"\int \frac{\left(a+c x^2\right) \left(A+B x+C x^2\right)}{d+e x} \, dx","Integrate[((a + c*x^2)*(A + B*x + C*x^2))/(d + e*x),x]","\frac{12 \left(a e^2+c d^2\right) \log (d+e x) \left(e (A e-B d)+C d^2\right)+e x \left(6 a e^2 (2 B e-2 C d+C e x)+2 c e \left(3 A e (e x-2 d)+B \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)+c C \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)}{12 e^5}","\frac{\left(a e^2+c d^2\right) \log (d+e x) \left(A e^2-B d e+C d^2\right)}{e^5}-\frac{x \left(a e^2 (C d-B e)+c d \left(C d^2-e (B d-A e)\right)\right)}{e^4}+\frac{x^2 \left(a C e^2+c \left(C d^2-e (B d-A e)\right)\right)}{2 e^3}-\frac{c x^3 (C d-B e)}{3 e^2}+\frac{c C x^4}{4 e}",1,"(e*x*(6*a*e^2*(-2*C*d + 2*B*e + C*e*x) + c*C*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*c*e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + 12*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x])/(12*e^5)","A",1
23,1,142,153,0.1536663,"\int \frac{\left(a+c x^2\right) \left(A+B x+C x^2\right)}{(d+e x)^2} \, dx","Integrate[((a + c*x^2)*(A + B*x + C*x^2))/(d + e*x)^2,x]","\frac{6 \log (d+e x) \left(a e^2 (B e-2 C d)+c d e (3 B d-2 A e)-4 c C d^3\right)+6 e x \left(a C e^2+c e (A e-2 B d)+3 c C d^2\right)-\frac{6 \left(a e^2+c d^2\right) \left(e (A e-B d)+C d^2\right)}{d+e x}+3 c e^2 x^2 (B e-2 C d)+2 c C e^3 x^3}{6 e^5}","-\frac{\left(a e^2+c d^2\right) \left(A e^2-B d e+C d^2\right)}{e^5 (d+e x)}-\frac{\log (d+e x) \left(a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)\right)}{e^5}+\frac{x \left(a C e^2+c \left(3 C d^2-e (2 B d-A e)\right)\right)}{e^4}-\frac{c x^2 (2 C d-B e)}{2 e^3}+\frac{c C x^3}{3 e^2}",1,"(6*e*(3*c*C*d^2 + a*C*e^2 + c*e*(-2*B*d + A*e))*x + 3*c*e^2*(-2*C*d + B*e)*x^2 + 2*c*C*e^3*x^3 - (6*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + 6*(-4*c*C*d^3 + c*d*e*(3*B*d - 2*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x])/(6*e^5)","A",1
24,1,176,156,0.0997665,"\int \frac{\left(a+c x^2\right) \left(A+B x+C x^2\right)}{(d+e x)^3} \, dx","Integrate[((a + c*x^2)*(A + B*x + C*x^2))/(d + e*x)^3,x]","\frac{\log (d+e x) \left(a C e^2+A c e^2-3 B c d e+6 c C d^2\right)}{e^5}+\frac{-a B e^3+2 a C d e^2+2 A c d e^2-3 B c d^2 e+4 c C d^3}{e^5 (d+e x)}+\frac{-a A e^4+a B d e^3-a C d^2 e^2-A c d^2 e^2+B c d^3 e-c C d^4}{2 e^5 (d+e x)^2}+\frac{c x (B e-3 C d)}{e^4}+\frac{c C x^2}{2 e^3}","\frac{a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)}{e^5 (d+e x)}-\frac{\left(a e^2+c d^2\right) \left(A e^2-B d e+C d^2\right)}{2 e^5 (d+e x)^2}+\frac{\log (d+e x) \left(a C e^2+c \left(6 C d^2-e (3 B d-A e)\right)\right)}{e^5}-\frac{c x (3 C d-B e)}{e^4}+\frac{c C x^2}{2 e^3}",1,"(c*(-3*C*d + B*e)*x)/e^4 + (c*C*x^2)/(2*e^3) + (-(c*C*d^4) + B*c*d^3*e - A*c*d^2*e^2 - a*C*d^2*e^2 + a*B*d*e^3 - a*A*e^4)/(2*e^5*(d + e*x)^2) + (4*c*C*d^3 - 3*B*c*d^2*e + 2*A*c*d*e^2 + 2*a*C*d*e^2 - a*B*e^3)/(e^5*(d + e*x)) + ((6*c*C*d^2 - 3*B*c*d*e + A*c*e^2 + a*C*e^2)*Log[d + e*x])/e^5","A",1
25,1,335,304,0.1309814,"\int (d+e x)^3 \left(a+c x^2\right)^2 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^2*(A + B*x + C*x^2),x]","\frac{1}{2} a^2 d^2 x^2 (3 A e+B d)+a^2 A d^3 x+\frac{1}{7} c x^7 \left(2 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right)+\frac{1}{8} c e x^8 \left(2 a C e^2+c e (A e+3 B d)+3 c C d^2\right)+\frac{1}{6} x^6 \left(A c e \left(2 a e^2+3 c d^2\right)+B c d \left(6 a e^2+c d^2\right)+a C e \left(a e^2+6 c d^2\right)\right)+\frac{1}{5} x^5 \left(A c d \left(6 a e^2+c d^2\right)+a \left(a e^2 (B e+3 C d)+2 c d^2 (3 B e+C d)\right)\right)+\frac{1}{3} a d x^3 \left(A \left(3 a e^2+2 c d^2\right)+a d (3 B e+C d)\right)+\frac{1}{4} a x^4 \left(a A e^3+3 a B d e^2+3 a C d^2 e+6 A c d^2 e+2 B c d^3\right)+\frac{1}{9} c^2 e^2 x^9 (B e+3 C d)+\frac{1}{10} c^2 C e^3 x^{10}","\frac{1}{4} a^2 e x^4 \left(e (A e+3 B d)+3 C d^2\right)+a^2 A d^3 x+\frac{1}{8} c e x^8 \left(2 a C e^2+c \left(e (A e+3 B d)+3 C d^2\right)\right)+\frac{1}{7} c x^7 \left(2 a e^2 (B e+3 C d)+c d \left(3 e (A e+B d)+C d^2\right)\right)+\frac{1}{6} a e x^6 \left(a C e^2+2 c \left(e (A e+3 B d)+3 C d^2\right)\right)+\frac{1}{5} x^5 \left(A c d \left(6 a e^2+c d^2\right)+a \left(a e^2 (B e+3 C d)+2 c d^2 (3 B e+C d)\right)\right)+\frac{1}{3} a d x^3 \left(A \left(3 a e^2+2 c d^2\right)+a d (3 B e+C d)\right)+\frac{d^2 \left(a+c x^2\right)^3 (3 A e+B d)}{6 c}+\frac{1}{9} c^2 e^2 x^9 (B e+3 C d)+\frac{1}{10} c^2 C e^3 x^{10}",1,"a^2*A*d^3*x + (a^2*d^2*(B*d + 3*A*e)*x^2)/2 + (a*d*(a*d*(C*d + 3*B*e) + A*(2*c*d^2 + 3*a*e^2))*x^3)/3 + (a*(2*B*c*d^3 + 6*A*c*d^2*e + 3*a*C*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + ((A*c*d*(c*d^2 + 6*a*e^2) + a*(a*e^2*(3*C*d + B*e) + 2*c*d^2*(C*d + 3*B*e)))*x^5)/5 + ((a*C*e*(6*c*d^2 + a*e^2) + A*c*e*(3*c*d^2 + 2*a*e^2) + B*c*d*(c*d^2 + 6*a*e^2))*x^6)/6 + (c*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 2*a*e^2*(3*C*d + B*e))*x^7)/7 + (c*e*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(3*B*d + A*e))*x^8)/8 + (c^2*e^2*(3*C*d + B*e)*x^9)/9 + (c^2*C*e^3*x^10)/10","A",1
26,1,241,217,0.0898283,"\int (d+e x)^2 \left(a+c x^2\right)^2 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^2*(A + B*x + C*x^2),x]","\frac{1}{2} a^2 d x^2 (2 A e+B d)+a^2 A d^2 x+\frac{1}{7} c x^7 \left(2 a C e^2+c e (A e+2 B d)+c C d^2\right)+\frac{1}{6} c x^6 \left(2 a B e^2+4 a C d e+2 A c d e+B c d^2\right)+\frac{1}{5} x^5 \left(A c \left(2 a e^2+c d^2\right)+a \left(a C e^2+2 c d (2 B e+C d)\right)\right)+\frac{1}{4} a x^4 \left(a B e^2+2 a C d e+4 A c d e+2 B c d^2\right)+\frac{1}{3} a x^3 \left(A \left(a e^2+2 c d^2\right)+a d (2 B e+C d)\right)+\frac{1}{8} c^2 e x^8 (B e+2 C d)+\frac{1}{9} c^2 C e^2 x^9","a^2 A d^2 x+\frac{1}{4} a^2 e x^4 (B e+2 C d)+\frac{1}{7} c x^7 \left(2 a C e^2+c \left(e (A e+2 B d)+C d^2\right)\right)+\frac{1}{5} x^5 \left(A c \left(2 a e^2+c d^2\right)+a \left(a C e^2+2 c d (2 B e+C d)\right)\right)+\frac{1}{3} a x^3 \left(A \left(a e^2+2 c d^2\right)+a d (2 B e+C d)\right)+\frac{d \left(a+c x^2\right)^3 (2 A e+B d)}{6 c}+\frac{1}{3} a c e x^6 (B e+2 C d)+\frac{1}{8} c^2 e x^8 (B e+2 C d)+\frac{1}{9} c^2 C e^2 x^9",1,"a^2*A*d^2*x + (a^2*d*(B*d + 2*A*e)*x^2)/2 + (a*(a*d*(C*d + 2*B*e) + A*(2*c*d^2 + a*e^2))*x^3)/3 + (a*(2*B*c*d^2 + 4*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x^4)/4 + ((A*c*(c*d^2 + 2*a*e^2) + a*(a*C*e^2 + 2*c*d*(C*d + 2*B*e)))*x^5)/5 + (c*(B*c*d^2 + 2*A*c*d*e + 4*a*C*d*e + 2*a*B*e^2)*x^6)/6 + (c*(c*C*d^2 + 2*a*C*e^2 + c*e*(2*B*d + A*e))*x^7)/7 + (c^2*e*(2*C*d + B*e)*x^8)/8 + (c^2*C*e^2*x^9)/9","A",1
27,1,144,128,0.0508455,"\int (d+e x) \left(a+c x^2\right)^2 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)*(a + c*x^2)^2*(A + B*x + C*x^2),x]","\frac{1}{2} a^2 x^2 (A e+B d)+a^2 A d x+\frac{1}{6} c x^6 (2 a C e+A c e+B c d)+\frac{1}{5} c x^5 (2 a B e+2 a C d+A c d)+\frac{1}{4} a x^4 (a C e+2 A c e+2 B c d)+\frac{1}{3} a x^3 (a B e+a C d+2 A c d)+\frac{1}{7} c^2 x^7 (B e+C d)+\frac{1}{8} c^2 C e x^8","a^2 A d x+\frac{1}{4} a^2 C e x^4+\frac{1}{5} c x^5 (2 a (B e+C d)+A c d)+\frac{1}{3} a x^3 (a B e+a C d+2 A c d)+\frac{\left(a+c x^2\right)^3 (A e+B d)}{6 c}+\frac{1}{3} a c C e x^6+\frac{1}{7} c^2 x^7 (B e+C d)+\frac{1}{8} c^2 C e x^8",1,"a^2*A*d*x + (a^2*(B*d + A*e)*x^2)/2 + (a*(2*A*c*d + a*C*d + a*B*e)*x^3)/3 + (a*(2*B*c*d + 2*A*c*e + a*C*e)*x^4)/4 + (c*(A*c*d + 2*a*C*d + 2*a*B*e)*x^5)/5 + (c*(B*c*d + A*c*e + 2*a*C*e)*x^6)/6 + (c^2*(C*d + B*e)*x^7)/7 + (c^2*C*e*x^8)/8","A",1
28,1,69,67,0.0307953,"\int \left(a+c x^2\right)^2 \left(A+B x+C x^2\right) \, dx","Integrate[(a + c*x^2)^2*(A + B*x + C*x^2),x]","\frac{1}{210} x \left(35 a^2 (6 A+x (3 B+2 C x))+7 a c x^2 (20 A+3 x (5 B+4 C x))+c^2 x^4 (42 A+5 x (7 B+6 C x))\right)","a^2 A x+\frac{1}{5} c x^5 (2 a C+A c)+\frac{1}{3} a x^3 (a C+2 A c)+\frac{B \left(a+c x^2\right)^3}{6 c}+\frac{1}{7} c^2 C x^7",1,"(x*(35*a^2*(6*A + x*(3*B + 2*C*x)) + 7*a*c*x^2*(20*A + 3*x*(5*B + 4*C*x)) + c^2*x^4*(42*A + 5*x*(7*B + 6*C*x))))/210","A",1
29,1,285,297,0.1686354,"\int \frac{\left(a+c x^2\right)^2 \left(A+B x+C x^2\right)}{d+e x} \, dx","Integrate[((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x),x]","\frac{e x \left(30 a^2 e^4 (2 B e-2 C d+C e x)+10 a c e^2 \left(2 e \left(3 A e (e x-2 d)+B \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)+C \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+c^2 \left(e \left(5 A e \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+B \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)+C \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)\right)+60 \left(a e^2+c d^2\right)^2 \log (d+e x) \left(e (A e-B d)+C d^2\right)}{60 e^7}","\frac{x^2 \left(a^2 C e^4+2 a c e^2 \left(C d^2-e (B d-A e)\right)+c^2 d^2 \left(C d^2-e (B d-A e)\right)\right)}{2 e^5}-\frac{x \left(a^2 e^4 (C d-B e)+2 a c d e^2 \left(C d^2-e (B d-A e)\right)+c^2 d^3 \left(C d^2-e (B d-A e)\right)\right)}{e^6}+\frac{\left(a e^2+c d^2\right)^2 \log (d+e x) \left(A e^2-B d e+C d^2\right)}{e^7}-\frac{c x^3 \left(2 a e^2 (C d-B e)+c d \left(C d^2-e (B d-A e)\right)\right)}{3 e^4}+\frac{c x^4 \left(2 a C e^2+c \left(C d^2-e (B d-A e)\right)\right)}{4 e^3}-\frac{c^2 x^5 (C d-B e)}{5 e^2}+\frac{c^2 C x^6}{6 e}",1,"(e*x*(30*a^2*e^4*(-2*C*d + 2*B*e + C*e*x) + 10*a*c*e^2*(C*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + c^2*(C*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(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)))) + 60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x])/(60*e^7)","A",1
30,1,272,292,0.2813224,"\int \frac{\left(a+c x^2\right)^2 \left(A+B x+C x^2\right)}{(d+e x)^2} \, dx","Integrate[((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x)^2,x]","\frac{60 e x \left(a^2 C e^4+2 a c e^2 \left(e (A e-2 B d)+3 C d^2\right)+c^2 \left(d^2 e (3 A e-4 B d)+5 C d^4\right)\right)-30 c e^2 x^2 \left(-2 a e^2 (B e-2 C d)+c d e (2 A e-3 B d)+4 c C d^3\right)-\frac{60 \left(a e^2+c d^2\right)^2 \left(e (A e-B d)+C d^2\right)}{d+e x}+20 c e^3 x^3 \left(2 a C e^2+c e (A e-2 B d)+3 c C d^2\right)-60 \left(a e^2+c d^2\right) \log (d+e x) \left(a e^2 (2 C d-B e)+c d e (4 A e-5 B d)+6 c C d^3\right)+15 c^2 e^4 x^4 (B e-2 C d)+12 c^2 C e^5 x^5}{60 e^7}","\frac{x \left(a^2 C e^4+2 a c e^2 \left(3 C d^2-e (2 B d-A e)\right)+c^2 d^2 \left(5 C d^2-e (4 B d-3 A e)\right)\right)}{e^6}-\frac{\left(a e^2+c d^2\right)^2 \left(A e^2-B d e+C d^2\right)}{e^7 (d+e x)}-\frac{\left(a e^2+c d^2\right) \log (d+e x) \left(a e^2 (2 C d-B e)+c d \left(6 C d^2-e (5 B d-4 A e)\right)\right)}{e^7}-\frac{c x^2 \left(2 a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)\right)}{2 e^5}+\frac{c x^3 \left(2 a C e^2+c \left(3 C d^2-e (2 B d-A e)\right)\right)}{3 e^4}-\frac{c^2 x^4 (2 C d-B e)}{4 e^3}+\frac{c^2 C x^5}{5 e^2}",1,"(60*e*(a^2*C*e^4 + 2*a*c*e^2*(3*C*d^2 + e*(-2*B*d + A*e)) + c^2*(5*C*d^4 + d^2*e*(-4*B*d + 3*A*e)))*x - 30*c*e^2*(4*c*C*d^3 + c*d*e*(-3*B*d + 2*A*e) - 2*a*e^2*(-2*C*d + B*e))*x^2 + 20*c*e^3*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(-2*B*d + A*e))*x^3 + 15*c^2*e^4*(-2*C*d + B*e)*x^4 + 12*c^2*C*e^5*x^5 - (60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) - 60*(c*d^2 + a*e^2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d + e*x])/(60*e^7)","A",1
31,1,274,295,0.1218383,"\int \frac{\left(a+c x^2\right)^2 \left(A+B x+C x^2\right)}{(d+e x)^3} \, dx","Integrate[((a + c*x^2)^2*(A + B*x + C*x^2))/(d + e*x)^3,x]","\frac{12 \log (d+e x) \left(a^2 C e^4+2 a c e^2 \left(e (A e-3 B d)+6 C d^2\right)+c^2 \left(2 d^2 e (3 A e-5 B d)+15 C d^4\right)\right)-12 c e x \left(-2 a e^2 (B e-3 C d)+3 c d e (A e-2 B d)+10 c C d^3\right)+6 c e^2 x^2 \left(2 a C e^2+c e (A e-3 B d)+6 c C d^2\right)-\frac{6 \left(a e^2+c d^2\right)^2 \left(e (A e-B d)+C d^2\right)}{(d+e x)^2}+\frac{12 \left(a e^2+c d^2\right) \left(a e^2 (2 C d-B e)+c d e (4 A e-5 B d)+6 c C d^3\right)}{d+e x}+4 c^2 e^3 x^3 (B e-3 C d)+3 c^2 C e^4 x^4}{12 e^7}","\frac{\log (d+e x) \left(a^2 C e^4+2 a c e^2 \left(6 C d^2-e (3 B d-A e)\right)+c^2 d^2 \left(15 C d^2-2 e (5 B d-3 A e)\right)\right)}{e^7}+\frac{\left(a e^2+c d^2\right) \left(a e^2 (2 C d-B e)+c d \left(6 C d^2-e (5 B d-4 A e)\right)\right)}{e^7 (d+e x)}-\frac{\left(a e^2+c d^2\right)^2 \left(A e^2-B d e+C d^2\right)}{2 e^7 (d+e x)^2}-\frac{c x \left(2 a e^2 (3 C d-B e)+c d \left(10 C d^2-3 e (2 B d-A e)\right)\right)}{e^6}+\frac{c x^2 \left(2 a C e^2+c \left(6 C d^2-e (3 B d-A e)\right)\right)}{2 e^5}-\frac{c^2 x^3 (3 C d-B e)}{3 e^4}+\frac{c^2 C x^4}{4 e^3}",1,"(-12*c*e*(10*c*C*d^3 + 3*c*d*e*(-2*B*d + A*e) - 2*a*e^2*(-3*C*d + B*e))*x + 6*c*e^2*(6*c*C*d^2 + 2*a*C*e^2 + c*e*(-3*B*d + A*e))*x^2 + 4*c^2*e^3*(-3*C*d + B*e)*x^3 + 3*c^2*C*e^4*x^4 - (6*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 + (12*(c*d^2 + a*e^2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 4*A*e) + a*e^2*(2*C*d - B*e)))/(d + e*x) + 12*(a^2*C*e^4 + 2*a*c*e^2*(6*C*d^2 + e*(-3*B*d + A*e)) + c^2*(15*C*d^4 + 2*d^2*e*(-5*B*d + 3*A*e)))*Log[d + e*x])/(12*e^7)","A",1
32,1,459,404,0.2080262,"\int (d+e x)^3 \left(a+c x^2\right)^3 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^3*(a + c*x^2)^3*(A + B*x + C*x^2),x]","\frac{1}{2} a^3 d^2 x^2 (3 A e+B d)+a^3 A d^3 x+\frac{1}{3} a^2 d x^3 \left(3 A \left(a e^2+c d^2\right)+a d (3 B e+C d)\right)+\frac{1}{4} a^2 x^4 \left(a A e^3+3 a B d e^2+3 a C d^2 e+9 A c d^2 e+3 B c d^3\right)+\frac{1}{9} c^2 x^9 \left(3 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right)+\frac{1}{10} c^2 e x^{10} \left(3 a C e^2+c e (A e+3 B d)+3 c C d^2\right)+\frac{1}{8} c x^8 \left(3 e \left(A c \left(a e^2+c d^2\right)+a C \left(a e^2+3 c d^2\right)\right)+B c d \left(9 a e^2+c d^2\right)\right)+\frac{1}{7} c x^7 \left(A c d \left(9 a e^2+c d^2\right)+3 a \left(a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right)\right)+\frac{1}{6} a x^6 \left(3 A c e \left(a e^2+3 c d^2\right)+3 B c d \left(3 a e^2+c d^2\right)+a C e \left(a e^2+9 c d^2\right)\right)+\frac{1}{5} a x^5 \left(3 A c d \left(3 a e^2+c d^2\right)+a \left(a e^2 (B e+3 C d)+3 c d^2 (3 B e+C d)\right)\right)+\frac{1}{11} c^3 e^2 x^{11} (B e+3 C d)+\frac{1}{12} c^3 C e^3 x^{12}","\frac{1}{4} a^3 e x^4 \left(e (A e+3 B d)+3 C d^2\right)+a^3 A d^3 x+\frac{1}{6} a^2 e x^6 \left(a C e^2+3 c \left(e (A e+3 B d)+3 C d^2\right)\right)+\frac{1}{3} a^2 d x^3 \left(3 A \left(a e^2+c d^2\right)+a d (3 B e+C d)\right)+\frac{1}{10} c^2 e x^{10} \left(3 a C e^2+c \left(e (A e+3 B d)+3 C d^2\right)\right)+\frac{1}{9} c^2 x^9 \left(3 a e^2 (B e+3 C d)+c d \left(3 e (A e+B d)+C d^2\right)\right)+\frac{3}{8} a c e x^8 \left(a C e^2+c \left(e (A e+3 B d)+3 C d^2\right)\right)+\frac{1}{7} c x^7 \left(A c d \left(9 a e^2+c d^2\right)+3 a \left(a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right)\right)+\frac{1}{5} a x^5 \left(3 A c d \left(3 a e^2+c d^2\right)+a \left(a e^2 (B e+3 C d)+3 c d^2 (3 B e+C d)\right)\right)+\frac{d^2 \left(a+c x^2\right)^4 (3 A e+B d)}{8 c}+\frac{1}{11} c^3 e^2 x^{11} (B e+3 C d)+\frac{1}{12} c^3 C e^3 x^{12}",1,"a^3*A*d^3*x + (a^3*d^2*(B*d + 3*A*e)*x^2)/2 + (a^2*d*(a*d*(C*d + 3*B*e) + 3*A*(c*d^2 + a*e^2))*x^3)/3 + (a^2*(3*B*c*d^3 + 9*A*c*d^2*e + 3*a*C*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + (a*(3*A*c*d*(c*d^2 + 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) + 3*c*d^2*(C*d + 3*B*e)))*x^5)/5 + (a*(3*A*c*e*(3*c*d^2 + a*e^2) + a*C*e*(9*c*d^2 + a*e^2) + 3*B*c*d*(c*d^2 + 3*a*e^2))*x^6)/6 + (c*(A*c*d*(c*d^2 + 9*a*e^2) + 3*a*(a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e)))*x^7)/7 + (c*(B*c*d*(c*d^2 + 9*a*e^2) + 3*e*(A*c*(c*d^2 + a*e^2) + a*C*(3*c*d^2 + a*e^2)))*x^8)/8 + (c^2*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 3*a*e^2*(3*C*d + B*e))*x^9)/9 + (c^2*e*(3*c*C*d^2 + 3*a*C*e^2 + c*e*(3*B*d + A*e))*x^10)/10 + (c^3*e^2*(3*C*d + B*e)*x^11)/11 + (c^3*C*e^3*x^12)/12","A",1
33,1,329,289,0.1302761,"\int (d+e x)^2 \left(a+c x^2\right)^3 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)^2*(a + c*x^2)^3*(A + B*x + C*x^2),x]","\frac{1}{2} a^3 d x^2 (2 A e+B d)+a^3 A d^2 x+\frac{1}{4} a^2 x^4 \left(a B e^2+2 a C d e+6 A c d e+3 B c d^2\right)+\frac{1}{3} a^2 x^3 \left(A \left(a e^2+3 c d^2\right)+a d (2 B e+C d)\right)+\frac{1}{9} c^2 x^9 \left(3 a C e^2+c e (A e+2 B d)+c C d^2\right)+\frac{1}{8} c^2 x^8 \left(3 a B e^2+6 a C d e+2 A c d e+B c d^2\right)+\frac{1}{7} c x^7 \left(A c \left(3 a e^2+c d^2\right)+3 a \left(a C e^2+c d (2 B e+C d)\right)\right)+\frac{1}{2} a c x^6 \left(2 d e (a C+A c)+B \left(a e^2+c d^2\right)\right)+\frac{1}{5} a x^5 \left(3 A c \left(a e^2+c d^2\right)+a \left(a C e^2+3 c d (2 B e+C d)\right)\right)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11}","a^3 A d^2 x+\frac{1}{4} a^3 e x^4 (B e+2 C d)+\frac{1}{3} a^2 x^3 \left(A \left(a e^2+3 c d^2\right)+a d (2 B e+C d)\right)+\frac{1}{2} a^2 c e x^6 (B e+2 C d)+\frac{1}{9} c^2 x^9 \left(3 a C e^2+c \left(e (A e+2 B d)+C d^2\right)\right)+\frac{1}{7} c x^7 \left(A c \left(3 a e^2+c d^2\right)+3 a \left(a C e^2+c d (2 B e+C d)\right)\right)+\frac{1}{5} a x^5 \left(3 A c \left(a e^2+c d^2\right)+a \left(a C e^2+3 c d (2 B e+C d)\right)\right)+\frac{d \left(a+c x^2\right)^4 (2 A e+B d)}{8 c}+\frac{3}{8} a c^2 e x^8 (B e+2 C d)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11}",1,"a^3*A*d^2*x + (a^3*d*(B*d + 2*A*e)*x^2)/2 + (a^2*(a*d*(C*d + 2*B*e) + A*(3*c*d^2 + a*e^2))*x^3)/3 + (a^2*(3*B*c*d^2 + 6*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x^4)/4 + (a*(3*A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*(C*d + 2*B*e)))*x^5)/5 + (a*c*(2*(A*c + a*C)*d*e + B*(c*d^2 + a*e^2))*x^6)/2 + (c*(A*c*(c*d^2 + 3*a*e^2) + 3*a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x^7)/7 + (c^2*(B*c*d^2 + 2*A*c*d*e + 6*a*C*d*e + 3*a*B*e^2)*x^8)/8 + (c^2*(c*C*d^2 + 3*a*C*e^2 + c*e*(2*B*d + A*e))*x^9)/9 + (c^3*e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x^11)/11","A",1
34,1,196,169,0.0703542,"\int (d+e x) \left(a+c x^2\right)^3 \left(A+B x+C x^2\right) \, dx","Integrate[(d + e*x)*(a + c*x^2)^3*(A + B*x + C*x^2),x]","\frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{1}{4} a^2 x^4 (a C e+3 A c e+3 B c d)+\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+\frac{1}{8} c^2 x^8 (3 a C e+A c e+B c d)+\frac{1}{7} c^2 x^7 (3 a B e+3 a C d+A c d)+\frac{1}{2} a c x^6 (a C e+A c e+B c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10}","a^3 A d x+\frac{1}{4} a^3 C e x^4+\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+\frac{1}{2} a^2 c C e x^6+\frac{1}{7} c^2 x^7 (3 a (B e+C d)+A c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{\left(a+c x^2\right)^4 (A e+B d)}{8 c}+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10}",1,"a^3*A*d*x + (a^3*(B*d + A*e)*x^2)/2 + (a^2*(3*A*c*d + a*C*d + a*B*e)*x^3)/3 + (a^2*(3*B*c*d + 3*A*c*e + a*C*e)*x^4)/4 + (3*a*c*(A*c*d + a*C*d + a*B*e)*x^5)/5 + (a*c*(B*c*d + A*c*e + a*C*e)*x^6)/2 + (c^2*(A*c*d + 3*a*C*d + 3*a*B*e)*x^7)/7 + (c^2*(B*c*d + A*c*e + 3*a*C*e)*x^8)/8 + (c^3*(C*d + B*e)*x^9)/9 + (c^3*C*e*x^10)/10","A",1
35,1,100,87,0.0305138,"\int \left(a+c x^2\right)^3 \left(A+B x+C x^2\right) \, dx","Integrate[(a + c*x^2)^3*(A + B*x + C*x^2),x]","\frac{1}{6} a^3 x (6 A+x (3 B+2 C x))+\frac{1}{20} a^2 c x^3 (20 A+3 x (5 B+4 C x))+\frac{1}{70} a c^2 x^5 (42 A+5 x (7 B+6 C x))+\frac{1}{504} c^3 x^7 (72 A+7 x (9 B+8 C x))","a^3 A x+\frac{1}{3} a^2 x^3 (a C+3 A c)+\frac{1}{7} c^2 x^7 (3 a C+A c)+\frac{3}{5} a c x^5 (a C+A c)+\frac{B \left(a+c x^2\right)^4}{8 c}+\frac{1}{9} c^3 C x^9",1,"(a^3*x*(6*A + x*(3*B + 2*C*x)))/6 + (a^2*c*x^3*(20*A + 3*x*(5*B + 4*C*x)))/20 + (a*c^2*x^5*(42*A + 5*x*(7*B + 6*C*x)))/70 + (c^3*x^7*(72*A + 7*x*(9*B + 8*C*x)))/504","A",1
36,1,498,490,0.4742529,"\int \frac{\left(a+c x^2\right)^3 \left(A+B x+C x^2\right)}{d+e x} \, dx","Integrate[((a + c*x^2)^3*(A + B*x + C*x^2))/(d + e*x),x]","\frac{x \left(420 a^3 e^6 (2 B e-2 C d+C e x)+210 a^2 c e^4 \left(2 e \left(3 A e (e x-2 d)+B \left(6 d^2-3 d e x+2 e^2 x^2\right)\right)+C \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)\right)+42 a c^2 e^2 \left(e \left(5 A e \left(-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right)+B \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)+C \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)\right)+c^3 \left(2 e \left(7 A e \left(-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right)+B \left(420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right)\right)+C \left(-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right)\right)\right)}{840 e^8}+\frac{\left(a e^2+c d^2\right)^3 \log (d+e x) \left(e (A e-B d)+C d^2\right)}{e^9}","\frac{c (d+e x)^4 \left(3 a^2 C e^4+3 a c e^2 \left(15 C d^2-e (5 B d-A e)\right)+5 c^2 d^2 \left(14 C d^2-e (7 B d-3 A e)\right)\right)}{4 e^9}+\frac{(d+e x)^2 \left(a e^2+c d^2\right) \left(a^2 C e^4+a c e^2 \left(17 C d^2-3 e (3 B d-A e)\right)+c^2 d^2 \left(28 C d^2-3 e (7 B d-5 A e)\right)\right)}{2 e^9}-\frac{c (d+e x)^3 \left(3 a^2 e^4 (4 C d-B e)+6 a c d e^2 \left(10 C d^2-e (5 B d-2 A e)\right)+c^2 d^3 \left(56 C d^2-5 e (7 B d-4 A e)\right)\right)}{3 e^9}+\frac{c^2 (d+e x)^6 \left(3 a C e^2+c \left(28 C d^2-e (7 B d-A e)\right)\right)}{6 e^9}-\frac{c^2 (d+e x)^5 \left(3 a e^2 (6 C d-B e)+c d \left(56 C d^2-3 e (7 B d-2 A e)\right)\right)}{5 e^9}+\frac{\left(a e^2+c d^2\right)^3 \log (d+e x) \left(A e^2-B d e+C d^2\right)}{e^9}-\frac{x \left(a e^2+c d^2\right)^2 \left(a e^2 (2 C d-B e)+c d \left(8 C d^2-e (7 B d-6 A e)\right)\right)}{e^8}-\frac{c^3 (d+e x)^7 (8 C d-B e)}{7 e^9}+\frac{c^3 C (d+e x)^8}{8 e^9}",1,"(x*(420*a^3*e^6*(-2*C*d + 2*B*e + C*e*x) + 210*a^2*c*e^4*(C*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + 42*a*c^2*e^2*(C*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(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))) + c^3*(C*(-840*d^7 + 420*d^6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7) + 2*e*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6)))))/(840*e^8) + ((c*d^2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x])/e^9","A",1
37,1,641,486,0.3968504,"\int \frac{\left(a+c x^2\right)^3 \left(A+B x+C x^2\right)}{(d+e x)^2} \, dx","Integrate[((a + c*x^2)^3*(A + B*x + C*x^2))/(d + e*x)^2,x]","\frac{420 a^3 e^6 \left(e (B d-A e)+C \left(-d^2+d e x+e^2 x^2\right)\right)+210 a^2 c e^4 \left(3 e \left(2 A e \left(-d^2+d e x+e^2 x^2\right)+B \left(2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right)\right)+2 C \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)\right)+21 a c^2 e^2 \left(5 e \left(4 A e \left(-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right)+B \left(12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right)\right)-6 C \left(10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right)\right)-420 (d+e x) \left(a e^2+c d^2\right)^2 \log (d+e x) \left(a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right)+c^3 \left(7 e \left(6 A e \left(-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right)+B \left(60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right)\right)-4 C \left(105 d^8-735 d^7 e x-420 d^6 e^2 x^2+140 d^5 e^3 x^3-70 d^4 e^4 x^4+42 d^3 e^5 x^5-28 d^2 e^6 x^6+20 d e^7 x^7-15 e^8 x^8\right)\right)}{420 e^9 (d+e x)}","\frac{c x^3 \left(3 a^2 C e^4+3 a c e^2 \left(3 C d^2-e (2 B d-A e)\right)+c^2 d^2 \left(5 C d^2-e (4 B d-3 A e)\right)\right)}{3 e^6}-\frac{c x^2 \left(3 a^2 e^4 (2 C d-B e)+3 a c d e^2 \left(4 C d^2-e (3 B d-2 A e)\right)+c^2 d^3 \left(6 C d^2-e (5 B d-4 A e)\right)\right)}{2 e^7}+\frac{x \left(a^3 C e^6+3 a^2 c e^4 \left(3 C d^2-e (2 B d-A e)\right)+3 a c^2 d^2 e^2 \left(5 C d^2-e (4 B d-3 A e)\right)+c^3 d^4 \left(7 C d^2-e (6 B d-5 A e)\right)\right)}{e^8}-\frac{c^2 x^4 \left(3 a e^2 (2 C d-B e)+c d \left(4 C d^2-e (3 B d-2 A e)\right)\right)}{4 e^5}+\frac{c^2 x^5 \left(3 a C e^2+c \left(3 C d^2-e (2 B d-A e)\right)\right)}{5 e^4}-\frac{\left(a e^2+c d^2\right)^3 \left(A e^2-B d e+C d^2\right)}{e^9 (d+e x)}-\frac{\left(a e^2+c d^2\right)^2 \log (d+e x) \left(a e^2 (2 C d-B e)+c d \left(8 C d^2-e (7 B d-6 A e)\right)\right)}{e^9}-\frac{c^3 x^6 (2 C d-B e)}{6 e^3}+\frac{c^3 C x^7}{7 e^2}",1,"(420*a^3*e^6*(e*(B*d - A*e) + C*(-d^2 + d*e*x + e^2*x^2)) + 210*a^2*c*e^4*(2*C*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 3*e*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3))) + 21*a*c^2*e^2*(-6*C*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6) + 5*e*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5))) + c^3*(-4*C*(105*d^8 - 735*d^7*e*x - 420*d^6*e^2*x^2 + 140*d^5*e^3*x^3 - 70*d^4*e^4*x^4 + 42*d^3*e^5*x^5 - 28*d^2*e^6*x^6 + 20*d*e^7*x^7 - 15*e^8*x^8) + 7*e*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7))) - 420*(c*d^2 + a*e^2)^2*(8*c*C*d^3 + c*d*e*(-7*B*d + 6*A*e) + a*e^2*(2*C*d - B*e))*(d + e*x)*Log[d + e*x])/(420*e^9*(d + e*x))","A",1
38,1,438,466,0.225439,"\int \frac{\left(a+c x^2\right)^3 \left(A+B x+C x^2\right)}{(d+e x)^3} \, dx","Integrate[((a + c*x^2)^3*(A + B*x + C*x^2))/(d + e*x)^3,x]","\frac{30 c e^2 x^2 \left(3 a^2 C e^4+3 a c e^2 \left(e (A e-3 B d)+6 C d^2\right)+c^2 \left(2 d^2 e (3 A e-5 B d)+15 C d^4\right)\right)+60 \left(a e^2+c d^2\right) \log (d+e x) \left(a^2 C e^4+a c e^2 \left(3 e (A e-3 B d)+17 C d^2\right)+c^2 \left(3 d^2 e (5 A e-7 B d)+28 C d^4\right)\right)-60 c e x \left(-3 a^2 e^4 (B e-3 C d)+3 a c d e^2 \left(3 e (A e-2 B d)+10 C d^2\right)+c^2 \left(5 d^3 e (2 A e-3 B d)+21 C d^5\right)\right)-20 c^2 e^3 x^3 \left(-3 a e^2 (B e-3 C d)+3 c d e (A e-2 B d)+10 c C d^3\right)+15 c^2 e^4 x^4 \left(3 a C e^2+c e (A e-3 B d)+6 c C d^2\right)-\frac{30 \left(a e^2+c d^2\right)^3 \left(e (A e-B d)+C d^2\right)}{(d+e x)^2}+\frac{60 \left(a e^2+c d^2\right)^2 \left(a e^2 (2 C d-B e)+c d e (6 A e-7 B d)+8 c C d^3\right)}{d+e x}+12 c^3 e^5 x^5 (B e-3 C d)+10 c^3 C e^6 x^6}{60 e^9}","\frac{\left(a e^2+c d^2\right) \log (d+e x) \left(a^2 C e^4+a c e^2 \left(17 C d^2-3 e (3 B d-A e)\right)+c^2 d^2 \left(28 C d^2-3 e (7 B d-5 A e)\right)\right)}{e^9}+\frac{c x^2 \left(3 a^2 C e^4+3 a c e^2 \left(6 C d^2-e (3 B d-A e)\right)+c^2 d^2 \left(15 C d^2-2 e (5 B d-3 A e)\right)\right)}{2 e^7}-\frac{c x \left(3 a^2 e^4 (3 C d-B e)+3 a c d e^2 \left(10 C d^2-3 e (2 B d-A e)\right)+c^2 d^3 \left(21 C d^2-5 e (3 B d-2 A e)\right)\right)}{e^8}-\frac{c^2 x^3 \left(3 a e^2 (3 C d-B e)+c d \left(10 C d^2-3 e (2 B d-A e)\right)\right)}{3 e^6}+\frac{c^2 x^4 \left(3 a C e^2+c \left(6 C d^2-e (3 B d-A e)\right)\right)}{4 e^5}+\frac{\left(a e^2+c d^2\right)^2 \left(a e^2 (2 C d-B e)+c d \left(8 C d^2-e (7 B d-6 A e)\right)\right)}{e^9 (d+e x)}-\frac{\left(a e^2+c d^2\right)^3 \left(A e^2-B d e+C d^2\right)}{2 e^9 (d+e x)^2}-\frac{c^3 x^5 (3 C d-B e)}{5 e^4}+\frac{c^3 C x^6}{6 e^3}",1,"(-60*c*e*(-3*a^2*e^4*(-3*C*d + B*e) + 3*a*c*d*e^2*(10*C*d^2 + 3*e*(-2*B*d + A*e)) + c^2*(21*C*d^5 + 5*d^3*e*(-3*B*d + 2*A*e)))*x + 30*c*e^2*(3*a^2*C*e^4 + 3*a*c*e^2*(6*C*d^2 + e*(-3*B*d + A*e)) + c^2*(15*C*d^4 + 2*d^2*e*(-5*B*d + 3*A*e)))*x^2 - 20*c^2*e^3*(10*c*C*d^3 + 3*c*d*e*(-2*B*d + A*e) - 3*a*e^2*(-3*C*d + B*e))*x^3 + 15*c^2*e^4*(6*c*C*d^2 + 3*a*C*e^2 + c*e*(-3*B*d + A*e))*x^4 + 12*c^3*e^5*(-3*C*d + B*e)*x^5 + 10*c^3*C*e^6*x^6 - (30*(c*d^2 + a*e^2)^3*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 + (60*(c*d^2 + a*e^2)^2*(8*c*C*d^3 + c*d*e*(-7*B*d + 6*A*e) + a*e^2*(2*C*d - B*e)))/(d + e*x) + 60*(c*d^2 + a*e^2)*(a^2*C*e^4 + a*c*e^2*(17*C*d^2 + 3*e*(-3*B*d + A*e)) + c^2*(28*C*d^4 + 3*d^2*e*(-7*B*d + 5*A*e)))*Log[d + e*x])/(60*e^9)","A",1
39,1,62,17,0.0373016,"\int \frac{\left(a+b x^2\right) \left(-a d+4 b c x+3 b d x^2\right)}{(c+d x)^2} \, dx","Integrate[((a + b*x^2)*(-(a*d) + 4*b*c*x + 3*b*d*x^2))/(c + d*x)^2,x]","\frac{a^2 d^4+2 a b d^2 \left(c^2+c d x+d^2 x^2\right)+b^2 \left(c^4+c^3 d x+d^4 x^4\right)}{d^4 (c+d x)}","\frac{\left(a+b x^2\right)^2}{c+d x}",1,"(a^2*d^4 + 2*a*b*d^2*(c^2 + c*d*x + d^2*x^2) + b^2*(c^4 + c^3*d*x + d^4*x^4))/(d^4*(c + d*x))","B",1
40,1,62,17,0.0159218,"\int \frac{\left(a+b x^2\right) (-a d+b x (4 c+3 d x))}{(c+d x)^2} \, dx","Integrate[((a + b*x^2)*(-(a*d) + b*x*(4*c + 3*d*x)))/(c + d*x)^2,x]","\frac{a^2 d^4+2 a b d^2 \left(c^2+c d x+d^2 x^2\right)+b^2 \left(c^4+c^3 d x+d^4 x^4\right)}{d^4 (c+d x)}","\frac{\left(a+b x^2\right)^2}{c+d x}",1,"(a^2*d^4 + 2*a*b*d^2*(c^2 + c*d*x + d^2*x^2) + b^2*(c^4 + c^3*d*x + d^4*x^4))/(d^4*(c + d*x))","B",1
41,1,90,17,0.0423199,"\int \frac{\left(a+b x^2\right)^2 \left(-a d+6 b c x+5 b d x^2\right)}{(c+d x)^2} \, dx","Integrate[((a + b*x^2)^2*(-(a*d) + 6*b*c*x + 5*b*d*x^2))/(c + d*x)^2,x]","\frac{a^3 d^6+3 a^2 b d^4 \left(c^2+c d x+d^2 x^2\right)+3 a b^2 d^2 \left(c^4+c^3 d x+d^4 x^4\right)+b^3 \left(c^6+c^5 d x+d^6 x^6\right)}{d^6 (c+d x)}","\frac{\left(a+b x^2\right)^3}{c+d x}",1,"(a^3*d^6 + 3*a^2*b*d^4*(c^2 + c*d*x + d^2*x^2) + 3*a*b^2*d^2*(c^4 + c^3*d*x + d^4*x^4) + b^3*(c^6 + c^5*d*x + d^6*x^6))/(d^6*(c + d*x))","B",1
42,1,90,17,0.02451,"\int \frac{\left(a+b x^2\right)^2 (-a d+b x (6 c+5 d x))}{(c+d x)^2} \, dx","Integrate[((a + b*x^2)^2*(-(a*d) + b*x*(6*c + 5*d*x)))/(c + d*x)^2,x]","\frac{a^3 d^6+3 a^2 b d^4 \left(c^2+c d x+d^2 x^2\right)+3 a b^2 d^2 \left(c^4+c^3 d x+d^4 x^4\right)+b^3 \left(c^6+c^5 d x+d^6 x^6\right)}{d^6 (c+d x)}","\frac{\left(a+b x^2\right)^3}{c+d x}",1,"(a^3*d^6 + 3*a^2*b*d^4*(c^2 + c*d*x + d^2*x^2) + 3*a*b^2*d^2*(c^4 + c^3*d*x + d^4*x^4) + b^3*(c^6 + c^5*d*x + d^6*x^6))/(d^6*(c + d*x))","B",1
43,1,223,240,0.2429726,"\int \frac{(d+e x)^3 \left(A+B x+C x^2\right)}{a+c x^2} \, dx","Integrate[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(c d^2-3 a e^2\right)+a \left(a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right)\right)}{\sqrt{a} c^{5/2}}+\frac{6 \log \left(a+c x^2\right) \left(e (A c-a C) \left(3 c d^2-a e^2\right)+B c d \left(c d^2-3 a e^2\right)\right)+c x \left(-6 a e^2 (2 B e+6 C d+C e x)+2 c e \left(3 A e (6 d+e x)+B \left(18 d^2+9 d e x+2 e^2 x^2\right)\right)+3 c C \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)\right)}{12 c^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(c d^2-3 a e^2\right)+a \left(a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right)\right)}{\sqrt{a} c^{5/2}}+\frac{\log \left(a+c x^2\right) \left(e (A c-a C) \left(3 c d^2-a e^2\right)+B c d \left(c d^2-3 a e^2\right)\right)}{2 c^3}-\frac{e x^2 \left(a C e^2-c \left(e (A e+3 B d)+3 C d^2\right)\right)}{2 c^2}-\frac{x \left(a e^2 (B e+3 C d)-c d \left(3 e (A e+B d)+C d^2\right)\right)}{c^2}+\frac{e^2 x^3 (B e+3 C d)}{3 c}+\frac{C e^3 x^4}{4 c}",1,"((A*c*d*(c*d^2 - 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) - c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/2)) + (c*x*(-6*a*e^2*(6*C*d + 2*B*e + C*e*x) + 3*c*C*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 2*c*e*(3*A*e*(6*d + e*x) + B*(18*d^2 + 9*d*e*x + 2*e^2*x^2))) + 6*(B*c*d*(c*d^2 - 3*a*e^2) + (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[a + c*x^2])/(12*c^3)","A",1
44,1,155,168,0.1673695,"\int \frac{(d+e x)^2 \left(A+B x+C x^2\right)}{a+c x^2} \, dx","Integrate[((d + e*x)^2*(A + B*x + C*x^2))/(a + c*x^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2-c d (2 B e+C d)\right)\right)}{\sqrt{a} c^{5/2}}+\frac{x \left(-6 a C e^2+3 c e (2 A e+4 B d+B e x)+2 c C \left(3 d^2+3 d e x+e^2 x^2\right)\right)+3 \log \left(a+c x^2\right) \left(-a B e^2-2 a C d e+2 A c d e+B c d^2\right)}{6 c^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2-c d (2 B e+C d)\right)\right)}{\sqrt{a} c^{5/2}}+\frac{\log \left(a+c x^2\right) \left(-a B e^2-2 a C d e+2 A c d e+B c d^2\right)}{2 c^2}-\frac{x \left(a C e^2-c \left(e (A e+2 B d)+C d^2\right)\right)}{c^2}+\frac{e x^2 (B e+2 C d)}{2 c}+\frac{C e^2 x^3}{3 c}",1,"((A*c*(c*d^2 - a*e^2) + a*(a*C*e^2 - c*d*(C*d + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/2)) + (x*(-6*a*C*e^2 + 3*c*e*(4*B*d + 2*A*e + B*e*x) + 2*c*C*(3*d^2 + 3*d*e*x + e^2*x^2)) + 3*(B*c*d^2 + 2*A*c*d*e - 2*a*C*d*e - a*B*e^2)*Log[a + c*x^2])/(6*c^2)","A",1
45,1,86,93,0.0871239,"\int \frac{(d+e x) \left(A+B x+C x^2\right)}{a+c x^2} \, dx","Integrate[((d + e*x)*(A + B*x + C*x^2))/(a + c*x^2),x]","\frac{\log \left(a+c x^2\right) (-a C e+A c e+B c d)-\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d-A c d)}{\sqrt{a}}+c x (2 B e+2 C d+C e x)}{2 c^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (A c d-a (B e+C d))}{\sqrt{a} c^{3/2}}+\frac{\log \left(a+c x^2\right) (-a C e+A c e+B c d)}{2 c^2}+\frac{x (B e+C d)}{c}+\frac{C e x^2}{2 c}",1,"(c*x*(2*C*d + 2*B*e + C*e*x) - (2*Sqrt[c]*(-(A*c*d) + a*C*d + a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + (B*c*d + A*c*e - a*C*e)*Log[a + c*x^2])/(2*c^2)","A",1
46,1,56,55,0.0409438,"\int \frac{A+B x+C x^2}{a+c x^2} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2),x]","-\frac{(a C-A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{B \log \left(a+c x^2\right)}{2 c}+\frac{C x}{c}","\frac{(A c-a C) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{a} c^{3/2}}+\frac{B \log \left(a+c x^2\right)}{2 c}+\frac{C x}{c}",1,"(C*x)/c - ((-(A*c) + a*C)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (B*Log[a + c*x^2])/(2*c)","A",1
47,1,120,133,0.1021717,"\int \frac{A+B x+C x^2}{(d+e x) \left(a+c x^2\right)} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)*(a + c*x^2)),x]","\frac{\sqrt{a} \left(e \log \left(a+c x^2\right) (a C e-A c e+B c d)+2 c \log (d+e x) \left(A e^2-B d e+C d^2\right)\right)+2 \sqrt{c} e \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e-a C d+A c d)}{2 \sqrt{a} c e \left(a e^2+c d^2\right)}","\frac{\log \left(a+c x^2\right) (a C e-A c e+B c d)}{2 c \left(a e^2+c d^2\right)}+\frac{\log (d+e x) \left(A e^2-B d e+C d^2\right)}{e \left(a e^2+c d^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e-a C d+A c d)}{\sqrt{a} \sqrt{c} \left(a e^2+c d^2\right)}",1,"(2*Sqrt[c]*e*(A*c*d - a*C*d + a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[a]*(2*c*(C*d^2 - B*d*e + A*e^2)*Log[d + e*x] + e*(B*c*d - A*c*e + a*C*e)*Log[a + c*x^2]))/(2*Sqrt[a]*c*e*(c*d^2 + a*e^2))","A",1
48,1,188,214,0.3121456,"\int \frac{A+B x+C x^2}{(d+e x)^2 \left(a+c x^2\right)} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^2*(a + c*x^2)),x]","\frac{\log \left(a+c x^2\right) \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)-\frac{2 \left(a e^2+c d^2\right) \left(e (A e-B d)+C d^2\right)}{e (d+e x)}+\log (d+e x) \left(2 a B e^2-4 a C d e+4 A c d e-2 B c d^2\right)+\frac{2 \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2+c d (2 B e-C d)\right)\right)}{\sqrt{a} \sqrt{c}}}{2 \left(a e^2+c d^2\right)^2}","\frac{\log \left(a+c x^2\right) \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)}{2 \left(a e^2+c d^2\right)^2}-\frac{A e^2-B d e+C d^2}{e (d+e x) \left(a e^2+c d^2\right)}-\frac{\log (d+e x) \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)}{\left(a e^2+c d^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2-c d (C d-2 B e)\right)\right)}{\sqrt{a} \sqrt{c} \left(a e^2+c d^2\right)^2}",1,"((-2*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(e*(d + e*x)) + (2*(A*c*(c*d^2 - a*e^2) + a*(a*C*e^2 + c*d*(-(C*d) + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (-2*B*c*d^2 + 4*A*c*d*e - 4*a*C*d*e + 2*a*B*e^2)*Log[d + e*x] + (B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2)*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)","A",1
49,1,277,305,0.3088574,"\int \frac{A+B x+C x^2}{(d+e x)^3 \left(a+c x^2\right)} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^3*(a + c*x^2)),x]","\frac{\log \left(a+c x^2\right) \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)-\frac{\left(a e^2+c d^2\right)^2 \left(e (A e-B d)+C d^2\right)}{e (d+e x)^2}+\frac{2 \left(a e^2+c d^2\right) \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)}{d+e x}-2 \log (d+e x) \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)+\frac{2 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(c d^2-3 a e^2\right)+a \left(a e^2 (3 C d-B e)+c d^2 (3 B e-C d)\right)\right)}{\sqrt{a}}}{2 \left(a e^2+c d^2\right)^3}","\frac{\log \left(a+c x^2\right) \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)}{2 \left(a e^2+c d^2\right)^3}-\frac{A e^2-B d e+C d^2}{2 e (d+e x)^2 \left(a e^2+c d^2\right)}+\frac{-a B e^2+2 a C d e-2 A c d e+B c d^2}{(d+e x) \left(a e^2+c d^2\right)^2}-\frac{\log (d+e x) \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)}{\left(a e^2+c d^2\right)^3}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(c d^2-3 a e^2\right)-a \left(c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right)\right)}{\sqrt{a} \left(a e^2+c d^2\right)^3}",1,"(-(((c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(e*(d + e*x)^2)) + (2*(c*d^2 + a*e^2)*(B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2))/(d + e*x) + (2*Sqrt[c]*(A*c*d*(c*d^2 - 3*a*e^2) + a*(a*e^2*(3*C*d - B*e) + c*d^2*(-(C*d) + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] - 2*(B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[d + e*x] + (B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)","A",1
50,1,233,216,0.204088,"\int \frac{(d+e x)^3 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^2} \, dx","Integrate[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2)^2,x]","\frac{\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+c d^2\right)+a \left(c d^2 (3 B e+C d)-3 a e^2 (B e+3 C d)\right)\right)}{a^{3/2}}+\frac{-a^3 C e^3+a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))-a c^2 d \left(3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right)+A c^3 d^3 x}{a \left(a+c x^2\right)}+e \log \left(a+c x^2\right) \left(-2 a C e^2+c e (A e+3 B d)+3 c C d^2\right)+2 c e^2 x (B e+3 C d)+c C e^3 x^2}{2 c^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+c d^2\right)-a \left(3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right)\right)}{2 a^{3/2} c^{5/2}}-\frac{e \log \left(a+c x^2\right) \left(2 a C e^2-c \left(e (A e+3 B d)+3 C d^2\right)\right)}{2 c^3}-\frac{3 e^2 x (A c d-a (B e+3 C d))}{2 a c^2}-\frac{(d+e x)^3 (a B-x (A c-a C))}{2 a c \left(a+c x^2\right)}-\frac{e^3 x^2 (A c-2 a C)}{2 a c^2}",1,"(2*c*e^2*(3*C*d + B*e)*x + c*C*e^3*x^2 + (-(a^3*C*e^3) + A*c^3*d^3*x - a*c^2*d*(C*d^2*x + 3*A*e*(d + e*x) + B*d*(d + 3*e*x)) + a^2*c*e*(3*C*d*(d + e*x) + e*(3*B*d + A*e + B*e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(A*c*d*(c*d^2 + 3*a*e^2) + a*(-3*a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + e*(3*c*C*d^2 - 2*a*C*e^2 + c*e*(3*B*d + A*e))*Log[a + c*x^2])/(2*c^3)","A",1
51,1,175,146,0.137936,"\int \frac{(d+e x)^2 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^2} \, dx","Integrate[((d + e*x)^2*(A + B*x + C*x^2))/(a + c*x^2)^2,x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(a e^2+c d^2\right)+a \left(c d (2 B e+C d)-3 a C e^2\right)\right)}{a^{3/2}}+\frac{\sqrt{c} \left(a^2 e (B e+2 C d+C e x)-a c \left(A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right)+A c^2 d^2 x\right)}{a \left(a+c x^2\right)}+\sqrt{c} e \log \left(a+c x^2\right) (B e+2 C d)+2 \sqrt{c} C e^2 x}{2 c^{5/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(c d (2 a B e+a C d+A c d)+a e^2 (A c-3 a C)\right)}{2 a^{3/2} c^{5/2}}-\frac{(d+e x)^2 (a B-x (A c-a C))}{2 a c \left(a+c x^2\right)}-\frac{e^2 x (A c-3 a C)}{2 a c^2}+\frac{e \log \left(a+c x^2\right) (B e+2 C d)}{2 c^2}",1,"(2*Sqrt[c]*C*e^2*x + (Sqrt[c]*(A*c^2*d^2*x + a^2*e*(2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + A*e*(2*d + e*x) + B*d*(d + 2*e*x))))/(a*(a + c*x^2)) + ((A*c*(c*d^2 + a*e^2) + a*(-3*a*C*e^2 + c*d*(C*d + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + Sqrt[c]*e*(2*C*d + B*e)*Log[a + c*x^2])/(2*c^(5/2))","A",1
52,1,102,97,0.0946016,"\int \frac{(d+e x) \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^2} \, dx","Integrate[((d + e*x)*(A + B*x + C*x^2))/(a + c*x^2)^2,x]","\frac{\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+A c d)}{a^{3/2}}+\frac{a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x}{a \left(a+c x^2\right)}+C e \log \left(a+c x^2\right)}{2 c^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+A c d)}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a B-x (A c-a C))}{2 a c \left(a+c x^2\right)}+\frac{C e \log \left(a+c x^2\right)}{2 c^2}",1,"((a^2*C*e + A*c^2*d*x - a*c*(A*e + C*d*x + B*(d + e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(A*c*d + a*C*d + a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + C*e*Log[a + c*x^2])/(2*c^2)","A",1
53,1,68,69,0.0517699,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^2,x]","\frac{(a C+A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{3/2}}+\frac{-a B-a C x+A c x}{2 a c \left(a+c x^2\right)}","\frac{(a C+A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{2 a^{3/2} c^{3/2}}-\frac{a B-x (A c-a C)}{2 a c \left(a+c x^2\right)}",1,"(-(a*B) + A*c*x - a*C*x)/(2*a*c*(a + c*x^2)) + ((A*c + a*C)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))","A",1
54,1,195,226,0.2239021,"\int \frac{A+B x+C x^2}{(d+e x) \left(a+c x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)*(a + c*x^2)^2),x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+c d^2\right)+a \left(c d^2-a e^2\right) (C d-B e)\right)}{a^{3/2} \sqrt{c}}+\frac{\left(a e^2+c d^2\right) \left(a^2 (-C) e+a c (A e-B d+B e x-C d x)+A c^2 d x\right)}{a c \left(a+c x^2\right)}-e \log \left(a+c x^2\right) \left(e (A e-B d)+C d^2\right)+2 e \log (d+e x) \left(e (A e-B d)+C d^2\right)}{2 \left(a e^2+c d^2\right)^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+c d^2\right)+a \left(c d^2-a e^2\right) (C d-B e)\right)}{2 a^{3/2} \sqrt{c} \left(a e^2+c d^2\right)^2}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left(a+c x^2\right) \left(a e^2+c d^2\right)}-\frac{e \log \left(a+c x^2\right) \left(A e^2-B d e+C d^2\right)}{2 \left(a e^2+c d^2\right)^2}+\frac{e \log (d+e x) \left(A e^2-B d e+C d^2\right)}{\left(a e^2+c d^2\right)^2}",1,"(((c*d^2 + a*e^2)*(-(a^2*C*e) + A*c^2*d*x + a*c*(-(B*d) + A*e - C*d*x + B*e*x)))/(a*c*(a + c*x^2)) + ((a*(C*d - B*e)*(c*d^2 - a*e^2) + A*c*d*(c*d^2 + 3*a*e^2))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[c]) + 2*e*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x] - e*(C*d^2 + e*(-(B*d) + A*e))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^2)","A",1
55,1,320,374,0.4130256,"\int \frac{A+B x+C x^2}{(d+e x)^2 \left(a+c x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^2*(a + c*x^2)^2),x]","\frac{\frac{\left(a e^2+c d^2\right) \left(a^2 e (B e-2 C d+C e x)-a c \left(A e (e x-2 d)+B d (d-2 e x)+C d^2 x\right)+A c^2 d^2 x\right)}{a \left(a+c x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right)+a \left(a^2 C e^4+6 a c d e^2 (B e-C d)+c^2 d^3 (C d-2 B e)\right)\right)}{a^{3/2} \sqrt{c}}-e \log \left(a+c x^2\right) \left(a e^2 (B e-2 C d)+c d e (4 A e-3 B d)+2 c C d^3\right)+2 e \log (d+e x) \left(a e^2 (B e-2 C d)+c d e (4 A e-3 B d)+2 c C d^3\right)-\frac{2 e \left(a e^2+c d^2\right) \left(e (A e-B d)+C d^2\right)}{d+e x}}{2 \left(a e^2+c d^2\right)^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right)+a \left(a^2 C e^4-6 a c d e^2 (C d-B e)+c^2 d^3 (C d-2 B e)\right)\right)}{2 a^{3/2} \sqrt{c} \left(a e^2+c d^2\right)^3}-\frac{a \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)-x \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2-c d (C d-2 B e)\right)\right)}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)^2}+\frac{e \log \left(a+c x^2\right) \left(a e^2 (2 C d-B e)-c d \left(2 C d^2-e (3 B d-4 A e)\right)\right)}{2 \left(a e^2+c d^2\right)^3}-\frac{e \left(A e^2-B d e+C d^2\right)}{(d+e x) \left(a e^2+c d^2\right)^2}-\frac{e \log (d+e x) \left(a e^2 (2 C d-B e)-c d \left(2 C d^2-e (3 B d-4 A e)\right)\right)}{\left(a e^2+c d^2\right)^3}",1,"((-2*e*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + ((c*d^2 + a*e^2)*(A*c^2*d^2*x + a^2*e*(-2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a*(a + c*x^2)) + ((A*c*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4) + a*(a^2*C*e^4 + c^2*d^3*(C*d - 2*B*e) + 6*a*c*d*e^2*(-(C*d) + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[c]) + 2*e*(2*c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x] - e*(2*c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)","A",1
56,1,466,524,0.6301827,"\int \frac{A+B x+C x^2}{(d+e x)^3 \left(a+c x^2\right)^2} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^3*(a + c*x^2)^2),x]","\frac{-\log \left(a+c x^2\right) \left(a^2 C e^5-2 a c e^3 \left(e (A e-3 B d)+4 C d^2\right)+c^2 d^2 e \left(2 e (5 A e-3 B d)+3 C d^2\right)\right)+2 \log (d+e x) \left(a^2 C e^5-2 a c e^3 \left(e (A e-3 B d)+4 C d^2\right)+c^2 d^2 e \left(2 e (5 A e-3 B d)+3 C d^2\right)\right)+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right)+a \left(-3 a^2 e^4 (B e-3 C d)-2 a c d^2 e^2 (7 C d-9 B e)+c^2 d^4 (C d-3 B e)\right)\right)}{a^{3/2}}+\frac{\left(a e^2+c d^2\right) \left(a^3 C e^3-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))-a c^2 d \left(3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right)+A c^3 d^3 x\right)}{a \left(a+c x^2\right)}-\frac{e \left(a e^2+c d^2\right)^2 \left(e (A e-B d)+C d^2\right)}{(d+e x)^2}-\frac{2 e \left(a e^2+c d^2\right) \left(a e^2 (B e-2 C d)+c d e (4 A e-3 B d)+2 c C d^3\right)}{d+e x}}{2 \left(a e^2+c d^2\right)^4}","-\frac{e \log \left(a+c x^2\right) \left(a^2 C e^4-2 a c e^2 \left(4 C d^2-e (3 B d-A e)\right)+c^2 d^2 \left(3 C d^2-2 e (3 B d-5 A e)\right)\right)}{2 \left(a e^2+c d^2\right)^4}+\frac{e \log (d+e x) \left(a^2 C e^4-2 a c e^2 \left(4 C d^2-e (3 B d-A e)\right)+c^2 d^2 \left(3 C d^2-2 e (3 B d-5 A e)\right)\right)}{\left(a e^2+c d^2\right)^4}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right)-a \left(-3 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (7 C d-9 B e)-c^2 d^4 (C d-3 B e)\right)\right)}{2 a^{3/2} \left(a e^2+c d^2\right)^4}-\frac{a \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)-c x \left(A c d \left(c d^2-3 a e^2\right)-a \left(c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right)\right)}{2 a \left(a+c x^2\right) \left(a e^2+c d^2\right)^3}-\frac{e \left(A e^2-B d e+C d^2\right)}{2 (d+e x)^2 \left(a e^2+c d^2\right)^2}+\frac{e \left(a e^2 (2 C d-B e)-c d \left(2 C d^2-e (3 B d-4 A e)\right)\right)}{(d+e x) \left(a e^2+c d^2\right)^3}",1,"(-((e*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2) - (2*e*(c*d^2 + a*e^2)*(2*c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e)))/(d + e*x) + ((c*d^2 + a*e^2)*(a^3*C*e^3 + A*c^3*d^3*x - a*c^2*d*(C*d^2*x + B*d*(d - 3*e*x) + 3*A*e*(-d + e*x)) - a^2*c*e*(3*C*d*(d - e*x) + e*(-3*B*d + A*e + B*e*x))))/(a*(a + c*x^2)) + (Sqrt[c]*(A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) + a*(-2*a*c*d^2*e^2*(7*C*d - 9*B*e) + c^2*d^4*(C*d - 3*B*e) - 3*a^2*e^4*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*(a^2*C*e^5 - 2*a*c*e^3*(4*C*d^2 + e*(-3*B*d + A*e)) + c^2*d^2*e*(3*C*d^2 + 2*e*(-3*B*d + 5*A*e)))*Log[d + e*x] - (a^2*C*e^5 - 2*a*c*e^3*(4*C*d^2 + e*(-3*B*d + A*e)) + c^2*d^2*e*(3*C*d^2 + 2*e*(-3*B*d + 5*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)","A",1
57,1,281,209,0.2468946,"\int \frac{(d+e x)^3 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^3} \, dx","Integrate[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2)^3,x]","\frac{\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 A c d \left(a e^2+c d^2\right)+a \left(3 a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right)\right)}{a^{5/2}}+\frac{-2 a^3 C e^3+2 a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))-2 a c^2 d \left(3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right)+2 A c^3 d^3 x}{a \left(a+c x^2\right)^2}+\frac{8 a^3 C e^3-a^2 c e (e (4 A e+12 B d+5 B e x)+3 C d (4 d+5 e x))+a c^2 d x \left(3 e (A e+B d)+C d^2\right)+3 A c^3 d^3 x}{a^2 \left(a+c x^2\right)}+4 C e^3 \log \left(a+c x^2\right)}{8 c^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(c d^2 (3 a B e+a C d+3 A c d)+3 a e^2 (a B e+3 a C d+A c d)\right)}{8 a^{5/2} c^{5/2}}-\frac{(d+e x) \left(a e (3 a B e+5 a C d+3 A c d)-x \left(3 A c^2 d^2-a \left(4 a C e^2-c d (3 B e+C d)\right)\right)\right)}{8 a^2 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^3 (a B-x (A c-a C))}{4 a c \left(a+c x^2\right)^2}+\frac{C e^3 \log \left(a+c x^2\right)}{2 c^3}",1,"((-2*a^3*C*e^3 + 2*A*c^3*d^3*x - 2*a*c^2*d*(C*d^2*x + 3*A*e*(d + e*x) + B*d*(d + 3*e*x)) + 2*a^2*c*e*(3*C*d*(d + e*x) + e*(3*B*d + A*e + B*e*x)))/(a*(a + c*x^2)^2) + (8*a^3*C*e^3 + 3*A*c^3*d^3*x + a*c^2*d*(C*d^2 + 3*e*(B*d + A*e))*x - a^2*c*e*(3*C*d*(4*d + 5*e*x) + e*(12*B*d + 4*A*e + 5*B*e*x)))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c*d^2 + a*e^2) + a*(3*a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*C*e^3*Log[a + c*x^2])/(8*c^3)","A",1
58,1,211,156,0.1385995,"\int \frac{(d+e x)^2 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^3} \, dx","Integrate[((d + e*x)^2*(A + B*x + C*x^2))/(a + c*x^2)^3,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(a e^2+3 c d^2\right)+a \left(3 a C e^2+c d (2 B e+C d)\right)\right)}{8 a^{5/2} c^{5/2}}+\frac{a^2 (-e) (4 B e+8 C d+5 C e x)+a c x \left(e (A e+2 B d)+C d^2\right)+3 A c^2 d^2 x}{8 a^2 c^2 \left(a+c x^2\right)}+\frac{a^2 e (B e+2 C d+C e x)-a c \left(A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right)+A c^2 d^2 x}{4 a c^2 \left(a+c x^2\right)^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(c d (2 a B e+a C d+3 A c d)+a e^2 (3 a C+A c)\right)}{8 a^{5/2} c^{5/2}}-\frac{(d+e x) (a e (3 a C+A c)-c x (2 a B e+a C d+3 A c d))}{8 a^2 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^2 (a B-x (A c-a C))}{4 a c \left(a+c x^2\right)^2}",1,"(3*A*c^2*d^2*x + a*c*(C*d^2 + e*(2*B*d + A*e))*x - a^2*e*(8*C*d + 4*B*e + 5*C*e*x))/(8*a^2*c^2*(a + c*x^2)) + (A*c^2*d^2*x + a^2*e*(2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + A*e*(2*d + e*x) + B*d*(d + 2*e*x)))/(4*a*c^2*(a + c*x^2)^2) + ((A*c*(3*c*d^2 + a*e^2) + a*(3*a*C*e^2 + c*d*(C*d + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2))","A",1
59,1,137,130,0.1010152,"\int \frac{(d+e x) \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^3} \, dx","Integrate[((d + e*x)*(A + B*x + C*x^2))/(a + c*x^2)^3,x]","\frac{\frac{\sqrt{a} \left(-4 a^2 C e+a c x (B e+C d)+3 A c^2 d x\right)}{a+c x^2}+\frac{2 a^{3/2} \left(a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x\right)}{\left(a+c x^2\right)^2}+\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+3 A c d)}{8 a^{5/2} c^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+3 A c d)}{8 a^{5/2} c^{3/2}}-\frac{2 a e (a C+A c)-c x (a B e+a C d+3 A c d)}{8 a^2 c^2 \left(a+c x^2\right)}-\frac{(d+e x) (a B-x (A c-a C))}{4 a c \left(a+c x^2\right)^2}",1,"((Sqrt[a]*(-4*a^2*C*e + 3*A*c^2*d*x + a*c*(C*d + B*e)*x))/(a + c*x^2) + (2*a^(3/2)*(a^2*C*e + A*c^2*d*x - a*c*(A*e + C*d*x + B*(d + e*x))))/(a + c*x^2)^2 + Sqrt[c]*(3*A*c*d + a*C*d + a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^2)","A",1
60,1,90,98,0.0696313,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^3,x]","\frac{(a C+3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{3/2}}+\frac{-a^2 (2 B+C x)+a c x \left(5 A+C x^2\right)+3 A c^2 x^3}{8 a^2 c \left(a+c x^2\right)^2}","\frac{(a C+3 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{8 a^{5/2} c^{3/2}}+\frac{x (a C+3 A c)}{8 a^2 c \left(a+c x^2\right)}-\frac{a B-x (A c-a C)}{4 a c \left(a+c x^2\right)^2}",1,"(3*A*c^2*x^3 - a^2*(2*B + C*x) + a*c*x*(5*A + C*x^2))/(8*a^2*c*(a + c*x^2)^2) + ((3*A*c + a*C)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))","A",1
61,1,321,353,0.424092,"\int \frac{A+B x+C x^2}{(d+e x) \left(a+c x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)*(a + c*x^2)^3),x]","\frac{\frac{2 \left(a e^2+c d^2\right)^2 \left(a^2 (-C) e+a c (A e-B d+B e x-C d x)+A c^2 d x\right)}{a c \left(a+c x^2\right)^2}+\frac{\left(a e^2+c d^2\right) \left(a^2 e (e (4 A e-4 B d+3 B e x)+C d (4 d-3 e x))+a c d x \left(e (7 A e-B d)+C d^2\right)+3 A c^2 d^3 x\right)}{a^2 \left(a+c x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right)+a \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) (C d-B e)\right)}{a^{5/2} \sqrt{c}}-4 e^3 \log \left(a+c x^2\right) \left(e (A e-B d)+C d^2\right)+8 e^3 \log (d+e x) \left(e (A e-B d)+C d^2\right)}{8 \left(a e^2+c d^2\right)^3}","\frac{4 a^2 e \left(A e^2-B d e+C d^2\right)+x \left(A c d \left(7 a e^2+3 c d^2\right)+a \left(c d^2-3 a e^2\right) (C d-B e)\right)}{8 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right)+a \left(-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right) (C d-B e)\right)}{8 a^{5/2} \sqrt{c} \left(a e^2+c d^2\right)^3}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{4 a c \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)}-\frac{e^3 \log \left(a+c x^2\right) \left(A e^2-B d e+C d^2\right)}{2 \left(a e^2+c d^2\right)^3}+\frac{e^3 \log (d+e x) \left(A e^2-B d e+C d^2\right)}{\left(a e^2+c d^2\right)^3}",1,"((2*(c*d^2 + a*e^2)^2*(-(a^2*C*e) + A*c^2*d*x + a*c*(-(B*d) + A*e - C*d*x + B*e*x)))/(a*c*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(3*A*c^2*d^3*x + a*c*d*(C*d^2 + e*(-(B*d) + 7*A*e))*x + a^2*e*(C*d*(4*d - 3*e*x) + e*(-4*B*d + 4*A*e + 3*B*e*x))))/(a^2*(a + c*x^2)) + ((a*(C*d - B*e)*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4) + A*c*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[c]) + 8*e^3*(C*d^2 + e*(-(B*d) + A*e))*Log[d + e*x] - 4*e^3*(C*d^2 + e*(-(B*d) + A*e))*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^3)","A",1
62,1,498,571,0.7642623,"\int \frac{A+B x+C x^2}{(d+e x)^2 \left(a+c x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^2*(a + c*x^2)^3),x]","\frac{\frac{2 \left(a e^2+c d^2\right)^2 \left(a^2 e (B e-2 C d+C e x)-a c \left(A e (e x-2 d)+B d (d-2 e x)+C d^2 x\right)+A c^2 d^2 x\right)}{a \left(a+c x^2\right)^2}+\frac{\left(a e^2+c d^2\right) \left(a^3 e^3 (4 B e-8 C d+3 C e x)+a^2 c e \left(e (A e (16 d-7 e x)-2 B d (6 d-7 e x))+4 C d^2 (2 d-3 e x)\right)+a c^2 d^2 x \left(2 e (6 A e-B d)+C d^2\right)+3 A c^3 d^4 x\right)}{a^2 \left(a+c x^2\right)}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 A c \left(-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right)+a \left(3 a^3 C e^6+3 a^2 c d e^4 (10 B e-11 C d)+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right)\right)}{a^{5/2} \sqrt{c}}-4 e^3 \log \left(a+c x^2\right) \left(a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right)+8 e^3 \log (d+e x) \left(a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right)-\frac{8 e^3 \left(a e^2+c d^2\right) \left(e (A e-B d)+C d^2\right)}{d+e x}}{8 \left(a e^2+c d^2\right)^4}","-\frac{4 a^2 e \left(a e^2 (2 C d-B e)-c d \left(2 C d^2-e (3 B d-4 A e)\right)\right)-x \left(A c \left(-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right)+a \left(3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right)\right)}{8 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 A c \left(-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right)+a \left(3 a^3 C e^6-3 a^2 c d e^4 (11 C d-10 B e)+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right)\right)}{8 a^{5/2} \sqrt{c} \left(a e^2+c d^2\right)^4}-\frac{a \left(-a B e^2+2 a C d e-2 A c d e+B c d^2\right)-x \left(A c \left(c d^2-a e^2\right)+a \left(a C e^2-c d (C d-2 B e)\right)\right)}{4 a \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)^2}+\frac{e^3 \log \left(a+c x^2\right) \left(a e^2 (2 C d-B e)-c d \left(4 C d^2-e (5 B d-6 A e)\right)\right)}{2 \left(a e^2+c d^2\right)^4}-\frac{e^3 \left(A e^2-B d e+C d^2\right)}{(d+e x) \left(a e^2+c d^2\right)^3}-\frac{e^3 \log (d+e x) \left(a e^2 (2 C d-B e)-c d \left(4 C d^2-e (5 B d-6 A e)\right)\right)}{\left(a e^2+c d^2\right)^4}",1,"((-8*e^3*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^2*(A*c^2*d^2*x + a^2*e*(-2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(3*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 2*e*(-(B*d) + 6*A*e))*x + a^3*e^3*(-8*C*d + 4*B*e + 3*C*e*x) + a^2*c*e*(4*C*d^2*(2*d - 3*e*x) + e*(-2*B*d*(6*d - 7*e*x) + A*e*(16*d - 7*e*x)))))/(a^2*(a + c*x^2)) + ((3*A*c*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6) + a*(3*a^3*C*e^6 + a*c^2*d^3*e^2*(13*C*d - 20*B*e) + c^3*d^5*(C*d - 2*B*e) + 3*a^2*c*d*e^4*(-11*C*d + 10*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[c]) + 8*e^3*(4*c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x] - 4*e^3*(4*c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e))*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)","A",1
63,1,672,753,1.0782562,"\int \frac{A+B x+C x^2}{(d+e x)^3 \left(a+c x^2\right)^3} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^3*(a + c*x^2)^3),x]","\frac{-4 \log \left(a+c x^2\right) \left(a^2 C e^7+a c e^5 \left(-3 A e^2+9 B d e-13 C d^2\right)+c^2 d^2 e^3 \left(3 e (7 A e-5 B d)+10 C d^2\right)\right)+8 \log (d+e x) \left(a^2 C e^7+a c e^5 \left(-3 A e^2+9 B d e-13 C d^2\right)+c^2 d^2 e^3 \left(3 e (7 A e-5 B d)+10 C d^2\right)\right)+\frac{2 \left(a e^2+c d^2\right)^2 \left(a^3 C e^3-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))-a c^2 d \left(3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right)+A c^3 d^3 x\right)}{a \left(a+c x^2\right)^2}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 A c d \left(-35 a^3 e^6+35 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right)+a \left(-15 a^3 e^6 (B e-3 C d)-5 a^2 c d^2 e^4 (25 C d-27 B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right)\right)}{a^{5/2}}+\frac{\left(a e^2+c d^2\right) \left(4 a^4 C e^5+a^3 c e^3 (e (-8 A e+24 B d-7 B e x)+C d (21 e x-32 d))+a^2 c^2 d e \left(e \left(40 A d e-33 A e^2 x-24 B d^2+38 B d e x\right)+2 C d^2 (6 d-13 e x)\right)+a c^3 d^3 x \left(3 e (6 A e-B d)+C d^2\right)+3 A c^4 d^5 x\right)}{a^2 \left(a+c x^2\right)}-\frac{4 e^3 \left(a e^2+c d^2\right)^2 \left(e (A e-B d)+C d^2\right)}{(d+e x)^2}-\frac{8 e^3 \left(a e^2+c d^2\right) \left(a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right)}{d+e x}}{8 \left(a e^2+c d^2\right)^5}","-\frac{e^3 \log \left(a+c x^2\right) \left(a^2 C e^4-a c e^2 \left(3 A e^2-9 B d e+13 C d^2\right)+c^2 d^2 \left(10 C d^2-3 e (5 B d-7 A e)\right)\right)}{2 \left(a e^2+c d^2\right)^5}+\frac{e^3 \log (d+e x) \left(a^2 C e^4-a c e^2 \left(3 A e^2-9 B d e+13 C d^2\right)+c^2 d^2 \left(10 C d^2-3 e (5 B d-7 A e)\right)\right)}{\left(a e^2+c d^2\right)^5}+\frac{4 a^2 e \left(a^2 C e^4-2 a c e^2 \left(4 C d^2-e (3 B d-A e)\right)+c^2 d^2 \left(3 C d^2-2 e (3 B d-5 A e)\right)\right)+c x \left(3 A c d \left(-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right)-a \left(-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right)\right)}{8 a^2 \left(a+c x^2\right) \left(a e^2+c d^2\right)^4}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 A c d \left(-35 a^3 e^6+35 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right)+a \left(15 a^3 e^6 (3 C d-B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right)\right)}{8 a^{5/2} \left(a e^2+c d^2\right)^5}-\frac{a \left(B c d \left(c d^2-3 a e^2\right)-e (A c-a C) \left(3 c d^2-a e^2\right)\right)-c x \left(A c d \left(c d^2-3 a e^2\right)-a \left(c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right)\right)}{4 a \left(a+c x^2\right)^2 \left(a e^2+c d^2\right)^3}+\frac{e^3 \left(a e^2 (2 C d-B e)-c d \left(4 C d^2-e (5 B d-6 A e)\right)\right)}{(d+e x) \left(a e^2+c d^2\right)^4}-\frac{e^3 \left(A e^2-B d e+C d^2\right)}{2 (d+e x)^2 \left(a e^2+c d^2\right)^3}",1,"((-4*e^3*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 - (8*e^3*(c*d^2 + a*e^2)*(4*c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^2*(a^3*C*e^3 + A*c^3*d^3*x - a*c^2*d*(C*d^2*x + B*d*(d - 3*e*x) + 3*A*e*(-d + e*x)) - a^2*c*e*(3*C*d*(d - e*x) + e*(-3*B*d + A*e + B*e*x))))/(a*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(4*a^4*C*e^5 + 3*A*c^4*d^5*x + a*c^3*d^3*(C*d^2 + 3*e*(-(B*d) + 6*A*e))*x + a^3*c*e^3*(C*d*(-32*d + 21*e*x) + e*(24*B*d - 8*A*e - 7*B*e*x)) + a^2*c^2*d*e*(2*C*d^2*(6*d - 13*e*x) + e*(-24*B*d^2 + 40*A*d*e + 38*B*d*e*x - 33*A*e^2*x))))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6 + 7*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25*C*d - 27*B*e) + c^3*d^6*(C*d - 3*B*e) - 15*a^3*e^6*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*(a^2*C*e^7 + a*c*e^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[d + e*x] - 4*(a^2*C*e^7 + a*c*e^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^5)","A",1
64,1,437,234,0.3063408,"\int \frac{(d+e x)^4 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^4} \, dx","Integrate[((d + e*x)^4*(A + B*x + C*x^2))/(a + c*x^2)^4,x]","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(a e^2+5 c d^2\right)+a \left(5 a C e^2+c d (4 B e+C d)\right)\right)}{16 a^{7/2} c^{7/2}}+\frac{-a^3 e^3 (8 B e+32 C d+11 C e x)+a^2 c e^2 x \left(e (A e+4 B d)+6 C d^2\right)+a c^2 d^2 x \left(6 A e^2+4 B d e+C d^2\right)+5 A c^3 d^4 x}{16 a^3 c^3 \left(a+c x^2\right)}+\frac{-a^3 e^3 (B e+4 C d+C e x)+a^2 c e \left(e (A e (4 d+e x)+2 B d (3 d+2 e x))+2 C d^2 (2 d+3 e x)\right)-a c^2 d^2 \left(4 A d e+6 A e^2 x+B d (d+4 e x)+C d^2 x\right)+A c^3 d^4 x}{6 a c^3 \left(a+c x^2\right)^3}+\frac{a^3 e^3 (12 B e+48 C d+13 C e x)-a^2 c e \left(e (A e (24 d+7 e x)+4 B d (9 d+7 e x))+6 C d^2 (4 d+7 e x)\right)+a c^2 d^2 x \left(6 A e^2+4 B d e+C d^2\right)+5 A c^3 d^4 x}{24 a^2 c^3 \left(a+c x^2\right)^2}","\frac{\left(a e^2+c d^2\right) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right)}{16 a^{7/2} c^{7/2}}-\frac{(d+e x) (a e-c d x) \left(c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right)}{16 a^3 c^3 \left(a+c x^2\right)}-\frac{(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{24 a^2 c^2 \left(a+c x^2\right)^2}-\frac{(d+e x)^4 (a B-x (A c-a C))}{6 a c \left(a+c x^2\right)^3}",1,"(5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^2*c*e^2*(6*C*d^2 + e*(4*B*d + A*e))*x - a^3*e^3*(32*C*d + 8*B*e + 11*C*e*x))/(16*a^3*c^3*(a + c*x^2)) + (A*c^3*d^4*x - a^3*e^3*(4*C*d + B*e + C*e*x) - a*c^2*d^2*(4*A*d*e + C*d^2*x + 6*A*e^2*x + B*d*(d + 4*e*x)) + a^2*c*e*(2*C*d^2*(2*d + 3*e*x) + e*(A*e*(4*d + e*x) + 2*B*d*(3*d + 2*e*x))))/(6*a*c^3*(a + c*x^2)^3) + (5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^3*e^3*(48*C*d + 12*B*e + 13*C*e*x) - a^2*c*e*(6*C*d^2*(4*d + 7*e*x) + e*(4*B*d*(9*d + 7*e*x) + A*e*(24*d + 7*e*x))))/(24*a^2*c^3*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(A*c*(5*c*d^2 + a*e^2) + a*(5*a*C*e^2 + c*d*(C*d + 4*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(7/2))","A",1
65,1,350,254,0.2972918,"\int \frac{(d+e x)^3 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^4} \, dx","Integrate[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2)^4,x]","\frac{-\frac{3 \sqrt{a} \left(8 a^3 C e^3-a^2 c e^2 x (B e+3 C d)-a c^2 d x \left(3 e (A e+B d)+C d^2\right)-5 A c^3 d^3 x\right)}{a+c x^2}-\frac{8 a^{5/2} \left(a^3 C e^3-a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))+a c^2 d \left(3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right)-A c^3 d^3 x\right)}{\left(a+c x^2\right)^3}+\frac{2 a^{3/2} \left(12 a^3 C e^3-a^2 c e (e (6 A e+18 B d+7 B e x)+3 C d (6 d+7 e x))+a c^2 d x \left(3 e (A e+B d)+C d^2\right)+5 A c^3 d^3 x\right)}{\left(a+c x^2\right)^2}+3 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+5 c d^2\right)+a \left(a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right)\right)}{48 a^{7/2} c^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c d \left(3 a e^2+5 c d^2\right)+a \left(a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right)\right)}{16 a^{7/2} c^{5/2}}-\frac{(d+e x) \left(a e (3 a B e+a C d+5 A c d)-x \left(3 c d (3 a B e+a C d+5 A c d)+4 a e^2 (2 a C+A c)\right)\right)}{48 a^3 c^2 \left(a+c x^2\right)}-\frac{(d+e x)^2 (2 a e (2 a C+A c)-c x (3 a B e+a C d+5 A c d))}{24 a^2 c^2 \left(a+c x^2\right)^2}-\frac{(d+e x)^3 (a B-x (A c-a C))}{6 a c \left(a+c x^2\right)^3}",1,"((-3*Sqrt[a]*(8*a^3*C*e^3 - 5*A*c^3*d^3*x - a^2*c*e^2*(3*C*d + B*e)*x - a*c^2*d*(C*d^2 + 3*e*(B*d + A*e))*x))/(a + c*x^2) - (8*a^(5/2)*(a^3*C*e^3 - A*c^3*d^3*x + a*c^2*d*(C*d^2*x + 3*A*e*(d + e*x) + B*d*(d + 3*e*x)) - a^2*c*e*(3*C*d*(d + e*x) + e*(3*B*d + A*e + B*e*x))))/(a + c*x^2)^3 + (2*a^(3/2)*(12*a^3*C*e^3 + 5*A*c^3*d^3*x + a*c^2*d*(C*d^2 + 3*e*(B*d + A*e))*x - a^2*c*e*(3*C*d*(6*d + 7*e*x) + e*(18*B*d + 6*A*e + 7*B*e*x))))/(a + c*x^2)^2 + 3*Sqrt[c]*(A*c*d*(5*c*d^2 + 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7/2)*c^3)","A",1
66,1,266,225,0.1610852,"\int \frac{(d+e x)^2 \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^4} \, dx","Integrate[((d + e*x)^2*(A + B*x + C*x^2))/(a + c*x^2)^4,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(A c \left(a e^2+5 c d^2\right)+a \left(a C e^2+c d (2 B e+C d)\right)\right)}{16 a^{7/2} c^{5/2}}+\frac{x \left(A c \left(a e^2+5 c d^2\right)+a \left(a C e^2+c d (2 B e+C d)\right)\right)}{16 a^3 c^2 \left(a+c x^2\right)}+\frac{a^2 (-e) (6 B e+12 C d+7 C e x)+a c x \left(e (A e+2 B d)+C d^2\right)+5 A c^2 d^2 x}{24 a^2 c^2 \left(a+c x^2\right)^2}+\frac{a^2 e (B e+2 C d+C e x)-a c \left(A e (2 d+e x)+B d (d+2 e x)+C d^2 x\right)+A c^2 d^2 x}{6 a c^2 \left(a+c x^2\right)^3}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(c d (2 a B e+a C d+5 A c d)+a e^2 (a C+A c)\right)}{16 a^{7/2} c^{5/2}}+\frac{x \left(c d (2 a B e+a C d+5 A c d)+a e^2 (a C+A c)\right)}{16 a^3 c^2 \left(a+c x^2\right)}-\frac{x \left(3 a e^2 (a C+A c)-c d (2 a B e+a C d+5 A c d)\right)+2 a e (a B e+2 a C d+4 A c d)}{24 a^2 c^2 \left(a+c x^2\right)^2}-\frac{(d+e x)^2 (a B-x (A c-a C))}{6 a c \left(a+c x^2\right)^3}",1,"((A*c*(5*c*d^2 + a*e^2) + a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x)/(16*a^3*c^2*(a + c*x^2)) + (5*A*c^2*d^2*x + a*c*(C*d^2 + e*(2*B*d + A*e))*x - a^2*e*(12*C*d + 6*B*e + 7*C*e*x))/(24*a^2*c^2*(a + c*x^2)^2) + (A*c^2*d^2*x + a^2*e*(2*C*d + B*e + C*e*x) - a*c*(C*d^2*x + A*e*(2*d + e*x) + B*d*(d + 2*e*x)))/(6*a*c^2*(a + c*x^2)^3) + ((A*c*(5*c*d^2 + a*e^2) + a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(5/2))","A",1
67,1,171,165,0.1341894,"\int \frac{(d+e x) \left(A+B x+C x^2\right)}{\left(a+c x^2\right)^4} \, dx","Integrate[((d + e*x)*(A + B*x + C*x^2))/(a + c*x^2)^4,x]","\frac{\frac{8 a^{5/2} \left(a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x\right)}{\left(a+c x^2\right)^3}+\frac{2 a^{3/2} \left(-6 a^2 C e+a c x (B e+C d)+5 A c^2 d x\right)}{\left(a+c x^2\right)^2}+\frac{3 \sqrt{a} c x (a B e+a C d+5 A c d)}{a+c x^2}+3 \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+5 A c d)}{48 a^{7/2} c^2}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a B e+a C d+5 A c d)}{16 a^{7/2} c^{3/2}}+\frac{x (a B e+a C d+5 A c d)}{16 a^3 c \left(a+c x^2\right)}-\frac{2 a e (a C+2 A c)-c x (a B e+a C d+5 A c d)}{24 a^2 c^2 \left(a+c x^2\right)^2}-\frac{(d+e x) (a B-x (A c-a C))}{6 a c \left(a+c x^2\right)^3}",1,"((2*a^(3/2)*(-6*a^2*C*e + 5*A*c^2*d*x + a*c*(C*d + B*e)*x))/(a + c*x^2)^2 + (3*Sqrt[a]*c*(5*A*c*d + a*C*d + a*B*e)*x)/(a + c*x^2) + (8*a^(5/2)*(a^2*C*e + A*c^2*d*x - a*c*(A*e + C*d*x + B*(d + e*x))))/(a + c*x^2)^3 + 3*Sqrt[c]*(5*A*c*d + a*C*d + a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7/2)*c^2)","A",1
68,1,112,126,0.088075,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^4} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^4,x]","\frac{(a C+5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{3/2}}+\frac{-a^3 (8 B+3 C x)+a^2 c x \left(33 A+8 C x^2\right)+a c^2 x^3 \left(40 A+3 C x^2\right)+15 A c^3 x^5}{48 a^3 c \left(a+c x^2\right)^3}","\frac{(a C+5 A c) \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{16 a^{7/2} c^{3/2}}+\frac{x (a C+5 A c)}{16 a^3 c \left(a+c x^2\right)}+\frac{x (a C+5 A c)}{24 a^2 c \left(a+c x^2\right)^2}-\frac{a B-x (A c-a C)}{6 a c \left(a+c x^2\right)^3}",1,"(15*A*c^3*x^5 - a^3*(8*B + 3*C*x) + a*c^2*x^3*(40*A + 3*C*x^2) + a^2*c*x*(33*A + 8*C*x^2))/(48*a^3*c*(a + c*x^2)^3) + ((5*A*c + a*C)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(3/2))","A",1
69,1,29,43,0.0187744,"\int \frac{x^3 \left(1+x+x^2\right)}{\left(1+x^2\right)^2} \, dx","Integrate[(x^3*(1 + x + x^2))/(1 + x^2)^2,x]","\frac{1}{2} \left(x \left(\frac{1}{x^2+1}+x+2\right)-\log \left(x^2+1\right)-3 \tan ^{-1}(x)\right)","\frac{x^2}{2}-\frac{1}{2} \log \left(x^2+1\right)-\frac{x^3}{2 \left(x^2+1\right)}+\frac{3 x}{2}-\frac{3}{2} \tan ^{-1}(x)",1,"(x*(2 + x + (1 + x^2)^(-1)) - 3*ArcTan[x] - Log[1 + x^2])/2","A",1
70,1,27,30,0.0210471,"\int \frac{x^2 \left(1+x+x^2\right)}{\left(1+x^2\right)^2} \, dx","Integrate[(x^2*(1 + x + x^2))/(1 + x^2)^2,x]","\frac{1}{2 \left(x^2+1\right)}+\frac{1}{2} \log \left(x^2+1\right)+x-\tan ^{-1}(x)","-\frac{x^2}{2 \left(x^2+1\right)}+\frac{1}{2} \log \left(x^2+1\right)+x-\tan ^{-1}(x)",1,"x + 1/(2*(1 + x^2)) - ArcTan[x] + Log[1 + x^2]/2","A",1
71,1,23,29,0.0085369,"\int \frac{x \left(1+x+x^2\right)}{\left(1+x^2\right)^2} \, dx","Integrate[(x*(1 + x + x^2))/(1 + x^2)^2,x]","\frac{1}{2} \left(-\frac{x}{x^2+1}+\log \left(x^2+1\right)+\tan ^{-1}(x)\right)","-\frac{x}{2 \left(x^2+1\right)}+\frac{1}{2} \log \left(x^2+1\right)+\frac{1}{2} \tan ^{-1}(x)",1,"(-(x/(1 + x^2)) + ArcTan[x] + Log[1 + x^2])/2","A",1
72,1,14,14,0.0070649,"\int \frac{1+x+x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(1 + x^2)^2,x]","\tan ^{-1}(x)-\frac{1}{2 \left(x^2+1\right)}","\tan ^{-1}(x)-\frac{1}{2 \left(x^2+1\right)}",1,"-1/2*1/(1 + x^2) + ArcTan[x]","A",1
73,1,28,31,0.0104374,"\int \frac{1+x+x^2}{x \left(1+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x*(1 + x^2)^2),x]","\frac{1}{2} \left(\frac{x}{x^2+1}-\log \left(x^2+1\right)+2 \log (x)+\tan ^{-1}(x)\right)","\frac{x}{2 \left(x^2+1\right)}-\frac{1}{2} \log \left(x^2+1\right)+\log (x)+\frac{1}{2} \tan ^{-1}(x)",1,"(x/(1 + x^2) + ArcTan[x] + 2*Log[x] - Log[1 + x^2])/2","A",1
74,1,33,33,0.0158831,"\int \frac{1+x+x^2}{x^2 \left(1+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x^2*(1 + x^2)^2),x]","\frac{1}{2 \left(x^2+1\right)}-\frac{1}{2} \log \left(x^2+1\right)-\frac{1}{x}+\log (x)-\tan ^{-1}(x)","\frac{1}{2 \left(x^2+1\right)}-\frac{1}{2} \log \left(x^2+1\right)-\frac{1}{x}+\log (x)-\tan ^{-1}(x)",1,"-x^(-1) + 1/(2*(1 + x^2)) - ArcTan[x] + Log[x] - Log[1 + x^2]/2","A",1
75,1,39,45,0.0163497,"\int \frac{1+x+x^2}{x^3 \left(1+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x^3*(1 + x^2)^2),x]","\frac{1}{2} \left(-\frac{x}{x^2+1}-\frac{1}{x^2}+\log \left(x^2+1\right)-\frac{2}{x}-2 \log (x)-3 \tan ^{-1}(x)\right)","-\frac{x}{2 \left(x^2+1\right)}-\frac{1}{2 x^2}+\frac{1}{2} \log \left(x^2+1\right)-\frac{1}{x}-\log (x)-\frac{3}{2} \tan ^{-1}(x)",1,"(-x^(-2) - 2/x - x/(1 + x^2) - 3*ArcTan[x] - 2*Log[x] + Log[1 + x^2])/2","A",1
76,1,12,12,0.0122199,"\int \frac{1+2 x+x^2}{\left(1+x^2\right)^2} \, dx","Integrate[(1 + 2*x + x^2)/(1 + x^2)^2,x]","\tan ^{-1}(x)-\frac{1}{x^2+1}","\tan ^{-1}(x)-\frac{1}{x^2+1}",1,"-(1 + x^2)^(-1) + ArcTan[x]","A",1
77,1,27,27,0.0109081,"\int \frac{2+12 x+3 x^2}{\left(4+x^2\right)^2} \, dx","Integrate[(2 + 12*x + 3*x^2)/(4 + x^2)^2,x]","\frac{-5 x-24}{4 \left(x^2+4\right)}+\frac{7}{8} \tan ^{-1}\left(\frac{x}{2}\right)","\frac{7}{8} \tan ^{-1}\left(\frac{x}{2}\right)-\frac{5 x+24}{4 \left(x^2+4\right)}",1,"(-24 - 5*x)/(4*(4 + x^2)) + (7*ArcTan[x/2])/8","A",1
78,1,362,390,0.4566557,"\int (g+h x)^3 \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^3*Sqrt[a + c*x^2]*(d + e*x + f*x^2),x]","\frac{105 a \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(a^2 h^2 (e h+3 f g)-2 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)+\sqrt{a+c x^2} \left(16 c x^2 \left(-4 a^2 f h^3+7 a c h \left(h (d h+3 e g)+3 f g^2\right)+35 c^2 g^2 (3 d h+e g)\right)+16 a \left(8 a^2 f h^3-14 a c h \left(h (d h+3 e g)+3 f g^2\right)+35 c^2 g^2 (3 d h+e g)\right)+105 c x \left(-a^2 h^2 (e h+3 f g)+2 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)+70 c^2 x^3 \left(a h^2 (e h+3 f g)+6 c \left(3 g h (d h+e g)+f g^3\right)\right)+48 c^2 h x^4 \left(a f h^2+7 c \left(h (d h+3 e g)+3 f g^2\right)\right)+280 c^3 h^2 x^5 (e h+3 f g)+240 c^3 f h^3 x^6\right)}{1680 c^3}","\frac{x \sqrt{a+c x^2} \left(a^2 h^2 (e h+3 f g)-2 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)}{16 c^2}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(a^2 h^2 (e h+3 f g)-2 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)}{16 c^{5/2}}+\frac{\left(a+c x^2\right)^{3/2} \left(8 \left(8 a^2 f h^4-2 a c h^2 \left(7 h (d h+3 e g)+15 f g^2\right)-c^2 g^2 \left(3 f g^2-7 h (12 d h+e g)\right)\right)-3 c h x \left(a h^2 (35 e h+41 f g)+2 c g \left(3 f g^2-7 h (7 d h+e g)\right)\right)\right)}{840 c^3 h}-\frac{\left(a+c x^2\right)^{3/2} (g+h x)^2 \left(8 a f h^2+c \left(3 f g^2-7 h (2 d h+e g)\right)\right)}{70 c^2 h}-\frac{\left(a+c x^2\right)^{3/2} (g+h x)^3 (3 f g-7 e h)}{42 c h}+\frac{f \left(a+c x^2\right)^{3/2} (g+h x)^4}{7 c h}",1,"(Sqrt[a + c*x^2]*(16*a*(8*a^2*f*h^3 + 35*c^2*g^2*(e*g + 3*d*h) - 14*a*c*h*(3*f*g^2 + h*(3*e*g + d*h))) + 105*c*(8*c^2*d*g^3 - a^2*h^2*(3*f*g + e*h) + 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x + 16*c*(-4*a^2*f*h^3 + 35*c^2*g^2*(e*g + 3*d*h) + 7*a*c*h*(3*f*g^2 + h*(3*e*g + d*h)))*x^2 + 70*c^2*(a*h^2*(3*f*g + e*h) + 6*c*(f*g^3 + 3*g*h*(e*g + d*h)))*x^3 + 48*c^2*h*(a*f*h^2 + 7*c*(3*f*g^2 + h*(3*e*g + d*h)))*x^4 + 280*c^3*h^2*(3*f*g + e*h)*x^5 + 240*c^3*f*h^3*x^6) + 105*a*Sqrt[c]*(8*c^2*d*g^3 + a^2*h^2*(3*f*g + e*h) - 2*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(1680*c^3)","A",1
79,1,256,280,0.6462634,"\int (g+h x)^2 \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^2*Sqrt[a + c*x^2]*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(\sqrt{c} \left(a^2 (-h) (32 e h+64 f g+15 f h x)+2 a c \left(5 d h (16 g+3 h x)+e \left(40 g^2+30 g h x+8 h^2 x^2\right)+f x \left(15 g^2+16 g h x+5 h^2 x^2\right)\right)+4 c^2 x \left(5 d \left(6 g^2+8 g h x+3 h^2 x^2\right)+x \left(2 e \left(10 g^2+15 g h x+6 h^2 x^2\right)+f x \left(15 g^2+24 g h x+10 h^2 x^2\right)\right)\right)\right)+\frac{15 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(a^2 f h^2-2 a c \left(h (d h+2 e g)+f g^2\right)+8 c^2 d g^2\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{240 c^{5/2}}","\frac{x \sqrt{a+c x^2} \left(a^2 f h^2-2 a c \left(h (d h+2 e g)+f g^2\right)+8 c^2 d g^2\right)}{16 c^2}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(a^2 f h^2-2 a c \left(h (d h+2 e g)+f g^2\right)+8 c^2 d g^2\right)}{16 c^{5/2}}-\frac{\left(a+c x^2\right)^{3/2} \left(8 \left(2 a h^2 (e h+2 f g)+c g \left(f g^2-2 h (5 d h+e g)\right)\right)-3 h x \left(5 h^2 (2 c d-a f)-2 c g (f g-2 e h)\right)\right)}{120 c^2 h}-\frac{\left(a+c x^2\right)^{3/2} (g+h x)^2 (f g-2 e h)}{10 c h}+\frac{f \left(a+c x^2\right)^{3/2} (g+h x)^3}{6 c h}",1,"(Sqrt[a + c*x^2]*(Sqrt[c]*(-(a^2*h*(64*f*g + 32*e*h + 15*f*h*x)) + 2*a*c*(5*d*h*(16*g + 3*h*x) + f*x*(15*g^2 + 16*g*h*x + 5*h^2*x^2) + e*(40*g^2 + 30*g*h*x + 8*h^2*x^2)) + 4*c^2*x*(5*d*(6*g^2 + 8*g*h*x + 3*h^2*x^2) + x*(2*e*(10*g^2 + 15*g*h*x + 6*h^2*x^2) + f*x*(15*g^2 + 24*g*h*x + 10*h^2*x^2)))) + (15*Sqrt[a]*(8*c^2*d*g^2 + a^2*f*h^2 - 2*a*c*(f*g^2 + h*(2*e*g + d*h)))*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(240*c^(5/2))","A",1
80,1,153,175,0.4171979,"\int (g+h x) \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)*Sqrt[a + c*x^2]*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(-16 a^2 f h-\frac{15 \sqrt{a} \sqrt{c} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a e h+a f g-4 c d g)}{\sqrt{\frac{c x^2}{a}+1}}+a c (40 d h+5 e (8 g+3 h x)+f x (15 g+8 h x))+2 c^2 x (10 d (3 g+2 h x)+x (5 e (4 g+3 h x)+3 f x (5 g+4 h x)))\right)}{120 c^2}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (-a e h-a f g+4 c d g)}{8 c^{3/2}}-\frac{\left(a+c x^2\right)^{3/2} \left(4 \left(2 a f h^2+c \left(3 f g^2-5 h (d h+e g)\right)\right)+3 c h x (3 f g-5 e h)\right)}{60 c^2 h}+\frac{x \sqrt{a+c x^2} (4 c d g-a (e h+f g))}{8 c}+\frac{f \left(a+c x^2\right)^{3/2} (g+h x)^2}{5 c h}",1,"(Sqrt[a + c*x^2]*(-16*a^2*f*h + a*c*(40*d*h + 5*e*(8*g + 3*h*x) + f*x*(15*g + 8*h*x)) + 2*c^2*x*(10*d*(3*g + 2*h*x) + x*(5*e*(4*g + 3*h*x) + 3*f*x*(5*g + 4*h*x))) - (15*Sqrt[a]*Sqrt[c]*(-4*c*d*g + a*f*g + a*e*h)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(120*c^2)","A",1
81,1,98,106,0.2037194,"\int \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[Sqrt[a + c*x^2]*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(\sqrt{c} (a (8 e+3 f x)+2 c x (6 d+x (4 e+3 f x)))-\frac{3 \sqrt{a} (a f-4 c d) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{24 c^{3/2}}","\frac{a (4 c d-a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} (4 c d-a f)}{8 c}+\frac{e \left(a+c x^2\right)^{3/2}}{3 c}+\frac{f x \left(a+c x^2\right)^{3/2}}{4 c}",1,"(Sqrt[a + c*x^2]*(Sqrt[c]*(a*(8*e + 3*f*x) + 2*c*x*(6*d + x*(4*e + 3*f*x))) - (3*Sqrt[a]*(-4*c*d + a*f)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(24*c^(3/2))","A",1
82,1,224,206,0.4357848,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{g+h x} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x),x]","\frac{\left(h (d h-e g)+f g^2\right) \left(-\sqrt{a h^2+c g^2} \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)-\sqrt{c} g \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+h \sqrt{a+c x^2}\right)}{h^4}+\frac{\sqrt{a+c x^2} \left(\sqrt{c} x \sqrt{\frac{c x^2}{a}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right) (e h-f g)}{2 \sqrt{c} h^2 \sqrt{\frac{c x^2}{a}+1}}+\frac{f \left(a+c x^2\right)^{3/2}}{3 c h}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(\left(a h^2+2 c g^2\right) (f g-e h)+2 c d g h^2\right)}{2 \sqrt{c} h^4}-\frac{\sqrt{a h^2+c g^2} \left(d h^2-e g h+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{h^4}+\frac{\sqrt{a+c x^2} \left(2 \left(d h^2-e g h+f g^2\right)-h x (f g-e h)\right)}{2 h^3}+\frac{f \left(a+c x^2\right)^{3/2}}{3 c h}",1,"(f*(a + c*x^2)^(3/2))/(3*c*h) + ((-(f*g) + e*h)*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(2*Sqrt[c]*h^2*Sqrt[1 + (c*x^2)/a]) + ((f*g^2 + h*(-(e*g) + d*h))*(h*Sqrt[a + c*x^2] - Sqrt[c]*g*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[c*g^2 + a*h^2]*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])]))/h^4","A",1
83,1,264,308,0.2580392,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^2} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^2,x]","\frac{\frac{h \sqrt{a+c x^2} \left(2 h (-d h+2 e g+e h x)+f \left(-6 g^2-3 g h x+h^2 x^2\right)\right)}{g+h x}+\frac{\log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(a f h^2+2 c h (d h-2 e g)+6 c f g^2\right)}{\sqrt{c}}+\frac{2 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(a h^2 (2 f g-e h)+c g h (d h-2 e g)+3 c f g^3\right)}{\sqrt{a h^2+c g^2}}-\frac{2 \log (g+h x) \left(a h^2 (2 f g-e h)+c g h (d h-2 e g)+3 c f g^3\right)}{\sqrt{a h^2+c g^2}}}{2 h^4}","-\frac{\left(a+c x^2\right)^{3/2} \left(d h^2-e g h+f g^2\right)}{h (g+h x) \left(a h^2+c g^2\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(a f h^2+2 c \left(3 f g^2-h (2 e g-d h)\right)\right)}{2 \sqrt{c} h^4}+\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a h^2 (2 f g-e h)+c g \left(3 f g^2-h (2 e g-d h)\right)\right)}{h^4 \sqrt{a h^2+c g^2}}-\frac{\sqrt{a+c x^2} \left(2 \left(a h^2 (2 f g-e h)+c g \left(3 f g^2-h (2 e g-d h)\right)\right)-h x \left(a f h^2+c \left(3 f g^2-2 h (e g-d h)\right)\right)\right)}{2 h^3 \left(a h^2+c g^2\right)}",1,"((h*Sqrt[a + c*x^2]*(2*h*(2*e*g - d*h + e*h*x) + f*(-6*g^2 - 3*g*h*x + h^2*x^2)))/(g + h*x) - (2*(3*c*f*g^3 + c*g*h*(-2*e*g + d*h) + a*h^2*(2*f*g - e*h))*Log[g + h*x])/Sqrt[c*g^2 + a*h^2] + ((6*c*f*g^2 + a*f*h^2 + 2*c*h*(-2*e*g + d*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c] + (2*(3*c*f*g^3 + c*g*h*(-2*e*g + d*h) + a*h^2*(2*f*g - e*h))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/Sqrt[c*g^2 + a*h^2])/(2*h^4)","A",1
84,1,318,296,0.5614156,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^3} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^3,x]","\frac{-\frac{\log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(2 a^2 f h^4+a c h^2 \left(h (d h-3 e g)+9 f g^2\right)+2 c^2 g^3 (3 f g-e h)\right)}{\left(a h^2+c g^2\right)^{3/2}}+\frac{\log (g+h x) \left(2 a^2 f h^4+a c h^2 \left(h (d h-3 e g)+9 f g^2\right)+2 c^2 g^3 (3 f g-e h)\right)}{\left(a h^2+c g^2\right)^{3/2}}+h \sqrt{a+c x^2} \left(\frac{-2 a h^2 (e h-2 f g)+c g h (d h-3 e g)+5 c f g^3}{(g+h x) \left(a h^2+c g^2\right)}+\frac{h (e g-d h)-f g^2}{(g+h x)^2}+2 f\right)+2 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) (e h-3 f g)}{2 h^4}","-\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(2 a^2 f h^4+a c h^2 \left(9 f g^2-h (3 e g-d h)\right)+2 c^2 g^3 (3 f g-e h)\right)}{2 h^4 \left(a h^2+c g^2\right)^{3/2}}-\frac{\left(a+c x^2\right)^{3/2} \left(d h^2-e g h+f g^2\right)}{2 h (g+h x)^2 \left(a h^2+c g^2\right)}+\frac{\sqrt{a+c x^2} \left(h x \left(2 a f h^2+c \left(3 f g^2-h (e g-d h)\right)\right)+2 \left(a h^2+c g^2\right) (3 f g-e h)\right)}{2 h^3 (g+h x) \left(a h^2+c g^2\right)}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (3 f g-e h)}{h^4}",1,"(h*Sqrt[a + c*x^2]*(2*f + (-(f*g^2) + h*(e*g - d*h))/(g + h*x)^2 + (5*c*f*g^3 + c*g*h*(-3*e*g + d*h) - 2*a*h^2*(-2*f*g + e*h))/((c*g^2 + a*h^2)*(g + h*x))) + ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h^2*(9*f*g^2 + h*(-3*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(3/2) + 2*Sqrt[c]*(-3*f*g + e*h)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - ((2*a^2*f*h^4 + 2*c^2*g^3*(3*f*g - e*h) + a*c*h^2*(9*f*g^2 + h*(-3*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(3/2))/(2*h^4)","A",1
85,1,382,314,0.8249753,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^4} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]","\frac{\frac{3 c \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(a^2 h^4 (4 f g-e h)+a c g h^2 \left(5 f g^2-d h^2\right)+2 c^2 f g^5\right)}{\left(a h^2+c g^2\right)^{5/2}}-\frac{3 c \log (g+h x) \left(a^2 h^4 (4 f g-e h)+a c g h^2 \left(5 f g^2-d h^2\right)+2 c^2 f g^5\right)}{\left(a h^2+c g^2\right)^{5/2}}+\frac{h \sqrt{a+c x^2} \left(-\frac{(g+h x)^2 \left(6 a^2 f h^4+a c h^2 \left(h (2 d h-5 e g)+20 f g^2\right)+c^2 \left(11 f g^4-g^2 h (d h+2 e g)\right)\right)}{\left(a h^2+c g^2\right)^2}+\frac{(g+h x) \left(-3 a h^2 (e h-2 f g)+c g h (d h-4 e g)+7 c f g^3\right)}{a h^2+c g^2}-2 \left(h (d h-e g)+f g^2\right)\right)}{(g+h x)^3}+6 \sqrt{c} f \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{6 h^4}","\frac{c \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a^2 h^4 (4 f g-e h)+a c g h^2 \left(5 f g^2-d h^2\right)+2 c^2 f g^5\right)}{2 h^4 \left(a h^2+c g^2\right)^{5/2}}-\frac{\sqrt{a+c x^2} \left(h x \left(2 a^2 f h^4+a c g h^2 (6 f g-e h)+c^2 \left(3 f g^4-d g^2 h^2\right)\right)+a^2 e h^5+a c g h^2 \left(d h^2+3 f g^2\right)+2 c^2 f g^5\right)}{2 h^3 (g+h x)^2 \left(a h^2+c g^2\right)^2}-\frac{\left(a+c x^2\right)^{3/2} \left(d h^2-e g h+f g^2\right)}{3 h (g+h x)^3 \left(a h^2+c g^2\right)}+\frac{\sqrt{c} f \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{h^4}",1,"((h*Sqrt[a + c*x^2]*(-2*(f*g^2 + h*(-(e*g) + d*h)) + ((7*c*f*g^3 + c*g*h*(-4*e*g + d*h) - 3*a*h^2*(-2*f*g + e*h))*(g + h*x))/(c*g^2 + a*h^2) - ((6*a^2*f*h^4 + c^2*(11*f*g^4 - g^2*h*(2*e*g + d*h)) + a*c*h^2*(20*f*g^2 + h*(-5*e*g + 2*d*h)))*(g + h*x)^2)/(c*g^2 + a*h^2)^2))/(g + h*x)^3 - (3*c*(2*c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) + a*c*g*h^2*(5*f*g^2 - d*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(5/2) + 6*Sqrt[c]*f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c*(2*c^2*f*g^5 + a^2*h^4*(4*f*g - e*h) + a*c*g*h^2*(5*f*g^2 - d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(5/2))/(6*h^4)","A",1
86,1,439,313,1.3050623,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^5} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^5,x]","\frac{1}{24} \left(-\frac{3 a c \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(4 a^2 f h^2-a c \left(h (d h-5 e g)+f g^2\right)+4 c^2 d g^2\right)}{\left(a h^2+c g^2\right)^{7/2}}+\frac{3 a c \log (g+h x) \left(4 a^2 f h^2-a c \left(h (d h-5 e g)+f g^2\right)+4 c^2 d g^2\right)}{\left(a h^2+c g^2\right)^{7/2}}-\frac{\sqrt{a+c x^2} \left((g+h x)^2 \left(a h^2+c g^2\right) \left(12 a^2 f h^4+a c h^2 \left(h (3 d h-7 e g)+35 f g^2\right)+2 c^2 \left(9 f g^4-g^2 h (d h+e g)\right)\right)-c (g+h x)^3 \left(4 a^2 h^4 (7 f g-2 e h)+a c g h^2 \left(h (9 e g-13 d h)+19 f g^2\right)+2 c^2 \left(g^3 h (d h+e g)+3 f g^5\right)\right)+6 \left(a h^2+c g^2\right)^3 \left(h (d h-e g)+f g^2\right)-2 (g+h x) \left(a h^2+c g^2\right)^2 \left(-4 a h^2 (e h-2 f g)+c g h (d h-5 e g)+9 c f g^3\right)\right)}{(g+h x)^4 \left(a h^3+c g^2 h\right)^3}\right)","-\frac{\sqrt{a+c x^2} (a h-c g x) \left(4 a^2 f h^2-a c \left(f g^2-h (5 e g-d h)\right)+4 c^2 d g^2\right)}{8 (g+h x)^2 \left(a h^2+c g^2\right)^3}-\frac{a c \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(4 a^2 f h^2-a c \left(f g^2-h (5 e g-d h)\right)+4 c^2 d g^2\right)}{8 \left(a h^2+c g^2\right)^{7/2}}+\frac{\left(a+c x^2\right)^{3/2} \left(4 a h^2 (2 f g-e h)+c g \left(h (e g-5 d h)+3 f g^2\right)\right)}{12 h (g+h x)^3 \left(a h^2+c g^2\right)^2}-\frac{\left(a+c x^2\right)^{3/2} \left(d h^2-e g h+f g^2\right)}{4 h (g+h x)^4 \left(a h^2+c g^2\right)}",1,"(-((Sqrt[a + c*x^2]*(6*(c*g^2 + a*h^2)^3*(f*g^2 + h*(-(e*g) + d*h)) - 2*(c*g^2 + a*h^2)^2*(9*c*f*g^3 + c*g*h*(-5*e*g + d*h) - 4*a*h^2*(-2*f*g + e*h))*(g + h*x) + (c*g^2 + a*h^2)*(12*a^2*f*h^4 + 2*c^2*(9*f*g^4 - g^2*h*(e*g + d*h)) + a*c*h^2*(35*f*g^2 + h*(-7*e*g + 3*d*h)))*(g + h*x)^2 - c*(4*a^2*h^4*(7*f*g - 2*e*h) + a*c*g*h^2*(19*f*g^2 + h*(9*e*g - 13*d*h)) + 2*c^2*(3*f*g^5 + g^3*h*(e*g + d*h)))*(g + h*x)^3))/((c*g^2*h + a*h^3)^3*(g + h*x)^4)) + (3*a*c*(4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 + h*(-5*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) - (3*a*c*(4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 + h*(-5*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/24","A",1
87,1,583,433,1.6188672,"\int \frac{\sqrt{a+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^6} \, dx","Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^6,x]","-\frac{a c^2 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(a^2 h^2 (6 f g-e h)-a c g \left(3 h (d h-2 e g)+f g^2\right)+4 c^2 d g^3\right)}{8 \left(a h^2+c g^2\right)^{9/2}}+\frac{a c^2 \log (g+h x) \left(a^2 h^2 (6 f g-e h)-a c g \left(3 h (d h-2 e g)+f g^2\right)+4 c^2 d g^3\right)}{8 \left(a h^2+c g^2\right)^{9/2}}-\frac{\sqrt{a+c x^2} \left(2 (g+h x)^2 \left(a h^2+c g^2\right)^2 \left(20 a^2 f h^4+a c h^2 \left(h (4 d h-9 e g)+54 f g^2\right)+c^2 \left(27 f g^4-g^2 h (3 d h+2 e g)\right)\right)-c (g+h x)^3 \left(a h^2+c g^2\right) \left(5 a^2 h^4 (10 f g-3 e h)+a c g h^2 \left(h (24 e g-29 d h)+21 f g^2\right)+c^2 \left(2 g^3 h (3 d h+2 e g)+6 f g^5\right)\right)-c (g+h x)^4 \left(-40 a^3 f h^6+a^2 c h^4 \left(h (16 d h-81 e g)+86 f g^2\right)+a c^2 g^2 h^2 \left(h (28 e g-83 d h)+27 f g^2\right)+c^3 \left(2 g^4 h (3 d h+2 e g)+6 f g^6\right)\right)+24 \left(a h^2+c g^2\right)^4 \left(h (d h-e g)+f g^2\right)-6 (g+h x) \left(a h^2+c g^2\right)^3 \left(-5 a h^2 (e h-2 f g)+c g h (d h-6 e g)+11 c f g^3\right)\right)}{120 h^3 (g+h x)^5 \left(a h^2+c g^2\right)^4}","-\frac{\left(a+c x^2\right)^{3/2} \left(20 a^2 f h^4-a c h^2 \left(18 f g^2-h (33 e g-8 d h)\right)-c^2 g^2 \left(h (2 e g-27 d h)+3 f g^2\right)\right)}{60 h (g+h x)^3 \left(a h^2+c g^2\right)^3}-\frac{c \sqrt{a+c x^2} (a h-c g x) \left(a^2 h^2 (6 f g-e h)-a c g \left(f g^2-3 h (2 e g-d h)\right)+4 c^2 d g^3\right)}{8 (g+h x)^2 \left(a h^2+c g^2\right)^4}-\frac{a c^2 \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a^2 h^2 (6 f g-e h)-a c g \left(f g^2-3 h (2 e g-d h)\right)+4 c^2 d g^3\right)}{8 \left(a h^2+c g^2\right)^{9/2}}+\frac{\left(a+c x^2\right)^{3/2} \left(5 a h^2 (2 f g-e h)+c g \left(h (2 e g-7 d h)+3 f g^2\right)\right)}{20 h (g+h x)^4 \left(a h^2+c g^2\right)^2}-\frac{\left(a+c x^2\right)^{3/2} \left(d h^2-e g h+f g^2\right)}{5 h (g+h x)^5 \left(a h^2+c g^2\right)}",1,"-1/120*(Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 6*(c*g^2 + a*h^2)^3*(11*c*f*g^3 + c*g*h*(-6*e*g + d*h) - 5*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(27*f*g^4 - g^2*h*(2*e*g + 3*d*h)) + a*c*h^2*(54*f*g^2 + h*(-9*e*g + 4*d*h)))*(g + h*x)^2 - c*(c*g^2 + a*h^2)*(5*a^2*h^4*(10*f*g - 3*e*h) + a*c*g*h^2*(21*f*g^2 + h*(24*e*g - 29*d*h)) + c^2*(6*f*g^5 + 2*g^3*h*(2*e*g + 3*d*h)))*(g + h*x)^3 - c*(-40*a^3*f*h^6 + a*c^2*g^2*h^2*(27*f*g^2 + h*(28*e*g - 83*d*h)) + c^3*(6*f*g^6 + 2*g^4*h*(2*e*g + 3*d*h)) + a^2*c*h^4*(86*f*g^2 + h*(-81*e*g + 16*d*h)))*(g + h*x)^4))/(h^3*(c*g^2 + a*h^2)^4*(g + h*x)^5) + (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log[g + h*x])/(8*(c*g^2 + a*h^2)^(9/2)) - (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(8*(c*g^2 + a*h^2)^(9/2))","A",1
88,1,481,462,0.540768,"\int (g+h x)^3 \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^3*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{315 a^2 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(3 a^2 h^2 (e h+3 f g)-8 a c g \left(3 h (d h+e g)+f g^2\right)+48 c^2 d g^3\right)+\sqrt{a+c x^2} \left(384 c^2 x^4 \left(a^2 f h^3+24 a c h \left(h (d h+3 e g)+3 f g^2\right)+21 c^2 g^2 (3 d h+e g)\right)+128 a c x^2 \left(-4 a^2 f h^3+9 a c h \left(h (d h+3 e g)+3 f g^2\right)+126 c^2 g^2 (3 d h+e g)\right)+128 a^2 \left(8 a^2 f h^3-18 a c h \left(h (d h+3 e g)+3 f g^2\right)+63 c^2 g^2 (3 d h+e g)\right)+210 c^2 x^3 \left(3 a^2 h^2 (e h+3 f g)+56 a c g \left(3 h (d h+e g)+f g^2\right)+48 c^2 d g^3\right)+315 a c x \left(-3 a^2 h^2 (e h+3 f g)+8 a c g \left(3 h (d h+e g)+f g^2\right)+80 c^2 d g^3\right)+840 c^3 x^5 \left(9 a h^2 (e h+3 f g)+8 c \left(3 g h (d h+e g)+f g^3\right)\right)+640 c^3 h x^6 \left(10 a f h^2+9 c \left(h (d h+3 e g)+3 f g^2\right)\right)+5040 c^4 h^2 x^7 (e h+3 f g)+4480 c^4 f h^3 x^8\right)}{40320 c^3}","\frac{x \left(a+c x^2\right)^{3/2} \left(3 a^2 h^2 (e h+3 f g)-8 a c g \left(3 h (d h+e g)+f g^2\right)+48 c^2 d g^3\right)}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left(3 a^2 h^2 (e h+3 f g)-8 a c g \left(3 h (d h+e g)+f g^2\right)+48 c^2 d g^3\right)}{128 c^2}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 h^2 (e h+3 f g)-8 a c g \left(3 h (d h+e g)+f g^2\right)+48 c^2 d g^3\right)}{128 c^{5/2}}+\frac{\left(a+c x^2\right)^{5/2} \left(4 \left(32 a^2 f h^4-8 a c h^2 \left(9 h (d h+3 e g)+17 f g^2\right)-3 c^2 g^2 \left(5 f g^2-3 h (64 d h+3 e g)\right)\right)-5 c h x \left(a h^2 (63 e h+61 f g)+2 c g \left(5 f g^2-9 h (12 d h+e g)\right)\right)\right)}{5040 c^3 h}+\frac{\left(a+c x^2\right)^{5/2} (g+h x)^2 \left(8 h^2 (9 c d-4 a f)-3 c g (5 f g-9 e h)\right)}{504 c^2 h}-\frac{\left(a+c x^2\right)^{5/2} (g+h x)^3 (5 f g-9 e h)}{72 c h}+\frac{f \left(a+c x^2\right)^{5/2} (g+h x)^4}{9 c h}",1,"(Sqrt[a + c*x^2]*(128*a^2*(8*a^2*f*h^3 + 63*c^2*g^2*(e*g + 3*d*h) - 18*a*c*h*(3*f*g^2 + h*(3*e*g + d*h))) + 315*a*c*(80*c^2*d*g^3 - 3*a^2*h^2*(3*f*g + e*h) + 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x + 128*a*c*(-4*a^2*f*h^3 + 126*c^2*g^2*(e*g + 3*d*h) + 9*a*c*h*(3*f*g^2 + h*(3*e*g + d*h)))*x^2 + 210*c^2*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) + 56*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*x^3 + 384*c^2*(a^2*f*h^3 + 21*c^2*g^2*(e*g + 3*d*h) + 24*a*c*h*(3*f*g^2 + h*(3*e*g + d*h)))*x^4 + 840*c^3*(9*a*h^2*(3*f*g + e*h) + 8*c*(f*g^3 + 3*g*h*(e*g + d*h)))*x^5 + 640*c^3*h*(10*a*f*h^2 + 9*c*(3*f*g^2 + h*(3*e*g + d*h)))*x^6 + 5040*c^4*h^2*(3*f*g + e*h)*x^7 + 4480*c^4*f*h^3*x^8) + 315*a^2*Sqrt[c]*(48*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 8*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(40320*c^3)","A",1
89,1,346,346,1.0949818,"\int (g+h x)^2 \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(-\frac{280 a \left(3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{c} x \left(5 a+2 c x^2\right) \sqrt{\frac{c x^2}{a}+1}\right) \left(h (d h+2 e g)+f g^2\right)}{c^{3/2} \sqrt{\frac{c x^2}{a}+1}}+1680 d g^2 \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right)+\frac{105 a f h^2 \left(\frac{3 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}-\sqrt{c} x \left(3 a^2+14 a c x^2+8 c^2 x^4\right)\right)}{c^{5/2}}+\frac{384 h \left(a+c x^2\right)^2 \left(5 c x^2-2 a\right) (e h+2 f g)}{c^2}+\frac{2240 x \left(a+c x^2\right)^2 \left(h (d h+2 e g)+f g^2\right)}{c}+\frac{2688 g \left(a+c x^2\right)^2 (2 d h+e g)}{c}+\frac{1680 f h^2 x^3 \left(a+c x^2\right)^2}{c}\right)}{13440}","\frac{x \left(a+c x^2\right)^{3/2} \left(3 a^2 f h^2-8 a c \left(h (d h+2 e g)+f g^2\right)+48 c^2 d g^2\right)}{192 c^2}+\frac{a x \sqrt{a+c x^2} \left(3 a^2 f h^2-8 a c \left(h (d h+2 e g)+f g^2\right)+48 c^2 d g^2\right)}{128 c^2}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 f h^2-8 a c \left(h (d h+2 e g)+f g^2\right)+48 c^2 d g^2\right)}{128 c^{5/2}}-\frac{\left(a+c x^2\right)^{5/2} \left(12 \left(8 a h^2 (e h+2 f g)+c g \left(5 f g^2-8 h (7 d h+e g)\right)\right)-5 h x \left(7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right)\right)}{1680 c^2 h}-\frac{\left(a+c x^2\right)^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac{f \left(a+c x^2\right)^{5/2} (g+h x)^3}{8 c h}",1,"(Sqrt[a + c*x^2]*((2688*g*(e*g + 2*d*h)*(a + c*x^2)^2)/c + (2240*(f*g^2 + h*(2*e*g + d*h))*x*(a + c*x^2)^2)/c + (1680*f*h^2*x^3*(a + c*x^2)^2)/c + (384*h*(2*f*g + e*h)*(a + c*x^2)^2*(-2*a + 5*c*x^2))/c^2 - (280*a*(f*g^2 + h*(2*e*g + d*h))*(Sqrt[c]*x*(5*a + 2*c*x^2)*Sqrt[1 + (c*x^2)/a] + 3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(c^(3/2)*Sqrt[1 + (c*x^2)/a]) + (105*a*f*h^2*(-(Sqrt[c]*x*(3*a^2 + 14*a*c*x^2 + 8*c^2*x^4)) + (3*a^(5/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/c^(5/2) + 1680*d*g^2*(5*a*x + 2*c*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/a]))))/13440","A",1
90,1,209,213,0.633104,"\int (g+h x) \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(-\frac{105 a^{5/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (a e h+a f g-6 c d g)}{c^{3/2} \left(a+c x^2\right)}-\frac{96 a^3 f h}{c^2}+\frac{3 a^2 (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))}{c}+2 a x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))+4 c x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))\right)}{1680}","\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac{\left(a+c x^2\right)^{5/2} \left(6 \left(2 a f h^2+c \left(5 f g^2-7 h (d h+e g)\right)\right)+5 c h x (5 f g-7 e h)\right)}{210 c^2 h}+\frac{x \left(a+c x^2\right)^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac{a x \sqrt{a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac{f \left(a+c x^2\right)^{5/2} (g+h x)^2}{7 c h}",1,"(Sqrt[a + c*x^2]*((-96*a^3*f*h)/c^2 + (3*a^2*(112*d*h + 7*e*(16*g + 5*h*x) + f*x*(35*g + 16*h*x)))/c + 4*c*x^3*(21*d*(5*g + 4*h*x) + 2*x*(7*e*(6*g + 5*h*x) + 5*f*x*(7*g + 6*h*x))) + 2*a*x*(21*d*(25*g + 16*h*x) + x*(7*e*(48*g + 35*h*x) + f*x*(245*g + 192*h*x))) - (105*a^(5/2)*(-6*c*d*g + a*f*g + a*e*h)*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(c^(3/2)*(a + c*x^2))))/1680","A",1
91,1,125,137,0.2572659,"\int \left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\sqrt{a+c x^2} \left(\sqrt{c} \left(3 a^2 (16 e+5 f x)+2 a c x (75 d+x (48 e+35 f x))+4 c^2 x^3 (15 d+2 x (6 e+5 f x))\right)-\frac{15 a^{3/2} (a f-6 c d) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{240 c^{3/2}}","\frac{a^2 (6 c d-a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{16 c^{3/2}}+\frac{x \left(a+c x^2\right)^{3/2} (6 c d-a f)}{24 c}+\frac{a x \sqrt{a+c x^2} (6 c d-a f)}{16 c}+\frac{e \left(a+c x^2\right)^{5/2}}{5 c}+\frac{f x \left(a+c x^2\right)^{5/2}}{6 c}",1,"(Sqrt[a + c*x^2]*(Sqrt[c]*(3*a^2*(16*e + 5*f*x) + 4*c^2*x^3*(15*d + 2*x*(6*e + 5*f*x)) + 2*a*c*x*(75*d + x*(48*e + 35*f*x))) - (15*a^(3/2)*(-6*c*d + a*f)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(240*c^(3/2))","A",1
92,1,348,326,1.2108964,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{g+h x} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x),x]","\frac{\sqrt{a+c x^2} \left(3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{c} x \left(5 a+2 c x^2\right) \sqrt{\frac{c x^2}{a}+1}\right) (e h-f g)}{8 \sqrt{c} h^2 \sqrt{\frac{c x^2}{a}+1}}-\frac{\left(h (d h-e g)+f g^2\right) \left(\sqrt{\frac{c x^2}{a}+1} \left(-h \sqrt{a+c x^2} \left(8 a h^2+6 c g^2-3 c g h x+2 c h^2 x^2\right)+6 \left(a h^2+c g^2\right)^{3/2} \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)+6 \sqrt{c} g \left(a h^2+c g^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)\right)+3 \sqrt{a} \sqrt{c} g h^2 \sqrt{a+c x^2} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)\right)}{6 h^6 \sqrt{\frac{c x^2}{a}+1}}+\frac{f \left(a+c x^2\right)^{5/2}}{5 c h}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 h^4 (f g-e h)+12 a c g h^2 \left(f g^2-h (e g-d h)\right)+8 c^2 g^3 \left(f g^2-h (e g-d h)\right)\right)}{8 \sqrt{c} h^6}-\frac{\left(a h^2+c g^2\right)^{3/2} \left(d h^2-e g h+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{h^6}+\frac{\sqrt{a+c x^2} \left(8 \left(a h^2+c g^2\right) \left(d h^2-e g h+f g^2\right)-h x \left(\left(3 a h^2+4 c g^2\right) (f g-e h)+4 c d g h^2\right)\right)}{8 h^5}+\frac{\left(a+c x^2\right)^{3/2} \left(4 \left(d h^2-e g h+f g^2\right)-3 h x (f g-e h)\right)}{12 h^3}+\frac{f \left(a+c x^2\right)^{5/2}}{5 c h}",1,"(f*(a + c*x^2)^(5/2))/(5*c*h) + ((-(f*g) + e*h)*Sqrt[a + c*x^2]*(Sqrt[c]*x*(5*a + 2*c*x^2)*Sqrt[1 + (c*x^2)/a] + 3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(8*Sqrt[c]*h^2*Sqrt[1 + (c*x^2)/a]) - ((f*g^2 + h*(-(e*g) + d*h))*(3*Sqrt[a]*Sqrt[c]*g*h^2*Sqrt[a + c*x^2]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]] + Sqrt[1 + (c*x^2)/a]*(-(h*Sqrt[a + c*x^2]*(6*c*g^2 + 8*a*h^2 - 3*c*g*h*x + 2*c*h^2*x^2)) + 6*Sqrt[c]*g*(c*g^2 + a*h^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + 6*(c*g^2 + a*h^2)^(3/2)*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])))/(6*h^6*Sqrt[1 + (c*x^2)/a])","A",1
93,1,392,432,0.52507,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^2} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^2,x]","\frac{\frac{3 \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(3 a^2 f h^4+12 a c h^2 \left(h (d h-2 e g)+3 f g^2\right)+8 c^2 \left(g^2 h (3 d h-4 e g)+5 f g^4\right)\right)}{\sqrt{c}}+24 \sqrt{a h^2+c g^2} \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(a h^2 (2 f g-e h)+c g h (3 d h-4 e g)+5 c f g^3\right)-24 \sqrt{a h^2+c g^2} \log (g+h x) \left(a h^2 (2 f g-e h)+c g h (3 d h-4 e g)+5 c f g^3\right)+h \sqrt{a+c x^2} \left(8 \left(4 a h^2 (e h-2 f g)-3 c \left(g h (2 d h-3 e g)+4 f g^3\right)\right)+3 h x \left(5 a f h^2+4 c \left(h (d h-2 e g)+3 f g^2\right)\right)-\frac{24 \left(a h^2+c g^2\right) \left(h (d h-e g)+f g^2\right)}{g+h x}+8 c h^2 x^2 (e h-2 f g)+6 c f h^3 x^3\right)}{24 h^6}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 f h^4+12 a c h^2 \left(3 f g^2-h (2 e g-d h)\right)+8 c^2 g^2 \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{8 \sqrt{c} h^6}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{h (g+h x) \left(a h^2+c g^2\right)}+\frac{\sqrt{a h^2+c g^2} \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a h^2 (2 f g-e h)+c g \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{h^6}-\frac{\sqrt{a+c x^2} \left(8 \left(a h^2 (2 f g-e h)+c g \left(5 f g^2-h (4 e g-3 d h)\right)\right)-h x \left(3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right)\right)}{8 h^5}-\frac{\left(a+c x^2\right)^{3/2} \left(4 \left(a h^2 (2 f g-e h)+c g \left(5 f g^2-h (4 e g-3 d h)\right)\right)-3 h x \left(a f h^2+c \left(5 f g^2-4 h (e g-d h)\right)\right)\right)}{12 h^3 \left(a h^2+c g^2\right)}",1,"(h*Sqrt[a + c*x^2]*(8*(4*a*h^2*(-2*f*g + e*h) - 3*c*(4*f*g^3 + g*h*(-3*e*g + 2*d*h))) + 3*h*(5*a*f*h^2 + 4*c*(3*f*g^2 + h*(-2*e*g + d*h)))*x + 8*c*h^2*(-2*f*g + e*h)*x^2 + 6*c*f*h^3*x^3 - (24*(c*g^2 + a*h^2)*(f*g^2 + h*(-(e*g) + d*h)))/(g + h*x)) - 24*Sqrt[c*g^2 + a*h^2]*(5*c*f*g^3 + c*g*h*(-4*e*g + 3*d*h) + a*h^2*(2*f*g - e*h))*Log[g + h*x] + (3*(3*a^2*f*h^4 + 12*a*c*h^2*(3*f*g^2 + h*(-2*e*g + d*h)) + 8*c^2*(5*f*g^4 + g^2*h*(-4*e*g + 3*d*h)))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c] + 24*Sqrt[c*g^2 + a*h^2]*(5*c*f*g^3 + c*g*h*(-4*e*g + 3*d*h) + a*h^2*(2*f*g - e*h))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(24*h^6)","A",1
94,1,435,488,0.6547682,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^3} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^3,x]","\frac{-\frac{3 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(2 a^2 f h^4+a c h^2 \left(3 h (d h-3 e g)+19 f g^2\right)+2 c^2 \left(3 g^2 h (d h-2 e g)+10 f g^4\right)\right)}{\sqrt{a h^2+c g^2}}+\frac{3 \log (g+h x) \left(2 a^2 f h^4+a c h^2 \left(3 h (d h-3 e g)+19 f g^2\right)+2 c^2 \left(3 g^2 h (d h-2 e g)+10 f g^4\right)\right)}{\sqrt{a h^2+c g^2}}-3 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(-3 a h^2 (e h-3 f g)+6 c g h (d h-2 e g)+20 c f g^3\right)+\frac{h \sqrt{a+c x^2} \left(a h^2 \left(f \left(17 g^2+28 g h x+8 h^2 x^2\right)-3 h (d h+e g+2 e h x)\right)+c \left(3 h \left(d h \left(6 g^2+9 g h x+2 h^2 x^2\right)+e \left(-12 g^3-18 g^2 h x-4 g h^2 x^2+h^3 x^3\right)\right)+f \left(60 g^4+90 g^3 h x+20 g^2 h^2 x^2-5 g h^3 x^3+2 h^4 x^4\right)\right)\right)}{(g+h x)^2}}{6 h^6}","-\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(2 a^2 f h^4+a c h^2 \left(19 f g^2-3 h (3 e g-d h)\right)+2 c^2 g^2 \left(10 f g^2-3 h (2 e g-d h)\right)\right)}{2 h^6 \sqrt{a h^2+c g^2}}+\frac{\sqrt{a+c x^2} \left(2 a^2 f h^4-c h x \left(a h^2 (7 f g-3 e h)+c g \left(10 f g^2-3 h (2 e g-d h)\right)\right)+a c h^2 \left(19 f g^2-3 h (3 e g-d h)\right)+2 c^2 g^2 \left(10 f g^2-3 h (2 e g-d h)\right)\right)}{2 h^5 \left(a h^2+c g^2\right)}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{2 h (g+h x)^2 \left(a h^2+c g^2\right)}-\frac{\left(a+c x^2\right)^{3/2} \left(2 \left(c g \left(-3 d h+6 e g-\frac{10 f g^2}{h}\right)-a h (7 f g-3 e h)\right)-x \left(2 a f h^2+c \left(5 f g^2-3 h (e g-d h)\right)\right)\right)}{6 h^2 (g+h x) \left(a h^2+c g^2\right)}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a h^2 (3 f g-e h)+2 c g \left(10 f g^2-3 h (2 e g-d h)\right)\right)}{2 h^6}",1,"((h*Sqrt[a + c*x^2]*(a*h^2*(-3*h*(e*g + d*h + 2*e*h*x) + f*(17*g^2 + 28*g*h*x + 8*h^2*x^2)) + c*(f*(60*g^4 + 90*g^3*h*x + 20*g^2*h^2*x^2 - 5*g*h^3*x^3 + 2*h^4*x^4) + 3*h*(d*h*(6*g^2 + 9*g*h*x + 2*h^2*x^2) + e*(-12*g^3 - 18*g^2*h*x - 4*g*h^2*x^2 + h^3*x^3)))))/(g + h*x)^2 + (3*(2*a^2*f*h^4 + a*c*h^2*(19*f*g^2 + 3*h*(-3*e*g + d*h)) + 2*c^2*(10*f*g^4 + 3*g^2*h*(-2*e*g + d*h)))*Log[g + h*x])/Sqrt[c*g^2 + a*h^2] - 3*Sqrt[c]*(20*c*f*g^3 + 6*c*g*h*(-2*e*g + d*h) - 3*a*h^2*(-3*f*g + e*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (3*(2*a^2*f*h^4 + a*c*h^2*(19*f*g^2 + 3*h*(-3*e*g + d*h)) + 2*c^2*(10*f*g^4 + 3*g^2*h*(-2*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/Sqrt[c*g^2 + a*h^2])/(6*h^6)","A",1
95,1,517,475,1.232855,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^4} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4,x]","\frac{\frac{3 c \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left(h (d h-4 e g)+11 f g^2\right)+2 c^2 \left(g^3 h (d h-4 e g)+10 f g^5\right)\right)}{\left(a h^2+c g^2\right)^{3/2}}-\frac{3 c \log (g+h x) \left(-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left(h (d h-4 e g)+11 f g^2\right)+2 c^2 \left(g^3 h (d h-4 e g)+10 f g^5\right)\right)}{\left(a h^2+c g^2\right)^{3/2}}-\frac{h \sqrt{a+c x^2} \left((g+h x)^2 \left(6 a^2 f h^4+a c h^2 \left(h (8 d h-23 e g)+50 f g^2\right)+c^2 \left(g^2 h (11 d h-26 e g)+47 f g^4\right)\right)+2 \left(a h^2+c g^2\right)^2 \left(h (d h-e g)+f g^2\right)-(g+h x) \left(a h^2+c g^2\right) \left(-3 a h^2 (e h-2 f g)+c g h (7 d h-10 e g)+13 c f g^3\right)+6 c (g+h x)^3 \left(a h^2+c g^2\right) (4 f g-e h)-3 c f h x (g+h x)^3 \left(a h^2+c g^2\right)\right)}{(g+h x)^3 \left(a h^2+c g^2\right)}+3 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(3 a f h^2+2 c h (d h-4 e g)+20 c f g^2\right)}{6 h^6}","\frac{c \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left(11 f g^2-h (4 e g-d h)\right)+2 c^2 g^3 \left(10 f g^2-h (4 e g-d h)\right)\right)}{2 h^6 \left(a h^2+c g^2\right)^{3/2}}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{3 h (g+h x)^3 \left(a h^2+c g^2\right)}-\frac{\left(a+c x^2\right)^{3/2} \left(-x \left(3 a f h^2+c \left(5 f g^2-2 h (e g-d h)\right)\right)-3 a h (3 f g-e h)+c g \left(-d h+4 e g-\frac{10 f g^2}{h}\right)\right)}{6 h^2 (g+h x)^2 \left(a h^2+c g^2\right)}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a f h^2+2 c \left(10 f g^2-h (4 e g-d h)\right)\right)}{2 h^6}-\frac{\sqrt{a+c x^2} \left(c h x \left(3 a h^2 (3 f g-e h)+c g \left(10 f g^2-h (4 e g-d h)\right)\right)+\left(a h^2+c g^2\right) \left(3 a f h^2+2 c \left(10 f g^2-h (4 e g-d h)\right)\right)\right)}{2 h^5 (g+h x) \left(a h^2+c g^2\right)}",1,"(-((h*Sqrt[a + c*x^2]*(2*(c*g^2 + a*h^2)^2*(f*g^2 + h*(-(e*g) + d*h)) - (c*g^2 + a*h^2)*(13*c*f*g^3 + c*g*h*(-10*e*g + 7*d*h) - 3*a*h^2*(-2*f*g + e*h))*(g + h*x) + (6*a^2*f*h^4 + a*c*h^2*(50*f*g^2 + h*(-23*e*g + 8*d*h)) + c^2*(47*f*g^4 + g^2*h*(-26*e*g + 11*d*h)))*(g + h*x)^2 + 6*c*(4*f*g - e*h)*(c*g^2 + a*h^2)*(g + h*x)^3 - 3*c*f*h*(c*g^2 + a*h^2)*x*(g + h*x)^3))/((c*g^2 + a*h^2)*(g + h*x)^3)) - (3*c*(-3*a^2*h^4*(-4*f*g + e*h) + 3*a*c*g*h^2*(11*f*g^2 + h*(-4*e*g + d*h)) + 2*c^2*(10*f*g^5 + g^3*h*(-4*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(3/2) + 3*Sqrt[c]*(20*c*f*g^2 + 3*a*f*h^2 + 2*c*h*(-4*e*g + d*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c*(-3*a^2*h^4*(-4*f*g + e*h) + 3*a*c*g*h^2*(11*f*g^2 + h*(-4*e*g + d*h)) + 2*c^2*(10*f*g^5 + g^3*h*(-4*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(3/2))/(6*h^6)","A",1
96,1,575,511,2.1542895,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^5} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]","-\frac{\frac{h \sqrt{a+c x^2} \left((g+h x)^2 \left(a h^2+c g^2\right) \left(12 a^2 f h^4+a c h^2 \left(h (15 d h-43 e g)+95 f g^2\right)+2 c^2 \left(g^2 h (9 d h-23 e g)+43 f g^4\right)\right)-c (g+h x)^3 \left(4 a^2 h^4 (31 f g-8 e h)+a c g h^2 \left(h (15 d h-91 e g)+287 f g^2\right)+2 c^2 \left(g^3 h (3 d h-25 e g)+77 f g^5\right)\right)+6 \left(a h^2+c g^2\right)^3 \left(h (d h-e g)+f g^2\right)-2 (g+h x) \left(a h^2+c g^2\right)^2 \left(-4 a h^2 (e h-2 f g)+c g h (9 d h-13 e g)+17 c f g^3\right)-24 c f (g+h x)^4 \left(a h^2+c g^2\right)^2\right)}{(g+h x)^4 \left(a h^2+c g^2\right)^2}+\frac{3 c \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(12 a^3 f h^6+3 a^2 c h^4 \left(h (d h-5 e g)+25 f g^2\right)+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right)}{\left(a h^2+c g^2\right)^{5/2}}-\frac{3 c \log (g+h x) \left(12 a^3 f h^6+3 a^2 c h^4 \left(h (d h-5 e g)+25 f g^2\right)+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right)}{\left(a h^2+c g^2\right)^{5/2}}+24 c^{3/2} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) (5 f g-e h)}{24 h^6}","\frac{\left(a+c x^2\right)^{3/2} \left(-3 h x \left(4 a^2 f h^4+a c h^2 \left(17 f g^2-h (5 e g-d h)\right)+2 c^2 g^2 \left(5 f g^2-h (d h+e g)\right)\right)+4 a^2 h^4 (f g-2 e h)-a c g h^2 \left(25 f g^2-h (5 e g-9 d h)\right)-4 c^2 g^4 (5 f g-e h)\right)}{24 h^3 (g+h x)^3 \left(a h^2+c g^2\right)^2}+\frac{c \sqrt{a+c x^2} \left(h x \left(12 a^2 f h^4+a c h^2 \left(35 f g^2-h (7 e g-3 d h)\right)+4 c^2 g^3 (5 f g-e h)\right)+8 \left(a h^2+c g^2\right)^2 (5 f g-e h)\right)}{8 h^5 (g+h x) \left(a h^2+c g^2\right)^2}-\frac{c \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(12 a^3 f h^6+3 a^2 c h^4 \left(25 f g^2-h (5 e g-d h)\right)+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right)}{8 h^6 \left(a h^2+c g^2\right)^{5/2}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (5 f g-e h)}{h^6}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{4 h (g+h x)^4 \left(a h^2+c g^2\right)}",1,"-1/24*((h*Sqrt[a + c*x^2]*(6*(c*g^2 + a*h^2)^3*(f*g^2 + h*(-(e*g) + d*h)) - 2*(c*g^2 + a*h^2)^2*(17*c*f*g^3 + c*g*h*(-13*e*g + 9*d*h) - 4*a*h^2*(-2*f*g + e*h))*(g + h*x) + (c*g^2 + a*h^2)*(12*a^2*f*h^4 + 2*c^2*(43*f*g^4 + g^2*h*(-23*e*g + 9*d*h)) + a*c*h^2*(95*f*g^2 + h*(-43*e*g + 15*d*h)))*(g + h*x)^2 - c*(4*a^2*h^4*(31*f*g - 8*e*h) + 2*c^2*(77*f*g^5 + g^3*h*(-25*e*g + 3*d*h)) + a*c*g*h^2*(287*f*g^2 + h*(-91*e*g + 15*d*h)))*(g + h*x)^3 - 24*c*f*(c*g^2 + a*h^2)^2*(g + h*x)^4))/((c*g^2 + a*h^2)^2*(g + h*x)^4) - (3*c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*g - e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h) + 3*a^2*c*h^4*(25*f*g^2 + h*(-5*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(5/2) + 24*c^(3/2)*(5*f*g - e*h)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c*(12*a^3*f*h^6 + 8*c^3*g^5*(5*f*g - e*h) + 20*a*c^2*g^3*h^2*(5*f*g - e*h) + 3*a^2*c*h^4*(25*f*g^2 + h*(-5*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(5/2))/h^6","A",1
97,1,639,507,2.268065,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^6} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]","\frac{\frac{15 c^2 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(-3 a^3 h^6 (e h-6 f g)+a^2 c g h^4 \left(35 f g^2-3 d h^2\right)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right)}{\left(a h^2+c g^2\right)^{7/2}}-\frac{15 c^2 \log (g+h x) \left(-3 a^3 h^6 (e h-6 f g)+a^2 c g h^4 \left(35 f g^2-3 d h^2\right)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right)}{\left(a h^2+c g^2\right)^{7/2}}-\frac{h \sqrt{a+c x^2} \left(2 (g+h x)^2 \left(a h^2+c g^2\right)^2 \left(20 a^2 f h^4+a c h^2 \left(3 h (8 d h-23 e g)+154 f g^2\right)+c^2 \left(9 g^2 h (3 d h-8 e g)+137 f g^4\right)\right)-c (g+h x)^3 \left(a h^2+c g^2\right) \left(5 a^2 h^4 (58 f g-15 e h)+a c g h^2 \left(3 h (7 d h-62 e g)+631 f g^2\right)+c^2 \left(6 g^3 h (d h-16 e g)+326 f g^5\right)\right)+c (g+h x)^4 \left(160 a^3 f h^6+3 a^2 c h^4 \left(h (8 d h-33 e g)+238 f g^2\right)+3 a c^2 g^2 h^2 \left(261 f g^2-h (9 d h+26 e g)\right)+c^3 \left(274 f g^6-6 g^4 h (d h+4 e g)\right)\right)+24 \left(a h^2+c g^2\right)^4 \left(h (d h-e g)+f g^2\right)-6 (g+h x) \left(a h^2+c g^2\right)^3 \left(-5 a h^2 (e h-2 f g)+c g h (11 d h-16 e g)+21 c f g^3\right)\right)}{(g+h x)^5 \left(a h^2+c g^2\right)^3}+120 c^{3/2} f \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{120 h^6}","-\frac{\left(a+c x^2\right)^{3/2} \left(h x \left(4 a^2 f h^4+a c g h^2 (14 f g-3 e h)+c^2 \left(7 f g^4-3 d g^2 h^2\right)\right)-a^2 h^4 (2 f g-3 e h)+a c g h^2 \left(3 d h^2+5 f g^2\right)+4 c^2 f g^5\right)}{12 h^3 (g+h x)^4 \left(a h^2+c g^2\right)^2}+\frac{c^2 \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(3 a^3 h^6 (6 f g-e h)+a^2 c g h^4 \left(35 f g^2-3 d h^2\right)+28 a c^2 f g^5 h^2+8 c^3 f g^7\right)}{8 h^6 \left(a h^2+c g^2\right)^{7/2}}-\frac{c \sqrt{a+c x^2} \left(-a^3 h^6 (2 f g-3 e h)+a^2 c g h^4 \left(3 d h^2+13 f g^2\right)+h x \left(8 a^3 f h^6+a^2 c g h^4 (34 f g-3 e h)+a c^2 g^2 h^2 \left(35 f g^2-3 d h^2\right)+12 c^3 f g^6\right)+20 a c^2 f g^5 h^2+8 c^3 f g^7\right)}{8 h^5 (g+h x)^2 \left(a h^2+c g^2\right)^3}+\frac{c^{3/2} f \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{h^6}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{5 h (g+h x)^5 \left(a h^2+c g^2\right)}",1,"(-((h*Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 6*(c*g^2 + a*h^2)^3*(21*c*f*g^3 + c*g*h*(-16*e*g + 11*d*h) - 5*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(137*f*g^4 + 9*g^2*h*(-8*e*g + 3*d*h)) + a*c*h^2*(154*f*g^2 + 3*h*(-23*e*g + 8*d*h)))*(g + h*x)^2 - c*(c*g^2 + a*h^2)*(5*a^2*h^4*(58*f*g - 15*e*h) + c^2*(326*f*g^5 + 6*g^3*h*(-16*e*g + d*h)) + a*c*g*h^2*(631*f*g^2 + 3*h*(-62*e*g + 7*d*h)))*(g + h*x)^3 + c*(160*a^3*f*h^6 + c^3*(274*f*g^6 - 6*g^4*h*(4*e*g + d*h)) + 3*a^2*c*h^4*(238*f*g^2 + h*(-33*e*g + 8*d*h)) + 3*a*c^2*g^2*h^2*(261*f*g^2 - h*(26*e*g + 9*d*h)))*(g + h*x)^4))/((c*g^2 + a*h^2)^3*(g + h*x)^5)) - (15*c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) + 120*c^(3/2)*f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*(8*c^3*f*g^7 + 28*a*c^2*f*g^5*h^2 - 3*a^3*h^6*(-6*f*g + e*h) + a^2*c*g*h^4*(35*f*g^2 - 3*d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/(120*h^6)","A",1
98,1,696,404,2.4717617,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^7} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]","\frac{1}{240} \left(-\frac{15 a^2 c^2 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(6 a^2 f h^2-a c \left(h (d h-7 e g)+f g^2\right)+6 c^2 d g^2\right)}{\left(a h^2+c g^2\right)^{9/2}}+\frac{15 a^2 c^2 \log (g+h x) \left(6 a^2 f h^2-a c \left(h (d h-7 e g)+f g^2\right)+6 c^2 d g^2\right)}{\left(a h^2+c g^2\right)^{9/2}}-\frac{\sqrt{a+c x^2} \left(2 (g+h x)^2 \left(a h^2+c g^2\right)^3 \left(30 a^2 f h^4+a c h^2 \left(h (35 d h-101 e g)+227 f g^2\right)+2 c^2 \left(g^2 h (19 d h-52 e g)+100 f g^4\right)\right)-2 c (g+h x)^3 \left(a h^2+c g^2\right)^2 \left(6 a^2 h^4 (31 f g-8 e h)+3 a c g h^2 \left(h (3 d h-37 e g)+131 f g^2\right)+2 c^2 \left(g^3 h (d h-28 e g)+100 f g^5\right)\right)+c (g+h x)^4 \left(a h^2+c g^2\right) \left(150 a^3 f h^6+3 a^2 c h^4 \left(h (5 d h-19 e g)+193 f g^2\right)+6 a c^2 g^2 h^2 \left(99 f g^2-h (4 d h+5 e g)\right)+4 c^3 \left(50 f g^6-g^4 h (d h+2 e g)\right)\right)-c^2 (g+h x)^5 \left(6 a^3 h^6 (41 f g-8 e h)+3 a^2 c g h^4 \left(h (29 e g-27 d h)+89 f g^2\right)+2 a c^2 g^3 h^2 \left(h (14 d h+19 e g)+83 f g^2\right)+4 c^3 \left(g^5 h (d h+2 e g)+10 f g^7\right)\right)+40 \left(a h^2+c g^2\right)^5 \left(h (d h-e g)+f g^2\right)-8 (g+h x) \left(a h^2+c g^2\right)^4 \left(-6 a h^2 (e h-2 f g)+c g h (13 d h-19 e g)+25 c f g^3\right)\right)}{h^5 (g+h x)^6 \left(a h^2+c g^2\right)^4}\right)","-\frac{\left(a+c x^2\right)^{3/2} (a h-c g x) \left(6 a^2 f h^2-a c \left(f g^2-h (7 e g-d h)\right)+6 c^2 d g^2\right)}{24 (g+h x)^4 \left(a h^2+c g^2\right)^3}-\frac{a c \sqrt{a+c x^2} (a h-c g x) \left(6 a^2 f h^2-a c \left(f g^2-h (7 e g-d h)\right)+6 c^2 d g^2\right)}{16 (g+h x)^2 \left(a h^2+c g^2\right)^4}-\frac{a^2 c^2 \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(6 a^2 f h^2-a c \left(f g^2-h (7 e g-d h)\right)+6 c^2 d g^2\right)}{16 \left(a h^2+c g^2\right)^{9/2}}+\frac{\left(a+c x^2\right)^{5/2} \left(6 a h^2 (2 f g-e h)+c g \left(h (e g-7 d h)+5 f g^2\right)\right)}{30 h (g+h x)^5 \left(a h^2+c g^2\right)^2}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{6 h (g+h x)^6 \left(a h^2+c g^2\right)}",1,"(-((Sqrt[a + c*x^2]*(40*(c*g^2 + a*h^2)^5*(f*g^2 + h*(-(e*g) + d*h)) - 8*(c*g^2 + a*h^2)^4*(25*c*f*g^3 + c*g*h*(-19*e*g + 13*d*h) - 6*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^3*(30*a^2*f*h^4 + 2*c^2*(100*f*g^4 + g^2*h*(-52*e*g + 19*d*h)) + a*c*h^2*(227*f*g^2 + h*(-101*e*g + 35*d*h)))*(g + h*x)^2 - 2*c*(c*g^2 + a*h^2)^2*(6*a^2*h^4*(31*f*g - 8*e*h) + 2*c^2*(100*f*g^5 + g^3*h*(-28*e*g + d*h)) + 3*a*c*g*h^2*(131*f*g^2 + h*(-37*e*g + 3*d*h)))*(g + h*x)^3 + c*(c*g^2 + a*h^2)*(150*a^3*f*h^6 + 4*c^3*(50*f*g^6 - g^4*h*(2*e*g + d*h)) + 6*a*c^2*g^2*h^2*(99*f*g^2 - h*(5*e*g + 4*d*h)) + 3*a^2*c*h^4*(193*f*g^2 + h*(-19*e*g + 5*d*h)))*(g + h*x)^4 - c^2*(6*a^3*h^6*(41*f*g - 8*e*h) + 3*a^2*c*g*h^4*(89*f*g^2 + h*(29*e*g - 27*d*h)) + 4*c^3*(10*f*g^7 + g^5*h*(2*e*g + d*h)) + 2*a*c^2*g^3*h^2*(83*f*g^2 + h*(19*e*g + 14*d*h)))*(g + h*x)^5))/(h^5*(c*g^2 + a*h^2)^4*(g + h*x)^6)) + (15*a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 + h*(-7*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(9/2) - (15*a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 + h*(-7*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(9/2))/240","A",1
99,1,863,532,2.4952176,"\int \frac{\left(a+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^8} \, dx","Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]","\frac{a^2 \left(6 c^2 d g^3-a c \left(f g^2+h (3 d h-8 e g)\right) g+a^2 h^2 (8 f g-e h)\right) \log (g+h x) c^3}{16 \left(c g^2+a h^2\right)^{11/2}}-\frac{a^2 \left(6 c^2 d g^3-a c \left(f g^2+h (3 d h-8 e g)\right) g+a^2 h^2 (8 f g-e h)\right) \log \left(a h-c g x+\sqrt{c g^2+a h^2} \sqrt{c x^2+a}\right) c^3}{16 \left(c g^2+a h^2\right)^{11/2}}-\frac{\sqrt{c x^2+a} \left(240 \left(f g^2+h (d h-e g)\right) \left(c g^2+a h^2\right)^6-40 \left(29 c f g^3+c h (15 d h-22 e g) g-7 a h^2 (e h-2 f g)\right) (g+h x) \left(c g^2+a h^2\right)^5+8 \left(42 a^2 f h^4+a c \left(314 f g^2+h (48 d h-139 e g)\right) h^2+c^2 \left(275 f g^4+h (51 d h-142 e g) g^2\right)\right) (g+h x)^2 \left(c g^2+a h^2\right)^4-2 c \left(7 a^2 (136 f g-35 e h) h^4+a c g \left(1979 f g^2+h (33 d h-544 e g)\right) h^2+2 c^2 \left(500 f g^5+h (3 d h-136 e g) g^3\right)\right) (g+h x)^3 \left(c g^2+a h^2\right)^3+2 c \left(336 a^3 f h^6+3 a^2 c \left(400 f g^2+h (8 d h-29 e g)\right) h^4+a c^2 g^2 \left(1201 f g^2-h (32 e g+45 d h)\right) h^2+c^3 \left(400 f g^6-2 g^4 h (4 e g+3 d h)\right)\right) (g+h x)^4 \left(c g^2+a h^2\right)^2-c^2 \left(21 a^3 (24 f g-5 e h) h^6+3 a^2 c g \left(109 f g^2+h (94 e g-73 d h)\right) h^4+2 a c^2 g^3 \left(89 f g^2+44 e h g+54 d h^2\right) h^2+4 c^3 \left(10 f g^7+h (4 e g+3 d h) g^5\right)\right) (g+h x)^5 \left(c g^2+a h^2\right)-c^2 \left(-336 a^4 f h^8+3 a^3 c \left(312 f g^2+h (32 d h-221 e g)\right) h^6+a^2 c^2 g^2 \left(505 f g^2+h (370 e g-741 d h)\right) h^4+2 a c^3 g^4 \left(109 f g^2+52 e h g+60 d h^2\right) h^2+4 c^4 \left(10 f g^8+h (4 e g+3 d h) g^6\right)\right) (g+h x)^6\right)}{1680 \left(a h^3+c g^2 h\right)^5 (g+h x)^7}","-\frac{\left(a+c x^2\right)^{5/2} \left(42 a^2 f h^4-a c h^2 \left(26 f g^2-h (61 e g-12 d h)\right)-c^2 g^2 \left(h (2 e g-51 d h)+5 f g^2\right)\right)}{210 h (g+h x)^5 \left(a h^2+c g^2\right)^3}-\frac{a c^2 \sqrt{a+c x^2} (a h-c g x) \left(a^2 h^2 (8 f g-e h)-a c g \left(f g^2-h (8 e g-3 d h)\right)+6 c^2 d g^3\right)}{16 (g+h x)^2 \left(a h^2+c g^2\right)^5}-\frac{c \left(a+c x^2\right)^{3/2} (a h-c g x) \left(a^2 h^2 (8 f g-e h)-a c g \left(f g^2-h (8 e g-3 d h)\right)+6 c^2 d g^3\right)}{24 (g+h x)^4 \left(a h^2+c g^2\right)^4}-\frac{a^2 c^3 \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a^2 h^2 (8 f g-e h)-a c g \left(f g^2-h (8 e g-3 d h)\right)+6 c^2 d g^3\right)}{16 \left(a h^2+c g^2\right)^{11/2}}+\frac{\left(a+c x^2\right)^{5/2} \left(7 a h^2 (2 f g-e h)+c g \left(h (2 e g-9 d h)+5 f g^2\right)\right)}{42 h (g+h x)^6 \left(a h^2+c g^2\right)^2}-\frac{\left(a+c x^2\right)^{5/2} \left(d h^2-e g h+f g^2\right)}{7 h (g+h x)^7 \left(a h^2+c g^2\right)}",1,"-1/1680*(Sqrt[a + c*x^2]*(240*(c*g^2 + a*h^2)^6*(f*g^2 + h*(-(e*g) + d*h)) - 40*(c*g^2 + a*h^2)^5*(29*c*f*g^3 + c*g*h*(-22*e*g + 15*d*h) - 7*a*h^2*(-2*f*g + e*h))*(g + h*x) + 8*(c*g^2 + a*h^2)^4*(42*a^2*f*h^4 + a*c*h^2*(314*f*g^2 + h*(-139*e*g + 48*d*h)) + c^2*(275*f*g^4 + g^2*h*(-142*e*g + 51*d*h)))*(g + h*x)^2 - 2*c*(c*g^2 + a*h^2)^3*(7*a^2*h^4*(136*f*g - 35*e*h) + 2*c^2*(500*f*g^5 + g^3*h*(-136*e*g + 3*d*h)) + a*c*g*h^2*(1979*f*g^2 + h*(-544*e*g + 33*d*h)))*(g + h*x)^3 + 2*c*(c*g^2 + a*h^2)^2*(336*a^3*f*h^6 + c^3*(400*f*g^6 - 2*g^4*h*(4*e*g + 3*d*h)) + 3*a^2*c*h^4*(400*f*g^2 + h*(-29*e*g + 8*d*h)) + a*c^2*g^2*h^2*(1201*f*g^2 - h*(32*e*g + 45*d*h)))*(g + h*x)^4 - c^2*(c*g^2 + a*h^2)*(21*a^3*h^6*(24*f*g - 5*e*h) + 2*a*c^2*g^3*h^2*(89*f*g^2 + 44*e*g*h + 54*d*h^2) + 3*a^2*c*g*h^4*(109*f*g^2 + h*(94*e*g - 73*d*h)) + 4*c^3*(10*f*g^7 + g^5*h*(4*e*g + 3*d*h)))*(g + h*x)^5 - c^2*(-336*a^4*f*h^8 + 2*a*c^3*g^4*h^2*(109*f*g^2 + 52*e*g*h + 60*d*h^2) + a^2*c^2*g^2*h^4*(505*f*g^2 + h*(370*e*g - 741*d*h)) + 4*c^4*(10*f*g^8 + g^6*h*(4*e*g + 3*d*h)) + 3*a^3*c*h^6*(312*f*g^2 + h*(-221*e*g + 32*d*h)))*(g + h*x)^6))/((c*g^2*h + a*h^3)^5*(g + h*x)^7) + (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 + h*(-8*e*g + 3*d*h)))*Log[g + h*x])/(16*(c*g^2 + a*h^2)^(11/2)) - (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 + h*(-8*e*g + 3*d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(16*(c*g^2 + a*h^2)^(11/2))","A",1
100,1,150,168,0.3113283,"\int \left(a+c x^2\right)^{5/2} \left(A+B x+C x^2\right) \, dx","Integrate[(a + c*x^2)^(5/2)*(A + B*x + C*x^2),x]","\frac{\sqrt{a+c x^2} \left(\sqrt{c} \left(3 a^3 (128 B+35 C x)+2 a^2 c x (924 A+x (576 B+413 C x))+8 a c^2 x^3 (182 A+x (144 B+119 C x))+16 c^3 x^5 (28 A+3 x (8 B+7 C x))\right)-\frac{105 a^{5/2} (a C-8 A c) \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{\sqrt{\frac{c x^2}{a}+1}}\right)}{2688 c^{3/2}}","\frac{5 a^3 (8 A c-a C) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} (8 A c-a C)}{128 c}+\frac{x \left(a+c x^2\right)^{5/2} (8 A c-a C)}{48 c}+\frac{5 a x \left(a+c x^2\right)^{3/2} (8 A c-a C)}{192 c}+\frac{B \left(a+c x^2\right)^{7/2}}{7 c}+\frac{C x \left(a+c x^2\right)^{7/2}}{8 c}",1,"(Sqrt[a + c*x^2]*(Sqrt[c]*(3*a^3*(128*B + 35*C*x) + 16*c^3*x^5*(28*A + 3*x*(8*B + 7*C*x)) + 8*a*c^2*x^3*(182*A + x*(144*B + 119*C*x)) + 2*a^2*c*x*(924*A + x*(576*B + 413*C*x))) - (105*a^(5/2)*(-8*A*c + a*C)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(2688*c^(3/2))","A",1
101,1,252,325,0.3549716,"\int \frac{(g+h x)^3 \left(d+e x+f x^2\right)}{\sqrt{a+c x^2}} \, dx","Integrate[((g + h*x)^3*(d + e*x + f*x^2))/Sqrt[a + c*x^2],x]","\frac{15 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(3 a^2 h^2 (e h+3 f g)-4 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)+\sqrt{a+c x^2} \left(8 \left(8 a^2 f h^3-10 a c h \left(h (d h+3 e g)+3 f g^2\right)+15 c^2 g^2 (3 d h+e g)\right)+15 c x \left(4 c \left(3 g h (d h+e g)+f g^3\right)-3 a h^2 (e h+3 f g)\right)+8 c h x^2 \left(5 c \left(h (d h+3 e g)+3 f g^2\right)-4 a f h^2\right)+30 c^2 h^2 x^3 (e h+3 f g)+24 c^2 f h^3 x^4\right)}{120 c^3}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 h^2 (e h+3 f g)-4 a c g \left(3 h (d h+e g)+f g^2\right)+8 c^2 d g^3\right)}{8 c^{5/2}}+\frac{\sqrt{a+c x^2} \left(4 \left(16 a^2 f h^4-4 a c h^2 \left(5 h (d h+3 e g)+13 f g^2\right)-c^2 g^2 \left(3 f g^2-5 h (16 d h+3 e g)\right)\right)-c h x \left(a h^2 (45 e h+71 f g)+2 c g \left(3 f g^2-5 h (10 d h+3 e g)\right)\right)\right)}{120 c^3 h}+\frac{\sqrt{a+c x^2} (g+h x)^2 \left(4 h^2 (5 c d-4 a f)-3 c g (f g-5 e h)\right)}{60 c^2 h}-\frac{\sqrt{a+c x^2} (g+h x)^3 (f g-5 e h)}{20 c h}+\frac{f \sqrt{a+c x^2} (g+h x)^4}{5 c h}",1,"(Sqrt[a + c*x^2]*(8*(8*a^2*f*h^3 + 15*c^2*g^2*(e*g + 3*d*h) - 10*a*c*h*(3*f*g^2 + h*(3*e*g + d*h))) + 15*c*(-3*a*h^2*(3*f*g + e*h) + 4*c*(f*g^3 + 3*g*h*(e*g + d*h)))*x + 8*c*h*(-4*a*f*h^2 + 5*c*(3*f*g^2 + h*(3*e*g + d*h)))*x^2 + 30*c^2*h^2*(3*f*g + e*h)*x^3 + 24*c^2*f*h^3*x^4) + 15*Sqrt[c]*(8*c^2*d*g^3 + 3*a^2*h^2*(3*f*g + e*h) - 4*a*c*g*(f*g^2 + 3*h*(e*g + d*h)))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(120*c^3)","A",1
102,1,164,223,0.2332972,"\int \frac{(g+h x)^2 \left(d+e x+f x^2\right)}{\sqrt{a+c x^2}} \, dx","Integrate[((g + h*x)^2*(d + e*x + f*x^2))/Sqrt[a + c*x^2],x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 f h^2-4 a c \left(h (d h+2 e g)+f g^2\right)+8 c^2 d g^2\right)+\sqrt{c} \sqrt{a+c x^2} \left(2 c \left(6 d h (4 g+h x)+4 e \left(3 g^2+3 g h x+h^2 x^2\right)+f x \left(6 g^2+8 g h x+3 h^2 x^2\right)\right)-a h (16 e h+32 f g+9 f h x)\right)}{24 c^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a^2 f h^2-4 a c \left(h (d h+2 e g)+f g^2\right)+8 c^2 d g^2\right)}{8 c^{5/2}}-\frac{\sqrt{a+c x^2} \left(4 \left(4 a h^2 (e h+2 f g)+c g \left(f g^2-4 h (3 d h+e g)\right)\right)-h x \left(3 h^2 (4 c d-3 a f)-2 c g (f g-4 e h)\right)\right)}{24 c^2 h}-\frac{\sqrt{a+c x^2} (g+h x)^2 (f g-4 e h)}{12 c h}+\frac{f \sqrt{a+c x^2} (g+h x)^3}{4 c h}",1,"(Sqrt[c]*Sqrt[a + c*x^2]*(-(a*h*(32*f*g + 16*e*h + 9*f*h*x)) + 2*c*(6*d*h*(4*g + h*x) + 4*e*(3*g^2 + 3*g*h*x + h^2*x^2) + f*x*(6*g^2 + 8*g*h*x + 3*h^2*x^2))) + 3*(8*c^2*d*g^2 + 3*a^2*f*h^2 - 4*a*c*(f*g^2 + h*(2*e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(24*c^(5/2))","A",1
103,1,96,136,0.1045049,"\int \frac{(g+h x) \left(d+e x+f x^2\right)}{\sqrt{a+c x^2}} \, dx","Integrate[((g + h*x)*(d + e*x + f*x^2))/Sqrt[a + c*x^2],x]","\frac{\sqrt{a+c x^2} \left(c \left(6 d h+6 e g+3 e h x+3 f g x+2 f h x^2\right)-4 a f h\right)+3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (2 c d g-a (e h+f g))}{6 c^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (2 c d g-a (e h+f g))}{2 c^{3/2}}-\frac{\sqrt{a+c x^2} \left(2 \left(2 a f h^2+c \left(f g^2-3 h (d h+e g)\right)\right)+c h x (f g-3 e h)\right)}{6 c^2 h}+\frac{f \sqrt{a+c x^2} (g+h x)^2}{3 c h}",1,"(Sqrt[a + c*x^2]*(-4*a*f*h + c*(6*e*g + 6*d*h + 3*f*g*x + 3*e*h*x + 2*f*h*x^2)) + 3*Sqrt[c]*(2*c*d*g - a*(f*g + e*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(6*c^2)","A",1
104,1,63,74,0.0381557,"\int \frac{d+e x+f x^2}{\sqrt{a+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/Sqrt[a + c*x^2],x]","\frac{(2 c d-a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)+\sqrt{c} \sqrt{a+c x^2} (2 e+f x)}{2 c^{3/2}}","\frac{(2 c d-a f) \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2}}{c}+\frac{f x \sqrt{a+c x^2}}{2 c}",1,"(Sqrt[c]*(2*e + f*x)*Sqrt[a + c*x^2] + (2*c*d - a*f)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2))","A",1
105,1,125,130,0.2193377,"\int \frac{d+e x+f x^2}{(g+h x) \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)*Sqrt[a + c*x^2]),x]","\frac{-\frac{\left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{\sqrt{a h^2+c g^2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (e h-f g)}{\sqrt{c}}+\frac{f h \sqrt{a+c x^2}}{c}}{h^2}","-\frac{\left(d h^2-e g h+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{h^2 \sqrt{a h^2+c g^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (f g-e h)}{\sqrt{c} h^2}+\frac{f \sqrt{a+c x^2}}{c h}",1,"((f*h*Sqrt[a + c*x^2])/c + ((-(f*g) + e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c] - ((f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/Sqrt[c*g^2 + a*h^2])/h^2","A",1
106,1,218,168,0.362826,"\int \frac{d+e x+f x^2}{(g+h x)^2 \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^2*Sqrt[a + c*x^2]),x]","\frac{-\frac{h \sqrt{a+c x^2} \left(h (d h-e g)+f g^2\right)}{(g+h x) \left(a h^2+c g^2\right)}+\frac{\log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(a h^2 (2 f g-e h)+c \left(f g^3-d g h^2\right)\right)}{\left(a h^2+c g^2\right)^{3/2}}+\frac{\log (g+h x) \left(a h^2 (e h-2 f g)+c \left(d g h^2-f g^3\right)\right)}{\left(a h^2+c g^2\right)^{3/2}}+\frac{f \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right)}{\sqrt{c}}}{h^2}","-\frac{\sqrt{a+c x^2} \left(d h^2-e g h+f g^2\right)}{h (g+h x) \left(a h^2+c g^2\right)}+\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a h^2 (2 f g-e h)+c \left(f g^3-d g h^2\right)\right)}{h^2 \left(a h^2+c g^2\right)^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{\sqrt{c} h^2}",1,"(-((h*(f*g^2 + h*(-(e*g) + d*h))*Sqrt[a + c*x^2])/((c*g^2 + a*h^2)*(g + h*x))) + ((a*h^2*(-2*f*g + e*h) + c*(-(f*g^3) + d*g*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(3/2) + (f*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/Sqrt[c] + ((a*h^2*(2*f*g - e*h) + c*(f*g^3 - d*g*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(3/2))/h^2","A",1
107,1,254,225,0.4548861,"\int \frac{d+e x+f x^2}{(g+h x)^3 \sqrt{a+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^3*Sqrt[a + c*x^2]),x]","\frac{(g+h x)^2 \log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(-2 a^2 f h^2+a c \left(h (d h-3 e g)+f g^2\right)-2 c^2 d g^2\right)+(g+h x)^2 \log (g+h x) \left(2 a^2 f h^2-a c \left(h (d h-3 e g)+f g^2\right)+2 c^2 d g^2\right)+\sqrt{a+c x^2} \sqrt{a h^2+c g^2} \left(c g \left(-d h (4 g+3 h x)+e g (2 g+h x)+f g^2 x\right)-a h (h (d h+e (g+2 h x))-f g (3 g+4 h x))\right)}{2 (g+h x)^2 \left(a h^2+c g^2\right)^{5/2}}","-\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(2 a^2 f h^2-a c \left(f g^2-h (3 e g-d h)\right)+2 c^2 d g^2\right)}{2 \left(a h^2+c g^2\right)^{5/2}}-\frac{\sqrt{a+c x^2} \left(d h^2-e g h+f g^2\right)}{2 h (g+h x)^2 \left(a h^2+c g^2\right)}+\frac{\sqrt{a+c x^2} \left(2 a h^2 (2 f g-e h)+c g \left(h (e g-3 d h)+f g^2\right)\right)}{2 h (g+h x) \left(a h^2+c g^2\right)^2}",1,"(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]*(c*g*(f*g^2*x + e*g*(2*g + h*x) - d*h*(4*g + 3*h*x)) - a*h*(-(f*g*(3*g + 4*h*x)) + h*(d*h + e*(g + 2*h*x)))) + (2*c^2*d*g^2 + 2*a^2*f*h^2 - a*c*(f*g^2 + h*(-3*e*g + d*h)))*(g + h*x)^2*Log[g + h*x] + (-2*c^2*d*g^2 - 2*a^2*f*h^2 + a*c*(f*g^2 + h*(-3*e*g + d*h)))*(g + h*x)^2*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(2*(c*g^2 + a*h^2)^(5/2)*(g + h*x)^2)","A",1
108,1,246,229,0.4498247,"\int \frac{(g+h x)^3 \left(d+e x+f x^2\right)}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)^3*(d + e*x + f*x^2))/(a + c*x^2)^(3/2),x]","\frac{\frac{-16 a^3 f h^3+a^2 c h \left(3 h (4 d h+3 e (4 g+h x))+f \left(36 g^2+27 g h x-8 h^2 x^2\right)\right)+a c^2 \left(6 d h \left(-3 g^2-3 g h x+h^2 x^2\right)-3 e \left(2 g^3+6 g^2 h x-6 g h^2 x^2-h^3 x^3\right)+f x \left(-6 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right)\right)+6 c^3 d g^3 x}{a \sqrt{a+c x^2}}+3 \sqrt{c} \log \left(\sqrt{c} \sqrt{a+c x^2}+c x\right) \left(-3 a h^2 (e h+3 f g)+6 c g h (d h+e g)+2 c f g^3\right)}{6 c^3}","-\frac{h \sqrt{a+c x^2} \left(4 \left(4 a^2 f h^2-a c \left(3 h (d h+3 e g)+7 f g^2\right)+3 c^2 d g^2\right)+c h x (-9 a e h-11 a f g+6 c d g)\right)}{6 a c^3}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(3 a h^2 (e h+3 f g)-2 c g \left(3 h (d h+e g)+f g^2\right)\right)}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (g+h x)^2 (3 c d-4 a f)}{3 a c^2}-\frac{(g+h x)^3 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}}",1,"((-16*a^3*f*h^3 + 6*c^3*d*g^3*x + a*c^2*(6*d*h*(-3*g^2 - 3*g*h*x + h^2*x^2) - 3*e*(2*g^3 + 6*g^2*h*x - 6*g*h^2*x^2 - h^3*x^3) + f*x*(-6*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3)) + a^2*c*h*(f*(36*g^2 + 27*g*h*x - 8*h^2*x^2) + 3*h*(4*d*h + 3*e*(4*g + h*x))))/(a*Sqrt[a + c*x^2]) + 3*Sqrt[c]*(2*c*f*g^3 + 6*c*g*h*(e*g + d*h) - 3*a*h^2*(3*f*g + e*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(6*c^3)","A",1
109,1,177,149,0.2841423,"\int \frac{(g+h x)^2 \left(d+e x+f x^2\right)}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)^2*(d + e*x + f*x^2))/(a + c*x^2)^(3/2),x]","\frac{\sqrt{c} \left(a^2 h (4 e h+8 f g+3 f h x)+a c \left(-2 d h (2 g+h x)-2 e \left(g^2+2 g h x-h^2 x^2\right)+f x \left(-2 g^2+4 g h x+h^2 x^2\right)\right)+2 c^2 d g^2 x\right)-a^{3/2} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) \left(3 a f h^2-2 c \left(h (d h+2 e g)+f g^2\right)\right)}{2 a c^{5/2} \sqrt{a+c x^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) \left(h^2 (2 c d-3 a f)+2 c g (2 e h+f g)\right)}{2 c^{5/2}}-\frac{h \sqrt{a+c x^2} (4 (c d g-a (e h+2 f g))+h x (2 c d-3 a f))}{2 a c^2}-\frac{(g+h x)^2 (a e-x (c d-a f))}{a c \sqrt{a+c x^2}}",1,"(Sqrt[c]*(2*c^2*d*g^2*x + a^2*h*(8*f*g + 4*e*h + 3*f*h*x) + a*c*(-2*d*h*(2*g + h*x) - 2*e*(g^2 + 2*g*h*x - h^2*x^2) + f*x*(-2*g^2 + 4*g*h*x + h^2*x^2))) - a^(3/2)*(3*a*f*h^2 - 2*c*(f*g^2 + h*(2*e*g + d*h)))*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(2*a*c^(5/2)*Sqrt[a + c*x^2])","A",1
110,1,102,100,0.1370843,"\int \frac{(g+h x) \left(d+e x+f x^2\right)}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)*(d + e*x + f*x^2))/(a + c*x^2)^(3/2),x]","\frac{a^{3/2} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right) (e h+f g)+2 a^2 f h-a c (d h+e (g+h x)+f x (g-h x))+c^2 d g x}{a c^2 \sqrt{a+c x^2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right) (e h+f g)}{c^{3/2}}-\frac{h \sqrt{a+c x^2} (c d-2 a f)}{a c^2}-\frac{(g+h x) (a e-x (c d-a f))}{a c \sqrt{a+c x^2}}",1,"(2*a^2*f*h + c^2*d*g*x - a*c*(d*h + f*x*(g - h*x) + e*(g + h*x)) + a^(3/2)*Sqrt[c]*(f*g + e*h)*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(a*c^2*Sqrt[a + c*x^2])","A",1
111,1,74,61,0.0618273,"\int \frac{d+e x+f x^2}{\left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/(a + c*x^2)^(3/2),x]","\frac{a^{3/2} f \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)+\sqrt{c} (c d x-a (e+f x))}{a c^{3/2} \sqrt{a+c x^2}}","\frac{f \tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right)}{c^{3/2}}-\frac{a e-x (c d-a f)}{a c \sqrt{a+c x^2}}",1,"(Sqrt[c]*(c*d*x - a*(e + f*x)) + a^(3/2)*f*Sqrt[1 + (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(a*c^(3/2)*Sqrt[a + c*x^2])","A",1
112,1,137,138,0.1769768,"\int \frac{d+e x+f x^2}{(g+h x) \left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)*(a + c*x^2)^(3/2)),x]","\frac{a^2 (-f) h+a c (d h-e g+e h x-f g x)+c^2 d g x}{a c \sqrt{a+c x^2} \left(a h^2+c g^2\right)}-\frac{\left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{\left(a h^2+c g^2\right)^{3/2}}","-\frac{a (a f h-c d h+c e g)-c x (a e h-a f g+c d g)}{a c \sqrt{a+c x^2} \left(a h^2+c g^2\right)}-\frac{\left(d h^2-e g h+f g^2\right) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right)}{\left(a h^2+c g^2\right)^{3/2}}",1,"(-(a^2*f*h) + c^2*d*g*x + a*c*(-(e*g) + d*h - f*g*x + e*h*x))/(a*c*(c*g^2 + a*h^2)*Sqrt[a + c*x^2]) - ((f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(c*g^2 + a*h^2)^(3/2)","A",1
113,1,285,239,0.6078484,"\int \frac{d+e x+f x^2}{(g+h x)^2 \left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^2*(a + c*x^2)^(3/2)),x]","\frac{\left(a h^2+c g^2\right)^{3/2} \left(-a^2 f h^2+a c (2 h (d h-e g+e h x)+f g (g-2 h x))+2 c^2 d g h x\right)+2 h \left(h \left(a+c x^2\right) \sqrt{a h^2+c g^2} \left(a^2 f h^2+a c \left(h (3 e g-2 d h)-2 f g^2\right)+c^2 d g^2\right)-a c \sqrt{a+c x^2} (g+h x) \tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a h^2 (e h-2 f g)+c g h (3 d h-2 e g)+c f g^3\right)\right)-a f \left(a h^2+c g^2\right)^{5/2}}{2 a c h \sqrt{a+c x^2} (g+h x) \left(a h^2+c g^2\right)^{5/2}}","-\frac{a (a h (2 f g-e h)+c g (e g-2 d h))-x \left(a^2 f h^2-a c \left(f g^2-h (2 e g-d h)\right)+c^2 d g^2\right)}{a \sqrt{a+c x^2} \left(a h^2+c g^2\right)^2}-\frac{h \sqrt{a+c x^2} \left(d h^2-e g h+f g^2\right)}{(g+h x) \left(a h^2+c g^2\right)^2}+\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(a h^2 (2 f g-e h)-c g \left(f g^2-h (2 e g-3 d h)\right)\right)}{\left(a h^2+c g^2\right)^{5/2}}",1,"(-(a*f*(c*g^2 + a*h^2)^(5/2)) + (c*g^2 + a*h^2)^(3/2)*(-(a^2*f*h^2) + 2*c^2*d*g*h*x + a*c*(f*g*(g - 2*h*x) + 2*h*(-(e*g) + d*h + e*h*x))) + 2*h*(h*Sqrt[c*g^2 + a*h^2]*(c^2*d*g^2 + a^2*f*h^2 + a*c*(-2*f*g^2 + h*(3*e*g - 2*d*h)))*(a + c*x^2) - a*c*(c*f*g^3 + c*g*h*(-2*e*g + 3*d*h) + a*h^2*(-2*f*g + e*h))*(g + h*x)*Sqrt[a + c*x^2]*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])]))/(2*a*c*h*(c*g^2 + a*h^2)^(5/2)*(g + h*x)*Sqrt[a + c*x^2])","A",1
114,1,404,374,1.1837409,"\int \frac{d+e x+f x^2}{(g+h x)^3 \left(a+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^3*(a + c*x^2)^(3/2)),x]","\frac{1}{2} \left(-\frac{\log \left(\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right) \left(2 a^2 f h^4+a c h^2 \left(-3 d h^2+9 e g h-11 f g^2\right)+2 c^2 g^2 \left(6 d h^2-3 e g h+f g^2\right)\right)}{\left(a h^2+c g^2\right)^{7/2}}+\frac{\log (g+h x) \left(2 a^2 f h^4+a c h^2 \left(-3 d h^2+9 e g h-11 f g^2\right)+2 c^2 g^2 \left(6 d h^2-3 e g h+f g^2\right)\right)}{\left(a h^2+c g^2\right)^{7/2}}-\frac{\sqrt{a+c x^2} \left(\frac{2 \left(-a^3 f h^3+a^2 c h (h (d h-3 e g+e h x)+3 f g (g-h x))+a c^2 g \left(3 d h (h x-g)+e g (g-3 h x)+f g^2 x\right)-c^3 d g^3 x\right)}{a \left(a+c x^2\right)}+\frac{h \left(2 a h^2 (e h-2 f g)+c g h (7 d h-5 e g)+3 c f g^3\right)}{g+h x}+\frac{h \left(a h^2+c g^2\right) \left(h (d h-e g)+f g^2\right)}{(g+h x)^2}\right)}{\left(a h^2+c g^2\right)^3}\right)","-\frac{\tanh ^{-1}\left(\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right) \left(2 a^2 f h^4-a c h^2 \left(3 d h^2-9 e g h+11 f g^2\right)+2 c^2 g^2 \left(6 d h^2-3 e g h+f g^2\right)\right)}{2 \left(a h^2+c g^2\right)^{7/2}}+\frac{a \left(a^2 f h^3-a c h \left(3 f g^2-h (3 e g-d h)\right)-c^2 g^2 (e g-3 d h)\right)+c x \left(a^2 h^2 (3 f g-e h)-a c g \left(f g^2-3 h (e g-d h)\right)+c^2 d g^3\right)}{a \sqrt{a+c x^2} \left(a h^2+c g^2\right)^3}-\frac{h \sqrt{a+c x^2} \left(d h^2-e g h+f g^2\right)}{2 (g+h x)^2 \left(a h^2+c g^2\right)^2}+\frac{h \sqrt{a+c x^2} \left(2 a h^2 (2 f g-e h)-c g \left(3 f g^2-h (5 e g-7 d h)\right)\right)}{2 (g+h x) \left(a h^2+c g^2\right)^3}",1,"(-((Sqrt[a + c*x^2]*((h*(c*g^2 + a*h^2)*(f*g^2 + h*(-(e*g) + d*h)))/(g + h*x)^2 + (h*(3*c*f*g^3 + c*g*h*(-5*e*g + 7*d*h) + 2*a*h^2*(-2*f*g + e*h)))/(g + h*x) + (2*(-(a^3*f*h^3) - c^3*d*g^3*x + a*c^2*g*(f*g^2*x + e*g*(g - 3*h*x) + 3*d*h*(-g + h*x)) + a^2*c*h*(3*f*g*(g - h*x) + h*(-3*e*g + d*h + e*h*x))))/(a*(a + c*x^2))))/(c*g^2 + a*h^2)^3) + ((2*a^2*f*h^4 + a*c*h^2*(-11*f*g^2 + 9*e*g*h - 3*d*h^2) + 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))*Log[g + h*x])/(c*g^2 + a*h^2)^(7/2) - ((2*a^2*f*h^4 + a*c*h^2*(-11*f*g^2 + 9*e*g*h - 3*d*h^2) + 2*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/2","A",1
115,1,50,67,0.0319481,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^{5/2}} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^(5/2),x]","\frac{-a^2 B+a c x \left(3 A+C x^2\right)+2 A c^2 x^3}{3 a^2 c \left(a+c x^2\right)^{3/2}}","\frac{x (a C+2 A c)}{3 a^2 c \sqrt{a+c x^2}}-\frac{a B-x (A c-a C)}{3 a c \left(a+c x^2\right)^{3/2}}",1,"(-(a^2*B) + 2*A*c^2*x^3 + a*c*x*(3*A + C*x^2))/(3*a^2*c*(a + c*x^2)^(3/2))","A",1
116,1,71,97,0.0497931,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^{7/2}} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^(7/2),x]","\frac{-3 a^3 B+5 a^2 c x \left(3 A+C x^2\right)+2 a c^2 x^3 \left(10 A+C x^2\right)+8 A c^3 x^5}{15 a^3 c \left(a+c x^2\right)^{5/2}}","\frac{2 x (a C+4 A c)}{15 a^3 c \sqrt{a+c x^2}}+\frac{x (a C+4 A c)}{15 a^2 c \left(a+c x^2\right)^{3/2}}-\frac{a B-x (A c-a C)}{5 a c \left(a+c x^2\right)^{5/2}}",1,"(-3*a^3*B + 8*A*c^3*x^5 + 5*a^2*c*x*(3*A + C*x^2) + 2*a*c^2*x^3*(10*A + C*x^2))/(15*a^3*c*(a + c*x^2)^(5/2))","A",1
117,1,92,127,0.0648498,"\int \frac{A+B x+C x^2}{\left(a+c x^2\right)^{9/2}} \, dx","Integrate[(A + B*x + C*x^2)/(a + c*x^2)^(9/2),x]","\frac{-15 a^4 B+35 a^3 c x \left(3 A+C x^2\right)+14 a^2 c^2 x^3 \left(15 A+2 C x^2\right)+8 a c^3 x^5 \left(21 A+C x^2\right)+48 A c^4 x^7}{105 a^4 c \left(a+c x^2\right)^{7/2}}","\frac{8 x (a C+6 A c)}{105 a^4 c \sqrt{a+c x^2}}+\frac{4 x (a C+6 A c)}{105 a^3 c \left(a+c x^2\right)^{3/2}}+\frac{x (a C+6 A c)}{35 a^2 c \left(a+c x^2\right)^{5/2}}-\frac{a B-x (A c-a C)}{7 a c \left(a+c x^2\right)^{7/2}}",1,"(-15*a^4*B + 48*A*c^4*x^7 + 35*a^3*c*x*(3*A + C*x^2) + 8*a*c^3*x^5*(21*A + C*x^2) + 14*a^2*c^2*x^3*(15*A + 2*C*x^2))/(105*a^4*c*(a + c*x^2)^(7/2))","A",1
118,1,54,106,0.0653184,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\sqrt{2+3 x^2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]","\frac{1}{405} \left(\sqrt{3 x^2+2} \left(864 x^4+2430 x^3+2292 x^2-135 x-1841\right)+225 \sqrt{3} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right)","\frac{2}{15} \sqrt{3 x^2+2} (2 x+1)^4+\frac{13}{60} \sqrt{3 x^2+2} (2 x+1)^3-\frac{19}{540} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{810} (2073 x+3937) \sqrt{3 x^2+2}+\frac{5 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{3 \sqrt{3}}",1,"(Sqrt[2 + 3*x^2]*(-1841 - 135*x + 2292*x^2 + 2430*x^3 + 864*x^4) + 225*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/405","A",1
119,1,48,82,0.0425582,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\sqrt{2+3 x^2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]","\frac{1}{27} \sqrt{3 x^2+2} \left(36 x^3+84 x^2+54 x-49\right)-\sqrt{3} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)","\frac{1}{6} \sqrt{3 x^2+2} (2 x+1)^3+\frac{5}{18} \sqrt{3 x^2+2} (2 x+1)^2-\frac{1}{27} (3 x+61) \sqrt{3 x^2+2}-\sqrt{3} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)",1,"(Sqrt[2 + 3*x^2]*(-49 + 54*x + 84*x^2 + 36*x^3))/27 - Sqrt[3]*ArcSinh[Sqrt[3/2]*x]","A",1
120,1,44,62,0.027188,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\sqrt{2+3 x^2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]","\frac{1}{27} \left(\sqrt{3 x^2+2} \left(24 x^2+45 x+13\right)-21 \sqrt{3} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right)","\frac{2}{9} \sqrt{3 x^2+2} (2 x+1)^2+\frac{7}{27} (3 x+1) \sqrt{3 x^2+2}-\frac{7 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{3 \sqrt{3}}",1,"(Sqrt[2 + 3*x^2]*(13 + 45*x + 24*x^2) - 21*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/27","A",1
121,1,60,67,0.0284645,"\int \frac{1+3 x+4 x^2}{(1+2 x) \sqrt{2+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*Sqrt[2 + 3*x^2]),x]","\frac{1}{66} \left(44 \sqrt{3 x^2+2}-3 \sqrt{11} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)+11 \sqrt{3} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)\right)","\frac{2}{3} \sqrt{3 x^2+2}-\frac{\tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{2 \sqrt{11}}+\frac{\sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{2 \sqrt{3}}",1,"(44*Sqrt[2 + 3*x^2] + 11*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 3*Sqrt[11]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/66","A",1
122,1,64,71,0.1007716,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \sqrt{2+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*Sqrt[2 + 3*x^2]),x]","-\frac{\sqrt{3 x^2+2}}{22 x+11}+\frac{4 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{11 \sqrt{11}}+\frac{\sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{3}}","-\frac{\sqrt{3 x^2+2}}{11 (2 x+1)}+\frac{4 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{11 \sqrt{11}}+\frac{\sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{3}}",1,"-(Sqrt[2 + 3*x^2]/(11 + 22*x)) + ArcSinh[Sqrt[3/2]*x]/Sqrt[3] + (4*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(11*Sqrt[11])","A",1
123,1,55,77,0.0639689,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \sqrt{2+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*Sqrt[2 + 3*x^2]),x]","\frac{\frac{11 (13 x+1) \sqrt{3 x^2+2}}{(2 x+1)^2}-103 \sqrt{11} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{1331}","\frac{13 \sqrt{3 x^2+2}}{242 (2 x+1)}-\frac{\sqrt{3 x^2+2}}{22 (2 x+1)^2}-\frac{103 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{121 \sqrt{11}}",1,"((11*(1 + 13*x)*Sqrt[2 + 3*x^2])/(1 + 2*x)^2 - 103*Sqrt[11]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/1331","A",1
124,1,58,87,0.05465,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(3/2),x]","\frac{576 x^4+1944 x^3+2136 x^2-684 \sqrt{9 x^2+6} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)+2133 x+2362}{162 \sqrt{3 x^2+2}}","\frac{32}{27} \sqrt{3 x^2+2} x^2+4 \sqrt{3 x^2+2} x+\frac{292}{81} \sqrt{3 x^2+2}+\frac{279 x+398}{54 \sqrt{3 x^2+2}}-\frac{38 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{3 \sqrt{3}}",1,"(2362 + 2133*x + 2136*x^2 + 1944*x^3 + 576*x^4 - 684*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(162*Sqrt[2 + 3*x^2])","A",1
125,1,53,71,0.0501416,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(3/2),x]","\frac{48 x^3+168 x^2+8 \sqrt{9 x^2+6} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)-15 x+182}{18 \sqrt{3 x^2+2}}","\frac{70-47 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} x \sqrt{3 x^2+2}+\frac{28}{9} \sqrt{3 x^2+2}+\frac{4 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{3 \sqrt{3}}",1,"(182 - 15*x + 168*x^2 + 48*x^3 + 8*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])","A",1
126,1,48,55,0.0286461,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(3/2),x]","\frac{48 x^2+20 \sqrt{9 x^2+6} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)-51 x+34}{18 \sqrt{3 x^2+2}}","\frac{2-51 x}{18 \sqrt{3 x^2+2}}+\frac{8}{9} \sqrt{3 x^2+2}+\frac{10 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{3 \sqrt{3}}",1,"(34 - 51*x + 48*x^2 + 20*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/(18*Sqrt[2 + 3*x^2])","A",1
127,1,51,53,0.0226801,"\int \frac{1+3 x+4 x^2}{(1+2 x) \left(2+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(3/2)),x]","\frac{-12 \sqrt{33 x^2+22} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)+231 x-418}{726 \sqrt{3 x^2+2}}","\frac{21 x-38}{66 \sqrt{3 x^2+2}}-\frac{2 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{11 \sqrt{11}}",1,"(-418 + 231*x - 12*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(726*Sqrt[2 + 3*x^2])","A",1
128,1,71,75,0.0420118,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left(2+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(3/2)),x]","\frac{11 \left(170 x^2+77 x-26\right)+8 (2 x+1) \sqrt{33 x^2+22} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{2662 (2 x+1) \sqrt{3 x^2+2}}","\frac{97 x-10}{242 \sqrt{3 x^2+2}}-\frac{4 \sqrt{3 x^2+2}}{121 (2 x+1)}+\frac{4 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{121 \sqrt{11}}",1,"(11*(-26 + 77*x + 170*x^2) + 8*(1 + 2*x)*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(2662*(1 + 2*x)*Sqrt[2 + 3*x^2])","A",1
129,1,78,97,0.0827171,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left(2+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 + 3*x^2)^(3/2)),x]","\frac{11 \left(1428 x^3+2716 x^2+1799 x+278\right)-644 (2 x+1)^2 \sqrt{33 x^2+22} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{29282 (2 x+1)^2 \sqrt{3 x^2+2}}","\frac{351 x+358}{2662 \sqrt{3 x^2+2}}+\frac{2 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{2 \sqrt{3 x^2+2}}{121 (2 x+1)^2}-\frac{322 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{1331 \sqrt{11}}",1,"(11*(278 + 1799*x + 2716*x^2 + 1428*x^3) - 644*(1 + 2*x)^2*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(29282*(1 + 2*x)^2*Sqrt[2 + 3*x^2])","A",1
130,1,63,73,0.0679722,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]","\frac{1728 x^4-4185 x^3+936 x^2+432 \sqrt{3} \left(3 x^2+2\right)^{3/2} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)-2511 x+254}{162 \left(3 x^2+2\right)^{3/2}}","\frac{279 x+398}{162 \left(3 x^2+2\right)^{3/2}}+\frac{32}{27} \sqrt{3 x^2+2}-\frac{465 x+152}{54 \sqrt{3 x^2+2}}+\frac{8 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{\sqrt{3}}",1,"(254 - 2511*x + 936*x^2 - 4185*x^3 + 1728*x^4 + 432*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(162*(2 + 3*x^2)^(3/2))","A",1
131,1,58,60,0.0511893,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]","\frac{-177 x^3-504 x^2+32 \sqrt{3} \left(3 x^2+2\right)^{3/2} \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)-165 x-266}{54 \left(3 x^2+2\right)^{3/2}}","\frac{70-47 x}{54 \left(3 x^2+2\right)^{3/2}}-\frac{59 x+168}{54 \sqrt{3 x^2+2}}+\frac{16 \sinh ^{-1}\left(\sqrt{\frac{3}{2}} x\right)}{9 \sqrt{3}}",1,"(-266 - 165*x - 504*x^2 - 177*x^3 + 32*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(54*(2 + 3*x^2)^(3/2))","A",1
132,1,30,41,0.0174099,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\left(2+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]","\frac{117 x^3-144 x^2+27 x-94}{54 \left(3 x^2+2\right)^{3/2}}","\frac{2-51 x}{54 \left(3 x^2+2\right)^{3/2}}-\frac{16-13 x}{18 \sqrt{3 x^2+2}}",1,"(-94 + 27*x - 144*x^2 + 117*x^3)/(54*(2 + 3*x^2)^(3/2))","A",1
133,1,58,73,0.0483941,"\int \frac{1+3 x+4 x^2}{(1+2 x) \left(2+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(5/2)),x]","\frac{855 x^3+216 x^2+801 x-274}{2178 \left(3 x^2+2\right)^{3/2}}-\frac{8 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{121 \sqrt{11}}","\frac{21 x-38}{198 \left(3 x^2+2\right)^{3/2}}+\frac{95 x+24}{726 \sqrt{3 x^2+2}}-\frac{8 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{121 \sqrt{11}}",1,"(-274 + 801*x + 216*x^2 + 855*x^3)/(2178*(2 + 3*x^2)^(3/2)) - (8*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(121*Sqrt[11])","A",1
134,1,91,95,0.0547149,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left(2+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(5/2)),x]","\frac{11 \left(4458 x^4+2805 x^3+4602 x^2+2717 x-446\right)-192 \sqrt{33 x^2+22} \left(6 x^3+3 x^2+4 x+2\right) \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{87846 (2 x+1) \left(3 x^2+2\right)^{3/2}}","\frac{97 x-10}{726 \left(3 x^2+2\right)^{3/2}}-\frac{16 \sqrt{3 x^2+2}}{1331 (2 x+1)}+\frac{887 x+24}{7986 \sqrt{3 x^2+2}}-\frac{32 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{1331 \sqrt{11}}",1,"(11*(-446 + 2717*x + 4602*x^2 + 2805*x^3 + 4458*x^4) - 192*Sqrt[22 + 33*x^2]*(2 + 4*x + 3*x^2 + 6*x^3)*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(87846*(1 + 2*x)*(2 + 3*x^2)^(3/2))","A",1
135,1,75,117,0.0997183,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left(2+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 + 3*x^2)^(5/2)),x]","\frac{\frac{11 \left(67284 x^5+111060 x^4+116937 x^3+109844 x^2+57371 x+7010\right)}{(2 x+1)^2 \left(3 x^2+2\right)^{3/2}}-7296 \sqrt{11} \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{33 x^2+22}}\right)}{966306}","\frac{351 x+358}{7986 \left(3 x^2+2\right)^{3/2}}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)}-\frac{8 \sqrt{3 x^2+2}}{1331 (2 x+1)^2}+\frac{2133 x+1216}{29282 \sqrt{3 x^2+2}}-\frac{1216 \tanh ^{-1}\left(\frac{4-3 x}{\sqrt{11} \sqrt{3 x^2+2}}\right)}{14641 \sqrt{11}}",1,"((11*(7010 + 57371*x + 109844*x^2 + 116937*x^3 + 111060*x^4 + 67284*x^5))/((1 + 2*x)^2*(2 + 3*x^2)^(3/2)) - 7296*Sqrt[11]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/966306","A",1
136,0,0,420,1.1479959,"\int (g+h x)^m \left(a+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^m*(a + c*x^2)^p*(d + e*x + f*x^2),x]","\int (g+h x)^m \left(a+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","-\frac{\left(a+c x^2\right)^p (g+h x)^{m+1} \left(1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right)^{-p} \left(1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right) \left(a f h^2 (m+1)-c \left(2 f g^2 (p+1)-h (m+2 p+3) (e g-d h)\right)\right)}{c h^3 (m+1) (m+2 p+3)}-\frac{\left(a+c x^2\right)^p (g+h x)^{m+2} \left(1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right)^{-p} \left(1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right)^{-p} (2 f g (p+1)-e h (m+2 p+3)) F_1\left(m+2;-p,-p;m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right)}{h^3 (m+2) (m+2 p+3)}+\frac{f \left(a+c x^2\right)^{p+1} (g+h x)^{m+1}}{c h (m+2 p+3)}",1,"Integrate[(g + h*x)^m*(a + c*x^2)^p*(d + e*x + f*x^2), x]","F",-1
137,0,0,403,0.6918292,"\int (g+h x)^m \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^m*Sqrt[a + c*x^2]*(d + e*x + f*x^2),x]","\int (g+h x)^m \sqrt{a+c x^2} \left(d+e x+f x^2\right) \, dx","-\frac{\sqrt{a+c x^2} (g+h x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right) \left(a f h^2 (m+1)-c \left(3 f g^2-h (m+4) (e g-d h)\right)\right)}{c h^3 (m+1) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}-\frac{\sqrt{a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left(m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right)}{h^3 (m+2) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}+\frac{f \left(a+c x^2\right)^{3/2} (g+h x)^{m+1}}{c h (m+4)}",1,"Integrate[(g + h*x)^m*Sqrt[a + c*x^2]*(d + e*x + f*x^2), x]","F",-1
138,0,0,474,2.8499326,"\int (g+h x)^{-3-2 p} \left(a+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^(-3 - 2*p)*(a + c*x^2)^p*(d + e*x + f*x^2),x]","\int (g+h x)^{-3-2 p} \left(a+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","-\frac{f \left(a+c x^2\right)^p (g+h x)^{-2 p} \left(1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right)^{-p} \left(1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right)}{2 h^3 p}-\frac{\left(a+c x^2\right)^{p+1} (g+h x)^{-2 (p+1)} \left(d h^2-e g h+f g^2\right)}{2 h (p+1) \left(a h^2+c g^2\right)}+\frac{\left(\sqrt{-a}-\sqrt{c} x\right) \left(a+c x^2\right)^p (g+h x)^{-2 p-1} \left(-\frac{\left(\sqrt{-a}+\sqrt{c} x\right) \left(\sqrt{-a} h+\sqrt{c} g\right)}{\left(\sqrt{-a}-\sqrt{c} x\right) \left(\sqrt{c} g-\sqrt{-a} h\right)}\right)^{-p} \left(a h^2 (2 f g-e h)+c \left(f g^3-d g h^2\right)\right) \, _2F_1\left(-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (g+h x)}{\left(\sqrt{c} g-\sqrt{-a} h\right) \left(\sqrt{-a}-\sqrt{c} x\right)}\right)}{h^2 (2 p+1) \left(\sqrt{-a} h+\sqrt{c} g\right) \left(a h^2+c g^2\right)}",1,"Integrate[(g + h*x)^(-3 - 2*p)*(a + c*x^2)^p*(d + e*x + f*x^2), x]","F",-1
139,1,165,222,0.3194246,"\int (d+e x)^m \left(-c d^2+b d e+b e^2 x+c e^2 x^2\right)^p \left((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2\right) \, dx","Integrate[(d + e*x)^m*(-(c*d^2) + b*d*e + b*e^2*x + c*e^2*x^2)^p*((-(c*d) + b*e)*f + (c*e*f - c*d*g + b*e*g)*x + c*e*g*x^2),x]","\frac{(d+e x)^m (b e-c d+c e x)^2 (-((d+e x) (c (d-e x)-b e)))^p \left(\frac{e \left(\frac{c (d+e x)}{2 c d-b e}\right)^{-m-p} (-b e g (m+p+1)+c d g (m-1)+c e f (m+2 p+3)) \, _2F_1\left(-m-p,p+2;p+3;\frac{-c d+b e+c e x}{b e-2 c d}\right)}{p+2}+c e g (d+e x)\right)}{c^2 e^3 (m+2 p+3)}","\frac{g (d+e x)^{m-1} \left(-d (c d-b e)+b e^2 x+c e^2 x^2\right)^{p+2}}{c e^2 (m+2 p+3)}-\frac{(d+e x)^m (-b e+c d-c e x)^2 \left(-d (c d-b e)+b e^2 x+c e^2 x^2\right)^p \left(\frac{c (d+e x)}{2 c d-b e}\right)^{-m-p} (b e g (m+p+1)+c (d g (1-m)-e f (m+2 p+3))) \, _2F_1\left(-m-p,p+2;p+3;\frac{c d-b e-c e x}{2 c d-b e}\right)}{c^2 e^2 (p+2) (m+2 p+3)}",1,"((d + e*x)^m*(-(c*d) + b*e + c*e*x)^2*(-((d + e*x)*(-(b*e) + c*(d - e*x))))^p*(c*e*g*(d + e*x) + (e*(c*d*g*(-1 + m) - b*e*g*(1 + m + p) + c*e*f*(3 + m + 2*p))*((c*(d + e*x))/(2*c*d - b*e))^(-m - p)*Hypergeometric2F1[-m - p, 2 + p, 3 + p, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(2 + p)))/(c^2*e^3*(3 + m + 2*p))","A",1
140,1,256,254,0.0941036,"\int \left(a+b x+c x^2\right)^4 \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^4*(A + C*x^2),x]","a^4 A x+2 a^3 A b x^2+a b x^4 \left(a^2 C+3 a A c+A b^2\right)+\frac{1}{3} a^2 x^3 \left(a^2 C+4 a A c+6 A b^2\right)+\frac{1}{7} x^7 \left(6 a^2 c^2 C+4 a A c^3+12 a b^2 c C+6 A b^2 c^2+b^4 C\right)+\frac{1}{5} x^5 \left(4 a^3 c C+6 a^2 A c^2+6 a^2 b^2 C+12 a A b^2 c+A b^4\right)+\frac{1}{9} c^2 x^9 \left(4 a c C+A c^2+6 b^2 C\right)+\frac{1}{2} b c x^8 \left(3 a c C+A c^2+b^2 C\right)+\frac{2}{3} b x^6 \left(3 a c+b^2\right) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11}","a^4 A x+2 a^3 A b x^2+a b x^4 \left(a^2 C+A \left(3 a c+b^2\right)\right)+\frac{1}{3} a^2 x^3 \left(a^2 C+4 a A c+6 A b^2\right)+\frac{1}{7} x^7 \left(C \left(6 a^2 c^2+12 a b^2 c+b^4\right)+2 A c^2 \left(2 a c+3 b^2\right)\right)+\frac{1}{5} x^5 \left(A \left(6 a^2 c^2+12 a b^2 c+b^4\right)+2 a^2 C \left(2 a c+3 b^2\right)\right)+\frac{1}{9} c^2 x^9 \left(4 a c C+A c^2+6 b^2 C\right)+\frac{1}{2} b c x^8 \left(C \left(3 a c+b^2\right)+A c^2\right)+\frac{2}{3} b x^6 \left(3 a c+b^2\right) (a C+A c)+\frac{2}{5} b c^3 C x^{10}+\frac{1}{11} c^4 C x^{11}",1,"a^4*A*x + 2*a^3*A*b*x^2 + (a^2*(6*A*b^2 + 4*a*A*c + a^2*C)*x^3)/3 + a*b*(A*b^2 + 3*a*A*c + a^2*C)*x^4 + ((A*b^4 + 12*a*A*b^2*c + 6*a^2*A*c^2 + 6*a^2*b^2*C + 4*a^3*c*C)*x^5)/5 + (2*b*(b^2 + 3*a*c)*(A*c + a*C)*x^6)/3 + ((6*A*b^2*c^2 + 4*a*A*c^3 + b^4*C + 12*a*b^2*c*C + 6*a^2*c^2*C)*x^7)/7 + (b*c*(A*c^2 + b^2*C + 3*a*c*C)*x^8)/2 + (c^2*(A*c^2 + 6*b^2*C + 4*a*c*C)*x^9)/9 + (2*b*c^3*C*x^10)/5 + (c^4*C*x^11)/11","A",1
141,1,163,161,0.0461959,"\int \left(a+b x+c x^2\right)^3 \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^3*(A + C*x^2),x]","a^3 A x+\frac{1}{4} b x^4 \left(3 a^2 C+6 a A c+A b^2\right)+\frac{1}{3} a x^3 \left(a^2 C+3 a A c+3 A b^2\right)+\frac{3}{2} a^2 A b x^2+\frac{1}{7} c x^7 \left(3 a c C+A c^2+3 b^2 C\right)+\frac{1}{6} b x^6 \left(6 a c C+3 A c^2+b^2 C\right)+\frac{3}{5} x^5 \left(a c+b^2\right) (a C+A c)+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9","a^3 A x+\frac{1}{4} b x^4 \left(3 a^2 C+A \left(6 a c+b^2\right)\right)+\frac{1}{3} a x^3 \left(a^2 C+3 A \left(a c+b^2\right)\right)+\frac{3}{2} a^2 A b x^2+\frac{1}{7} c x^7 \left(3 C \left(a c+b^2\right)+A c^2\right)+\frac{1}{6} b x^6 \left(C \left(6 a c+b^2\right)+3 A c^2\right)+\frac{3}{5} x^5 \left(a c+b^2\right) (a C+A c)+\frac{3}{8} b c^2 C x^8+\frac{1}{9} c^3 C x^9",1,"a^3*A*x + (3*a^2*A*b*x^2)/2 + (a*(3*A*b^2 + 3*a*A*c + a^2*C)*x^3)/3 + (b*(A*b^2 + 6*a*A*c + 3*a^2*C)*x^4)/4 + (3*(b^2 + a*c)*(A*c + a*C)*x^5)/5 + (b*(3*A*c^2 + b^2*C + 6*a*c*C)*x^6)/6 + (c*(A*c^2 + 3*b^2*C + 3*a*c*C)*x^7)/7 + (3*b*c^2*C*x^8)/8 + (c^3*C*x^9)/9","A",1
142,1,96,96,0.0234885,"\int \left(a+b x+c x^2\right)^2 \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^2*(A + C*x^2),x]","\frac{1}{3} x^3 \left(a^2 C+2 a A c+A b^2\right)+a^2 A x+\frac{1}{5} x^5 \left(2 a c C+A c^2+b^2 C\right)+\frac{1}{2} b x^4 (a C+A c)+a A b x^2+\frac{1}{3} b c C x^6+\frac{1}{7} c^2 C x^7","\frac{1}{3} x^3 \left(a^2 C+A \left(2 a c+b^2\right)\right)+a^2 A x+\frac{1}{5} x^5 \left(C \left(2 a c+b^2\right)+A c^2\right)+\frac{1}{2} b x^4 (a C+A c)+a A b x^2+\frac{1}{3} b c C x^6+\frac{1}{7} c^2 C x^7",1,"a^2*A*x + a*A*b*x^2 + ((A*b^2 + 2*a*A*c + a^2*C)*x^3)/3 + (b*(A*c + a*C)*x^4)/2 + ((A*c^2 + b^2*C + 2*a*c*C)*x^5)/5 + (b*c*C*x^6)/3 + (c^2*C*x^7)/7","A",1
143,1,46,46,0.0088129,"\int \left(a+b x+c x^2\right) \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)*(A + C*x^2),x]","\frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} A b x^2+\frac{1}{4} b C x^4+\frac{1}{5} c C x^5","\frac{1}{3} x^3 (a C+A c)+a A x+\frac{1}{2} A b x^2+\frac{1}{4} b C x^4+\frac{1}{5} c C x^5",1,"a*A*x + (A*b*x^2)/2 + ((A*c + a*C)*x^3)/3 + (b*C*x^4)/4 + (c*C*x^5)/5","A",1
144,1,84,81,0.0922366,"\int \frac{A+C x^2}{a+b x+c x^2} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2),x]","\frac{\left(-2 a c C+2 A c^2+b^2 C\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{c^2 \sqrt{4 a c-b^2}}-\frac{b C \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{C x}{c}","-\frac{\left(C \left(b^2-2 a c\right)+2 A c^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{b C \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{C x}{c}",1,"(C*x)/c + ((2*A*c^2 + b^2*C - 2*a*c*C)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]) - (b*C*Log[a + b*x + c*x^2])/(2*c^2)","A",1
145,1,98,100,0.0810696,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^2,x]","\frac{a C (b-2 c x)+A c (b+2 c x)+b^2 C x}{c \left(4 a c-b^2\right) (a+x (b+c x))}+\frac{4 (a C+A c) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{3/2}}","\frac{4 (a C+A c) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{3/2}}-\frac{x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)}{c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}",1,"(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*(A*c + a*C)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
146,1,160,161,0.2049561,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^3,x]","\frac{1}{2} \left(\frac{(b+2 c x) \left(C \left(2 a c+b^2\right)+6 A c^2\right)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))}+\frac{4 \left(C \left(2 a c+b^2\right)+6 A c^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{a C (b-2 c x)+A c (b+2 c x)+b^2 C x}{c \left(4 a c-b^2\right) (a+x (b+c x))^2}\right)","-\frac{x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)}{2 c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}-\frac{2 \left(C \left(2 a c+b^2\right)+6 A c^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{5/2}}+\frac{(b+2 c x) \left(2 a C+6 A c+\frac{b^2 C}{c}\right)}{2 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}",1,"(((6*A*c^2 + (b^2 + 2*a*c)*C)*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (4*(6*A*c^2 + (b^2 + 2*a*c)*C)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2","A",1
147,1,204,206,0.4082422,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^4} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^4,x]","\frac{1}{3} \left(-\frac{6 (b+2 c x) \left(C \left(a c+b^2\right)+5 A c^2\right)}{\left(b^2-4 a c\right)^3 (a+x (b+c x))}+\frac{(b+2 c x) \left(C \left(a c+b^2\right)+5 A c^2\right)}{c \left(b^2-4 a c\right)^2 (a+x (b+c x))^2}+\frac{24 c \left(C \left(a c+b^2\right)+5 A c^2\right) \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right)}{\left(4 a c-b^2\right)^{7/2}}+\frac{a C (b-2 c x)+A c (b+2 c x)+b^2 C x}{c \left(4 a c-b^2\right) (a+x (b+c x))^3}\right)","-\frac{2 (b+2 c x) \left(C \left(a c+b^2\right)+5 A c^2\right)}{\left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)}-\frac{x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)}{3 c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^3}+\frac{8 c \left(C \left(a c+b^2\right)+5 A c^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{\left(b^2-4 a c\right)^{7/2}}+\frac{(b+2 c x) \left(C \left(a+\frac{b^2}{c}\right)+5 A c\right)}{3 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^2}",1,"(((5*A*c^2 + (b^2 + a*c)*C)*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*(5*A*c^2 + (b^2 + a*c)*C)*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (24*c*(5*A*c^2 + (b^2 + a*c)*C)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3","A",1
148,1,585,591,0.650713,"\int \frac{(d+e x)^3 \left(f+g x+h x^2\right)}{a+b x+c x^2} \, dx","Integrate[((d + e*x)^3*(f + g*x + h*x^2))/(a + b*x + c*x^2),x]","\frac{6 \log (a+x (b+c x)) \left(c^2 e \left(a^2 e^2 h+2 a b e (3 d h+e g)+b^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)-b^2 c e^2 (3 a e h+3 b d h+b e g)-c^3 \left(a e \left(3 d^2 h+3 d e g+e^2 f\right)+b d \left(d^2 h+3 d e g+3 e^2 f\right)\right)+b^4 e^3 h+c^4 d^2 (d g+3 e f)\right)+\frac{12 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(c^3 \left(2 a^2 e^2 (3 d h+e g)+3 a b e \left(3 d^2 h+3 d e g+e^2 f\right)+b^2 d \left(d^2 h+3 d e g+3 e^2 f\right)\right)-b c^2 e \left(5 a^2 e^2 h+4 a b e (3 d h+e g)+b^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)+b^3 c e^2 (5 a e h+3 b d h+b e g)-c^4 d \left(2 a \left(d^2 h+3 d e g+3 e^2 f\right)+b d (d g+3 e f)\right)+b^5 \left(-e^3\right) h+2 c^5 d^3 f\right)}{\sqrt{4 a c-b^2}}+12 c x \left(-c^2 e \left(a e (3 d h+e g)+b \left(3 d^2 h+3 d e g+e^2 f\right)\right)+b c e^2 (2 a e h+3 b d h+b e g)-b^3 e^3 h+c^3 d \left(d^2 h+3 d e g+3 e^2 f\right)\right)+6 c^2 e x^2 \left(-c e (a e h+3 b d h+b e g)+b^2 e^2 h+c^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)+4 c^3 e^2 x^3 (-b e h+3 c d h+c e g)+3 c^4 e^3 h x^4}{12 c^5}","\frac{\log \left(a+b x+c x^2\right) \left(c^2 e \left(a^2 e^2 h+2 a b e (3 d h+e g)+b^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)-b^2 c e^2 (3 a e h+3 b d h+b e g)-c^3 \left(a e \left(3 d^2 h+3 d e g+e^2 f\right)+b d \left(d^2 h+3 d e g+3 e^2 f\right)\right)+b^4 e^3 h+c^4 d^2 (d g+3 e f)\right)}{2 c^5}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(c^3 \left(2 a^2 e^2 (3 d h+e g)+3 a b e \left(3 d^2 h+3 d e g+e^2 f\right)+b^2 d \left(d^2 h+3 d e g+3 e^2 f\right)\right)-b c^2 e \left(5 a^2 e^2 h+4 a b e (3 d h+e g)+b^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)+b^3 c e^2 (5 a e h+3 b d h+b e g)-c^4 d \left(2 a \left(d^2 h+3 d e g+3 e^2 f\right)+b d (d g+3 e f)\right)+b^5 \left(-e^3\right) h+2 c^5 d^3 f\right)}{c^5 \sqrt{b^2-4 a c}}-\frac{x \left(c^2 e \left(a e (3 d h+e g)+b \left(3 d^2 h+3 d e g+e^2 f\right)\right)-b c e^2 (2 a e h+3 b d h+b e g)+b^3 e^3 h+c^3 (-d) \left(d^2 h+3 d e g+3 e^2 f\right)\right)}{c^4}+\frac{e x^2 \left(-c e (a e h+3 b d h+b e g)+b^2 e^2 h+c^2 \left(3 d^2 h+3 d e g+e^2 f\right)\right)}{2 c^3}+\frac{e^2 x^3 (-b e h+3 c d h+c e g)}{3 c^2}+\frac{e^3 h x^4}{4 c}",1,"(12*c*(-(b^3*e^3*h) + c^3*d*(3*e^2*f + 3*d*e*g + d^2*h) + b*c*e^2*(b*e*g + 3*b*d*h + 2*a*e*h) - c^2*e*(a*e*(e*g + 3*d*h) + b*(e^2*f + 3*d*e*g + 3*d^2*h)))*x + 6*c^2*e*(b^2*e^2*h + c^2*(e^2*f + 3*d*e*g + 3*d^2*h) - c*e*(b*e*g + 3*b*d*h + a*e*h))*x^2 + 4*c^3*e^2*(c*e*g + 3*c*d*h - b*e*h)*x^3 + 3*c^4*e^3*h*x^4 + (12*(2*c^5*d^3*f - b^5*e^3*h + b^3*c*e^2*(b*e*g + 3*b*d*h + 5*a*e*h) - c^4*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f + 3*d*e*g + d^2*h)) - b*c^2*e*(5*a^2*e^2*h + 4*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) + c^3*(2*a^2*e^2*(e*g + 3*d*h) + b^2*d*(3*e^2*f + 3*d*e*g + d^2*h) + 3*a*b*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(c^4*d^2*(3*e*f + d*g) + b^4*e^3*h - b^2*c*e^2*(b*e*g + 3*b*d*h + 3*a*e*h) + c^2*e*(a^2*e^2*h + 2*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) - c^3*(b*d*(3*e^2*f + 3*d*e*g + d^2*h) + a*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*Log[a + x*(b + c*x)])/(12*c^5)","A",1
149,1,345,348,0.3803908,"\int \frac{(d+e x)^2 \left(f+g x+h x^2\right)}{a+b x+c x^2} \, dx","Integrate[((d + e*x)^2*(f + g*x + h*x^2))/(a + b*x + c*x^2),x]","\frac{\frac{6 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(c^2 \left(2 a^2 e^2 h+3 a b e (2 d h+e g)+b^2 \left(d^2 h+2 d e g+e^2 f\right)\right)-b^2 c e (4 a e h+2 b d h+b e g)-c^3 \left(2 a \left(d^2 h+2 d e g+e^2 f\right)+b d (d g+2 e f)\right)+b^4 e^2 h+2 c^4 d^2 f\right)}{\sqrt{4 a c-b^2}}+3 \log (a+x (b+c x)) \left(-c^2 \left(a e (2 d h+e g)+b \left(d^2 h+2 d e g+e^2 f\right)\right)+b c e (2 a e h+2 b d h+b e g)+b^3 \left(-e^2\right) h+c^3 d (d g+2 e f)\right)+6 c x \left(-c e (a e h+2 b d h+b e g)+b^2 e^2 h+c^2 \left(d^2 h+2 d e g+e^2 f\right)\right)+3 c^2 e x^2 (-b e h+2 c d h+c e g)+2 c^3 e^2 h x^3}{6 c^4}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(c^2 \left(2 a^2 e^2 h+3 a b e (2 d h+e g)+b^2 \left(d^2 h+2 d e g+e^2 f\right)\right)-b^2 c e (4 a e h+2 b d h+b e g)-c^3 \left(2 a \left(d^2 h+2 d e g+e^2 f\right)+b d (d g+2 e f)\right)+b^4 e^2 h+2 c^4 d^2 f\right)}{c^4 \sqrt{b^2-4 a c}}+\frac{\log \left(a+b x+c x^2\right) \left(-c^2 \left(a e (2 d h+e g)+b \left(d^2 h+2 d e g+e^2 f\right)\right)+b c e (2 a e h+2 b d h+b e g)+b^3 \left(-e^2\right) h+c^3 d (d g+2 e f)\right)}{2 c^4}+\frac{x \left(-c e (a e h+2 b d h+b e g)+b^2 e^2 h+c^2 \left(d^2 h+2 d e g+e^2 f\right)\right)}{c^3}+\frac{e x^2 (-b e h+2 c d h+c e g)}{2 c^2}+\frac{e^2 h x^3}{3 c}",1,"(6*c*(b^2*e^2*h + c^2*(e^2*f + 2*d*e*g + d^2*h) - c*e*(b*e*g + 2*b*d*h + a*e*h))*x + 3*c^2*e*(c*e*g + 2*c*d*h - b*e*h)*x^2 + 2*c^3*e^2*h*x^3 + (6*(2*c^4*d^2*f + b^4*e^2*h - b^2*c*e*(b*e*g + 2*b*d*h + 4*a*e*h) - c^3*(b*d*(2*e*f + d*g) + 2*a*(e^2*f + 2*d*e*g + d^2*h)) + c^2*(2*a^2*e^2*h + 3*a*b*e*(e*g + 2*d*h) + b^2*(e^2*f + 2*d*e*g + d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 3*(c^3*d*(2*e*f + d*g) - b^3*e^2*h + b*c*e*(b*e*g + 2*b*d*h + 2*a*e*h) - c^2*(a*e*(e*g + 2*d*h) + b*(e^2*f + 2*d*e*g + d^2*h)))*Log[a + x*(b + c*x)])/(6*c^4)","A",1
150,1,173,177,0.2046132,"\int \frac{(d+e x) \left(f+g x+h x^2\right)}{a+b x+c x^2} \, dx","Integrate[((d + e*x)*(f + g*x + h*x^2))/(a + b*x + c*x^2),x]","\frac{\log (a+x (b+c x)) \left(-c (a e h+b d h+b e g)+b^2 e h+c^2 (d g+e f)\right)-\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(c^2 (2 a d h+2 a e g+b d g+b e f)-b c (3 a e h+b d h+b e g)+b^3 e h-2 c^3 d f\right)}{\sqrt{4 a c-b^2}}+2 c x (-b e h+c d h+c e g)+c^2 e h x^2}{2 c^3}","\frac{\log \left(a+b x+c x^2\right) \left(-c (a e h+b d h+b e g)+b^2 e h+c^2 (d g+e f)\right)}{2 c^3}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-c^2 (2 a d h+2 a e g+b d g+b e f)+b c (3 a e h+b d h+b e g)+b^3 (-e) h+2 c^3 d f\right)}{c^3 \sqrt{b^2-4 a c}}+\frac{x (-b e h+c d h+c e g)}{c^2}+\frac{e h x^2}{2 c}",1,"(2*c*(c*e*g + c*d*h - b*e*h)*x + c^2*e*h*x^2 - (2*(-2*c^3*d*f + b^3*e*h + c^2*(b*e*f + b*d*g + 2*a*e*g + 2*a*d*h) - b*c*(b*e*g + b*d*h + 3*a*e*h))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c^2*(e*f + d*g) + b^2*e*h - c*(b*e*g + b*d*h + a*e*h))*Log[a + x*(b + c*x)])/(2*c^3)","A",1
151,1,95,92,0.0733951,"\int \frac{f+g x+h x^2}{a+b x+c x^2} \, dx","Integrate[(f + g*x + h*x^2)/(a + b*x + c*x^2),x]","\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-2 a c h+b^2 h-b c g+2 c^2 f\right)}{c^2 \sqrt{4 a c-b^2}}+\frac{(c g-b h) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{h x}{c}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-2 a c h+b^2 h-b c g+2 c^2 f\right)}{c^2 \sqrt{b^2-4 a c}}+\frac{(c g-b h) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{h x}{c}",1,"(h*x)/c + ((2*c^2*f - b*c*g + b^2*h - 2*a*c*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]) + ((c*g - b*h)*Log[a + b*x + c*x^2])/(2*c^2)","A",1
152,1,193,196,0.2055321,"\int \frac{f+g x+h x^2}{(d+e x) \left(a+b x+c x^2\right)} \, dx","Integrate[(f + g*x + h*x^2)/((d + e*x)*(a + b*x + c*x^2)),x]","\frac{-2 e \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(c (2 a d h-2 a e g+b d g+b e f)+b h (a e-b d)-2 c^2 d f\right)+2 c \sqrt{4 a c-b^2} \log (d+e x) \left(d^2 h-d e g+e^2 f\right)-e \sqrt{4 a c-b^2} \log (a+x (b+c x)) (-a e h+b d h-c d g+c e f)}{2 c e \sqrt{4 a c-b^2} \left(e (a e-b d)+c d^2\right)}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-c (2 a d h-2 a e g+b d g+b e f)+b h (b d-a e)+2 c^2 d f\right)}{c \sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)}-\frac{\log \left(a+b x+c x^2\right) (-a e h+b d h-c d g+c e f)}{2 c \left(a e^2-b d e+c d^2\right)}+\frac{\log (d+e x) \left(d^2 h-d e g+e^2 f\right)}{e \left(a e^2-b d e+c d^2\right)}",1,"(-2*e*(-2*c^2*d*f + b*(-(b*d) + a*e)*h + c*(b*e*f + b*d*g - 2*a*e*g + 2*a*d*h))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + 2*c*Sqrt[-b^2 + 4*a*c]*(e^2*f - d*e*g + d^2*h)*Log[d + e*x] - Sqrt[-b^2 + 4*a*c]*e*(c*e*f - c*d*g + b*d*h - a*e*h)*Log[a + x*(b + c*x)])/(2*c*Sqrt[-b^2 + 4*a*c]*e*(c*d^2 + e*(-(b*d) + a*e)))","A",1
153,1,281,316,0.5361549,"\int \frac{f+g x+h x^2}{(d+e x)^2 \left(a+b x+c x^2\right)} \, dx","Integrate[(f + g*x + h*x^2)/((d + e*x)^2*(a + b*x + c*x^2)),x]","\frac{\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(2 a^2 e^2 h-c \left(2 a \left(d^2 h-2 d e g+e^2 f\right)+b d (d g+2 e f)\right)-a b e (2 d h+e g)+b^2 \left(d^2 h+e^2 f\right)+2 c^2 d^2 f\right)}{\sqrt{4 a c-b^2}}-\frac{2 \left(e (a e-b d)+c d^2\right) \left(d^2 h-d e g+e^2 f\right)}{e (d+e x)}+2 \log (d+e x) \left(a e (e g-2 d h)+b \left(d^2 h-e^2 f\right)+c d (2 e f-d g)\right)+\log (a+x (b+c x)) \left(a e (2 d h-e g)+b \left(e^2 f-d^2 h\right)+c d (d g-2 e f)\right)}{2 \left(e (a e-b d)+c d^2\right)^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 a^2 e^2 h-c \left(2 a \left(d^2 h-2 d e g+e^2 f\right)+b d (d g+2 e f)\right)-a b e (2 d h+e g)+b^2 \left(d^2 h+e^2 f\right)+2 c^2 d^2 f\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)^2}-\frac{\log \left(a+b x+c x^2\right) \left(a e (e g-2 d h)-b \left(e^2 f-d^2 h\right)+c d (2 e f-d g)\right)}{2 \left(a e^2-b d e+c d^2\right)^2}-\frac{d^2 h-d e g+e^2 f}{e (d+e x) \left(a e^2-b d e+c d^2\right)}+\frac{\log (d+e x) \left(a e (e g-2 d h)-b \left(e^2 f-d^2 h\right)+c d (2 e f-d g)\right)}{\left(a e^2-b d e+c d^2\right)^2}",1,"((-2*(c*d^2 + e*(-(b*d) + a*e))*(e^2*f - d*e*g + d^2*h))/(e*(d + e*x)) + (2*(2*c^2*d^2*f + 2*a^2*e^2*h - a*b*e*(e*g + 2*d*h) + b^2*(e^2*f + d^2*h) - c*(b*d*(2*e*f + d*g) + 2*a*(e^2*f - 2*d*e*g + d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(c*d*(2*e*f - d*g) + a*e*(e*g - 2*d*h) + b*(-(e^2*f) + d^2*h))*Log[d + e*x] + (c*d*(-2*e*f + d*g) + a*e*(-(e*g) + 2*d*h) + b*(e^2*f - d^2*h))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
154,1,504,509,0.7663942,"\int \frac{f+g x+h x^2}{(d+e x)^3 \left(a+b x+c x^2\right)} \, dx","Integrate[(f + g*x + h*x^2)/((d + e*x)^3*(a + b*x + c*x^2)),x]","-\frac{\log (d+e x) \left(-\left(e^3 \left(a^2 h-a b g+b^2 f\right)\right)+a c e \left(3 d^2 h-3 d e g+e^2 f\right)+b c \left(3 d e^2 f-d^3 h\right)+c^2 d^2 (d g-3 e f)\right)}{\left(e (a e-b d)+c d^2\right)^3}+\frac{\log (a+x (b+c x)) \left(-\left(e^3 \left(a^2 h-a b g+b^2 f\right)\right)+a c e \left(3 d^2 h-3 d e g+e^2 f\right)+b c \left(3 d e^2 f-d^3 h\right)+c^2 d^2 (d g-3 e f)\right)}{2 \left(e (a e-b d)+c d^2\right)^3}+\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-c \left(-2 a^2 e^2 (e g-3 d h)+3 a b e \left(d^2 (-h)-d e g+e^2 f\right)+b^2 \left(d^3 h+3 d e^2 f\right)\right)+b e^3 \left(a^2 h-a b g+b^2 f\right)+c^2 d \left(2 a \left(d^2 h-3 d e g+3 e^2 f\right)+b d (d g+3 e f)\right)-2 c^3 d^3 f\right)}{\sqrt{4 a c-b^2} \left(e (b d-a e)-c d^2\right)^3}-\frac{d^2 h-d e g+e^2 f}{2 e (d+e x)^2 \left(e (a e-b d)+c d^2\right)}+\frac{a e (2 d h-e g)+b \left(e^2 f-d^2 h\right)+c d (d g-2 e f)}{(d+e x) \left(e (a e-b d)+c d^2\right)^2}","-\frac{\log \left(a+b x+c x^2\right) \left(e^3 \left(a^2 h-a b g+b^2 f\right)-c \left(a e \left(3 d^2 h-3 d e g+e^2 f\right)+b \left(3 d e^2 f-d^3 h\right)\right)+c^2 d^2 (3 e f-d g)\right)}{2 \left(a e^2-b d e+c d^2\right)^3}+\frac{\log (d+e x) \left(e^3 \left(a^2 h-a b g+b^2 f\right)-c \left(a e \left(3 d^2 h-3 d e g+e^2 f\right)+b \left(3 d e^2 f-d^3 h\right)\right)+c^2 d^2 (3 e f-d g)\right)}{\left(a e^2-b d e+c d^2\right)^3}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-c \left(2 a^2 e^2 (e g-3 d h)-3 a b e \left(d^2 (-h)-d e g+e^2 f\right)-\left(b^2 \left(d^3 h+3 d e^2 f\right)\right)\right)-b e^3 \left(a^2 h-a b g+b^2 f\right)-c^2 d \left(2 a \left(d^2 h-3 d e g+3 e^2 f\right)+b d (d g+3 e f)\right)+2 c^3 d^3 f\right)}{\sqrt{b^2-4 a c} \left(a e^2-b d e+c d^2\right)^3}-\frac{d^2 h-d e g+e^2 f}{2 e (d+e x)^2 \left(a e^2-b d e+c d^2\right)}-\frac{a e (e g-2 d h)-b \left(e^2 f-d^2 h\right)+c d (2 e f-d g)}{(d+e x) \left(a e^2-b d e+c d^2\right)^2}",1,"-1/2*(e^2*f - d*e*g + d^2*h)/(e*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (c*d*(-2*e*f + d*g) + a*e*(-(e*g) + 2*d*h) + b*(e^2*f - d^2*h))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + ((-2*c^3*d^3*f + b*e^3*(b^2*f - a*b*g + a^2*h) + c^2*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f - 3*d*e*g + d^2*h)) - c*(-2*a^2*e^2*(e*g - 3*d*h) + 3*a*b*e*(e^2*f - d*e*g - d^2*h) + b^2*(3*d*e^2*f + d^3*h)))*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))^3) - ((c^2*d^2*(-3*e*f + d*g) - e^3*(b^2*f - a*b*g + a^2*h) + a*c*e*(e^2*f - 3*d*e*g + 3*d^2*h) + b*c*(3*d*e^2*f - d^3*h))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + ((c^2*d^2*(-3*e*f + d*g) - e^3*(b^2*f - a*b*g + a^2*h) + a*c*e*(e^2*f - 3*d*e*g + 3*d^2*h) + b*c*(3*d*e^2*f - d^3*h))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)","A",1
155,1,398,288,0.8112191,"\int \frac{(d+e x)^2 \left(f+g x+h x^2\right)}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[((d + e*x)^2*(f + g*x + h*x^2))/(a + b*x + c*x^2)^2,x]","\frac{-\frac{2 \left(b c \left(-3 a^2 e^2 h+a c \left(d^2 h+2 d e (g+3 h x)+e^2 (f+3 g x)\right)+c^2 d (d (f-g x)-2 e f x)\right)+2 c^2 \left(a^2 e (2 d h+e (g+h x))-a c \left(d^2 (g+h x)+2 d e (f+g x)+e^2 f x\right)+c^2 d^2 f x\right)+b^3 e (a e h-c x (2 d h+e g))+b^2 c \left(c x \left(d^2 h+2 d e g+e^2 f\right)-a e (2 d h+e g+4 e h x)\right)+b^4 e^2 h x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(b^2 c e (12 a e h+2 b d h+b e g)+c^3 \left(4 a \left(d^2 h+2 d e g+e^2 f\right)-2 b d (d g+2 e f)\right)-6 a c^2 e (2 a e h+2 b d h+b e g)-2 b^4 e^2 h+4 c^4 d^2 f\right)}{\left(4 a c-b^2\right)^{3/2}}+e \log (a+x (b+c x)) (-2 b e h+2 c d h+c e g)+2 c e^2 h x}{2 c^3}","\frac{(d+e x)^2 \left(c \left(2 a g-b \left(\frac{a h}{c}+f\right)\right)-x \left(-2 a c h+b^2 h-b c g+2 c^2 f\right)\right)}{c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{e^2 x \left(-6 a c h+2 b^2 h-b c g+2 c^2 f\right)}{c^2 \left(b^2-4 a c\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(b^2 c e (12 a e h+2 b d h+b e g)-c^3 \left(2 b d (d g+2 e f)-4 a \left(d^2 h+2 d e g+e^2 f\right)\right)-6 a c^2 e (2 a e h+2 b d h+b e g)-2 b^4 e^2 h+4 c^4 d^2 f\right)}{c^3 \left(b^2-4 a c\right)^{3/2}}+\frac{e \log \left(a+b x+c x^2\right) (-2 b e h+2 c d h+c e g)}{2 c^3}",1,"(2*c*e^2*h*x - (2*(b^4*e^2*h*x + b^3*e*(a*e*h - c*(e*g + 2*d*h)*x) + b^2*c*(c*(e^2*f + 2*d*e*g + d^2*h)*x - a*e*(e*g + 2*d*h + 4*e*h*x)) + 2*c^2*(c^2*d^2*f*x - a*c*(e^2*f*x + 2*d*e*(f + g*x) + d^2*(g + h*x)) + a^2*e*(2*d*h + e*(g + h*x))) + b*c*(-3*a^2*e^2*h + c^2*d*(-2*e*f*x + d*(f - g*x)) + a*c*(d^2*h + e^2*(f + 3*g*x) + 2*d*e*(g + 3*h*x)))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(4*c^4*d^2*f - 2*b^4*e^2*h - 6*a*c^2*e*(b*e*g + 2*b*d*h + 2*a*e*h) + b^2*c*e*(b*e*g + 2*b*d*h + 12*a*e*h) + c^3*(-2*b*d*(2*e*f + d*g) + 4*a*(e^2*f + 2*d*e*g + d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e*(c*e*g + 2*c*d*h - 2*b*e*h)*Log[a + x*(b + c*x)])/(2*c^3)","A",1
156,1,225,178,0.4619246,"\int \frac{(d+e x) \left(f+g x+h x^2\right)}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[((d + e*x)*(f + g*x + h*x^2))/(a + b*x + c*x^2)^2,x]","\frac{-\frac{2 \left(2 c \left(a^2 e h-a c (d (g+h x)+e (f+g x))+c^2 d f x\right)+b^2 (c x (d h+e g)-a e h)+b c (a d h+a e (g+3 h x)+c d (f-g x)-c e f x)+b^3 (-e) h x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))}+\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-2 c^2 (b (d g+e f)-2 a (d h+e g))-6 a b c e h+b^3 e h+4 c^3 d f\right)}{\left(4 a c-b^2\right)^{3/2}}+e h \log (a+x (b+c x))}{2 c^2}","\frac{(d+e x) \left(c \left(2 a g-b \left(\frac{a h}{c}+f\right)\right)-x \left(-2 a c h+b^2 h-b c g+2 c^2 f\right)\right)}{c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-2 c^2 (b (d g+e f)-2 a (d h+e g))-6 a b c e h+b^3 e h+4 c^3 d f\right)}{c^2 \left(b^2-4 a c\right)^{3/2}}+\frac{e h \log \left(a+b x+c x^2\right)}{2 c^2}",1,"((-2*(-(b^3*e*h*x) + b^2*(-(a*e*h) + c*(e*g + d*h)*x) + b*c*(a*d*h - c*e*f*x + c*d*(f - g*x) + a*e*(g + 3*h*x)) + 2*c*(a^2*e*h + c^2*d*f*x - a*c*(e*(f + g*x) + d*(g + h*x)))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*(4*c^3*d*f + b^3*e*h - 6*a*b*c*e*h - 2*c^2*(b*(e*f + d*g) - 2*a*(e*g + d*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + e*h*Log[a + x*(b + c*x)])/(2*c^2)","A",1
157,1,114,118,0.105889,"\int \frac{f+g x+h x^2}{\left(a+b x+c x^2\right)^2} \, dx","Integrate[(f + g*x + h*x^2)/(a + b*x + c*x^2)^2,x]","\frac{a b h-2 a c (g+h x)+b^2 h x+b c (f-g x)+2 c^2 f x}{c \left(4 a c-b^2\right) (a+x (b+c x))}-\frac{2 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) (-2 a h+b g-2 c f)}{\left(4 a c-b^2\right)^{3/2}}","\frac{c \left(2 a g-b \left(\frac{a h}{c}+f\right)\right)-x \left(-2 a c h+b^2 h-b c g+2 c^2 f\right)}{c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)}+\frac{2 \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) (2 a h-b g+2 c f)}{\left(b^2-4 a c\right)^{3/2}}",1,"(a*b*h + 2*c^2*f*x + b^2*h*x + b*c*(f - g*x) - 2*a*c*(g + h*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) - (2*(-2*c*f + b*g - 2*a*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)","A",1
158,1,405,407,0.9047251,"\int \frac{f+g x+h x^2}{(d+e x) \left(a+b x+c x^2\right)^2} \, dx","Integrate[(f + g*x + h*x^2)/((d + e*x)*(a + b*x + c*x^2)^2),x]","\frac{-2 a^2 e h+a b (d h+e (g-h x))+2 a c (e (f+g x)-d (g+h x))+b^2 (d h x-e f)+b c (d (f-g x)-e f x)+2 c^2 d f x}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (b d-a e)-c d^2\right)}-\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(2 c e \left(2 a^2 e (d h-e g)+a b \left(d^2 h+d e g+3 e^2 f\right)-2 b^2 d^2 g\right)+b e \left(2 a^2 e^2 h-4 a b d e h+b^2 \left(d^2 h+d e g-e^2 f\right)\right)+2 c^2 d \left(b d (d g+3 e f)-2 a \left(d^2 h-d e g+3 e^2 f\right)\right)-4 c^3 d^3 f\right)}{\left(4 a c-b^2\right)^{3/2} \left(e (a e-b d)+c d^2\right)^2}+\frac{e \log (d+e x) \left(d^2 h-d e g+e^2 f\right)}{\left(e (a e-b d)+c d^2\right)^2}-\frac{e \log (a+x (b+c x)) \left(d^2 h-d e g+e^2 f\right)}{2 \left(e (a e-b d)+c d^2\right)^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 c e \left(2 a^2 e (e g-d h)-a b \left(d^2 h+d e g+3 e^2 f\right)+2 b^2 d^2 g\right)+b e \left(-2 a^2 e^2 h+4 a b d e h+b^2 \left(d^2 (-h)-d e g+e^2 f\right)\right)-2 c^2 d \left(b d (d g+3 e f)-2 a \left(d^2 h-d e g+3 e^2 f\right)\right)+4 c^3 d^3 f\right)}{\left(b^2-4 a c\right)^{3/2} \left(a e^2-b d e+c d^2\right)^2}+\frac{-x \left(-c (2 a d h-2 a e g+b d g+b e f)+b h (b d-a e)+2 c^2 d f\right)-b (a d h+a e g+c d f)-2 a (-a e h-c d g+c e f)+b^2 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)}-\frac{e \log \left(a+b x+c x^2\right) \left(d^2 h-d e g+e^2 f\right)}{2 \left(a e^2-b d e+c d^2\right)^2}+\frac{e \log (d+e x) \left(d^2 h-d e g+e^2 f\right)}{\left(a e^2-b d e+c d^2\right)^2}",1,"(-2*a^2*e*h + 2*c^2*d*f*x + b^2*(-(e*f) + d*h*x) + b*c*(-(e*f*x) + d*(f - g*x)) + a*b*(d*h + e*(g - h*x)) + 2*a*c*(e*(f + g*x) - d*(g + h*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - ((-4*c^3*d^3*f + 2*c^2*d*(b*d*(3*e*f + d*g) - 2*a*(3*e^2*f - d*e*g + d^2*h)) + b*e*(-4*a*b*d*e*h + 2*a^2*e^2*h + b^2*(-(e^2*f) + d*e*g + d^2*h)) + 2*c*e*(-2*b^2*d^2*g + 2*a^2*e*(-(e*g) + d*h) + a*b*(3*e^2*f + d*e*g + d^2*h)))*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) + (e*(e^2*f - d*e*g + d^2*h)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^2 - (e*(e^2*f - d*e*g + d^2*h)*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)","A",1
159,1,650,673,2.205762,"\int \frac{f+g x+h x^2}{(d+e x)^2 \left(a+b x+c x^2\right)^2} \, dx","Integrate[(f + g*x + h*x^2)/((d + e*x)^2*(a + b*x + c*x^2)^2),x]","\frac{b \left(-a^2 e^2 h+a c \left(d^2 (-h)-2 d e (g-h x)+e^2 (3 f+g x)\right)+c^2 d (-d f+d g x+2 e f x)\right)+2 c \left(a^2 (-e) (e (g+h x)-2 d h)+a c \left(d^2 (g+h x)-2 d e (f+g x)+e^2 f x\right)-c^2 d^2 f x\right)+b^2 \left(a e^2 g-c \left(d^2 h x-2 d e f+e^2 f x\right)\right)+b^3 \left(-e^2\right) f}{\left(b^2-4 a c\right) (a+x (b+c x)) \left(e (a e-b d)+c d^2\right)^2}-\frac{\tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(-6 c^2 e \left(2 a^2 e \left(2 d^2 h-2 d e g+e^2 f\right)+4 a b d e^2 f-b^2 d^3 g\right)+c e \left(4 a^3 e^3 h-6 a^2 b e^3 g+6 a b^2 e \left(2 d^2 h-d e g+2 e^2 f\right)+b^3 d \left(-2 d^2 h-3 d e g+4 e^2 f\right)\right)+b^3 e^3 (-2 a d h+a e g+b d g-2 b e f)-2 c^3 d^2 \left(b d (d g+4 e f)-2 a \left(d^2 h-2 d e g+6 e^2 f\right)\right)+4 c^4 d^4 f\right)}{\left(4 a c-b^2\right)^{3/2} \left(e (b d-a e)-c d^2\right)^3}-\frac{e \left(d^2 h-d e g+e^2 f\right)}{(d+e x) \left(e (a e-b d)+c d^2\right)^2}+\frac{\log (d+e x) \left(e^3 (-2 a d h+a e g+b d g-2 b e f)+c d e \left(2 d^2 h-3 d e g+4 e^2 f\right)\right)}{\left(e (a e-b d)+c d^2\right)^3}-\frac{\log (a+x (b+c x)) \left(e^3 (-2 a d h+a e g+b d g-2 b e f)+c d e \left(2 d^2 h-3 d e g+4 e^2 f\right)\right)}{2 \left(e (a e-b d)+c d^2\right)^3}","-\frac{c x \left(2 a^2 e^2 h-c \left(2 a \left(d^2 h-2 d e g+e^2 f\right)+b d (d g+2 e f)\right)-a b e (2 d h+e g)+b^2 \left(d^2 h+e^2 f\right)+2 c^2 d^2 f\right)+b \left(a^2 e^2 h-a c \left(d^2 (-h)-2 d e g+3 e^2 f\right)+c^2 d^2 f\right)-b^2 e (a e g+2 c d f)+2 a c (a e (e g-2 d h)+c d (2 e f-d g))+b^3 e^2 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)^2}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(-6 c^2 e \left(2 a^2 e \left(2 d^2 h-2 d e g+e^2 f\right)+4 a b d e^2 f-b^2 d^3 g\right)-c e \left(-4 a^3 e^3 h+6 a^2 b e^3 g-6 a b^2 e \left(2 d^2 h-d e g+2 e^2 f\right)+b^3 (-d) \left(-2 d^2 h-3 d e g+4 e^2 f\right)\right)-b^3 e^3 (2 a d h-a e g-b d g+2 b e f)-2 c^3 d^2 \left(b d (d g+4 e f)-2 a \left(d^2 h-2 d e g+6 e^2 f\right)\right)+4 c^4 d^4 f\right)}{\left(b^2-4 a c\right)^{3/2} \left(a e^2-b d e+c d^2\right)^3}+\frac{e \log \left(a+b x+c x^2\right) \left(e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left(2 d^2 h-3 d e g+4 e^2 f\right)\right)}{2 \left(a e^2-b d e+c d^2\right)^3}-\frac{e \left(d^2 h-d e g+e^2 f\right)}{(d+e x) \left(a e^2-b d e+c d^2\right)^2}-\frac{e \log (d+e x) \left(e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left(2 d^2 h-3 d e g+4 e^2 f\right)\right)}{\left(a e^2-b d e+c d^2\right)^3}",1,"-((e*(e^2*f - d*e*g + d^2*h))/((c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x))) + (-(b^3*e^2*f) + b^2*(a*e^2*g - c*(-2*d*e*f + e^2*f*x + d^2*h*x)) + b*(-(a^2*e^2*h) + c^2*d*(-(d*f) + 2*e*f*x + d*g*x) + a*c*(-(d^2*h) + e^2*(3*f + g*x) - 2*d*e*(g - h*x))) + 2*c*(-(c^2*d^2*f*x) + a*c*(e^2*f*x - 2*d*e*(f + g*x) + d^2*(g + h*x)) - a^2*e*(-2*d*h + e*(g + h*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) - ((4*c^4*d^4*f + b^3*e^3*(-2*b*e*f + b*d*g + a*e*g - 2*a*d*h) - 2*c^3*d^2*(b*d*(4*e*f + d*g) - 2*a*(6*e^2*f - 2*d*e*g + d^2*h)) - 6*c^2*e*(4*a*b*d*e^2*f - b^2*d^3*g + 2*a^2*e*(e^2*f - 2*d*e*g + 2*d^2*h)) + c*e*(-6*a^2*b*e^3*g + 4*a^3*e^3*h + b^3*d*(4*e^2*f - 3*d*e*g - 2*d^2*h) + 6*a*b^2*e*(2*e^2*f - d*e*g + 2*d^2*h)))*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))^3) + ((e^3*(-2*b*e*f + b*d*g + a*e*g - 2*a*d*h) + c*d*e*(4*e^2*f - 3*d*e*g + 2*d^2*h))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - ((e^3*(-2*b*e*f + b*d*g + a*e*g - 2*a*d*h) + c*d*e*(4*e^2*f - 3*d*e*g + 2*d^2*h))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^3)","A",1
160,1,60,62,0.0359971,"\int \frac{x^3 \left(1+x+x^2\right)}{\left(1-x+x^2\right)^2} \, dx","Integrate[(x^3*(1 + x + x^2))/(1 - x + x^2)^2,x]","\frac{x^2}{2}-\frac{2 (x-2)}{3 \left(x^2-x+1\right)}+2 \log \left(x^2-x+1\right)+3 x-\frac{10 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","\frac{x^2}{2}+\frac{2 (2-x)}{3 \left(x^2-x+1\right)}+2 \log \left(x^2-x+1\right)+3 x+\frac{10 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"3*x + x^2/2 - (2*(-2 + x))/(3*(1 - x + x^2)) - (10*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + 2*Log[1 - x + x^2]","A",1
161,1,55,55,0.0263002,"\int \frac{x^2 \left(1+x+x^2\right)}{\left(1-x+x^2\right)^2} \, dx","Integrate[(x^2*(1 + x + x^2))/(1 - x + x^2)^2,x]","-\frac{2 (2 x-1)}{3 \left(x^2-x+1\right)}+\frac{3}{2} \log \left(x^2-x+1\right)+x+\frac{7 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","\frac{2 (1-2 x)}{3 \left(x^2-x+1\right)}+\frac{3}{2} \log \left(x^2-x+1\right)+x-\frac{7 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"x - (2*(-1 + 2*x))/(3*(1 - x + x^2)) + (7*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (3*Log[1 - x + x^2])/2","A",1
162,1,52,52,0.021202,"\int \frac{x \left(1+x+x^2\right)}{\left(1-x+x^2\right)^2} \, dx","Integrate[(x*(1 + x + x^2))/(1 - x + x^2)^2,x]","-\frac{2 (x+1)}{3 \left(x^2-x+1\right)}+\frac{1}{2} \log \left(x^2-x+1\right)+\frac{11 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{2 (x+1)}{3 \left(x^2-x+1\right)}+\frac{1}{2} \log \left(x^2-x+1\right)-\frac{11 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(-2*(1 + x))/(3*(1 - x + x^2)) + (11*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + Log[1 - x + x^2]/2","A",1
163,1,39,41,0.021841,"\int \frac{1+x+x^2}{\left(1-x+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(1 - x + x^2)^2,x]","\frac{2 (x-2)}{3 \left(x^2-x+1\right)}+\frac{10 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{2 (2-x)}{3 \left(x^2-x+1\right)}-\frac{10 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(2*(-2 + x))/(3*(1 - x + x^2)) + (10*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3])","A",1
164,1,56,56,0.0260829,"\int \frac{1+x+x^2}{x \left(1-x+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x*(1 - x + x^2)^2),x]","\frac{2 (2 x-1)}{3 \left(x^2-x+1\right)}-\frac{1}{2} \log \left(x^2-x+1\right)+\log (x)+\frac{11 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","-\frac{2 (1-2 x)}{3 \left(x^2-x+1\right)}-\frac{1}{2} \log \left(x^2-x+1\right)+\log (x)-\frac{11 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"(2*(-1 + 2*x))/(3*(1 - x + x^2)) + (11*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + Log[x] - Log[1 - x + x^2]/2","A",1
165,1,61,61,0.0234709,"\int \frac{1+x+x^2}{x^2 \left(1-x+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x^2*(1 - x + x^2)^2),x]","\frac{2 (x+1)}{3 \left(x^2-x+1\right)}-\frac{3}{2} \log \left(x^2-x+1\right)-\frac{1}{x}+3 \log (x)+\frac{7 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","\frac{2 (x+1)}{3 \left(x^2-x+1\right)}-\frac{3}{2} \log \left(x^2-x+1\right)-\frac{1}{x}+3 \log (x)-\frac{7 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"-x^(-1) + (2*(1 + x))/(3*(1 - x + x^2)) + (7*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + 3*Log[x] - (3*Log[1 - x + x^2])/2","A",1
166,1,66,68,0.0297499,"\int \frac{1+x+x^2}{x^3 \left(1-x+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(x^3*(1 - x + x^2)^2),x]","-\frac{2 (x-2)}{3 \left(x^2-x+1\right)}-\frac{1}{2 x^2}-2 \log \left(x^2-x+1\right)-\frac{3}{x}+4 \log (x)-\frac{10 \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{3 \sqrt{3}}","\frac{2 (2-x)}{3 \left(x^2-x+1\right)}-\frac{1}{2 x^2}-2 \log \left(x^2-x+1\right)-\frac{3}{x}+4 \log (x)+\frac{10 \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"-1/2*1/x^2 - 3/x - (2*(-2 + x))/(3*(1 - x + x^2)) - (10*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + 4*Log[x] - 2*Log[1 - x + x^2]","A",1
167,1,10,10,0.0059021,"\int \frac{1-x^2}{\left(1+x+x^2\right)^2} \, dx","Integrate[(1 - x^2)/(1 + x + x^2)^2,x]","\frac{x}{x^2+x+1}","\frac{x}{x^2+x+1}",1,"x/(1 + x + x^2)","A",1
168,1,31,31,0.0079613,"\int \frac{1+x^2}{1+x+x^2} \, dx","Integrate[(1 + x^2)/(1 + x + x^2),x]","-\frac{1}{2} \log \left(x^2+x+1\right)+x+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}","-\frac{1}{2} \log \left(x^2+x+1\right)+x+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"x + ArcTan[(1 + 2*x)/Sqrt[3]]/Sqrt[3] - Log[1 + x + x^2]/2","A",1
169,1,23,23,0.0052537,"\int \frac{-1+x^2}{25-6 x+x^2} \, dx","Integrate[(-1 + x^2)/(25 - 6*x + x^2),x]","3 \log \left(x^2-6 x+25\right)+x-2 \tan ^{-1}\left(\frac{x-3}{4}\right)","3 \log \left(x^2-6 x+25\right)+x-2 \tan ^{-1}\left(\frac{x-3}{4}\right)",1,"x - 2*ArcTan[(-3 + x)/4] + 3*Log[25 - 6*x + x^2]","A",1
170,1,19,21,0.0093501,"\int \frac{-10+3 x^2}{4-4 x+x^2} \, dx","Integrate[(-10 + 3*x^2)/(4 - 4*x + x^2),x]","3 (x-2)-\frac{2}{x-2}+12 \log (x-2)","3 x+\frac{2}{2-x}+12 \log (2-x)",1,"-2/(-2 + x) + 3*(-2 + x) + 12*Log[-2 + x]","A",1
171,1,18,18,0.0047129,"\int \frac{8+x^2}{6-5 x+x^2} \, dx","Integrate[(8 + x^2)/(6 - 5*x + x^2),x]","x-12 \log (2-x)+17 \log (3-x)","x-12 \log (2-x)+17 \log (3-x)",1,"x - 12*Log[2 - x] + 17*Log[3 - x]","A",1
172,1,14,14,0.0062868,"\int \frac{-4+3 x+x^2}{-8-2 x+x^2} \, dx","Integrate[(-4 + 3*x + x^2)/(-8 - 2*x + x^2),x]","x+4 \log (4-x)+\log (x+2)","x+4 \log (4-x)+\log (x+2)",1,"x + 4*Log[4 - x] + Log[2 + x]","A",1
173,1,31,27,0.0054034,"\int \frac{7+5 x+4 x^2}{5+4 x+4 x^2} \, dx","Integrate[(7 + 5*x + 4*x^2)/(5 + 4*x + 4*x^2),x]","\frac{1}{8} \log \left(4 x^2+4 x+5\right)+x+\frac{3}{8} \tan ^{-1}\left(\frac{1}{2} (2 x+1)\right)","\frac{1}{8} \log \left(4 x^2+4 x+5\right)+x+\frac{3}{8} \tan ^{-1}\left(x+\frac{1}{2}\right)",1,"x + (3*ArcTan[(1 + 2*x)/2])/8 + Log[5 + 4*x + 4*x^2]/8","A",1
174,1,48,48,0.0753494,"\int \frac{2-x+x^2}{-5+2 x+x^2} \, dx","Integrate[(2 - x + x^2)/(-5 + 2*x + x^2),x]","x+\frac{1}{6} \left(5 \sqrt{6}-9\right) \log \left(-x+\sqrt{6}-1\right)+\frac{1}{6} \left(-9-5 \sqrt{6}\right) \log \left(x+\sqrt{6}+1\right)","x-\frac{1}{6} \left(9-5 \sqrt{6}\right) \log \left(x-\sqrt{6}+1\right)-\frac{1}{6} \left(9+5 \sqrt{6}\right) \log \left(x+\sqrt{6}+1\right)",1,"x + ((-9 + 5*Sqrt[6])*Log[-1 + Sqrt[6] - x])/6 + ((-9 - 5*Sqrt[6])*Log[1 + Sqrt[6] + x])/6","A",1
175,1,21,21,0.0077483,"\int \frac{1+4 x+3 x^2}{\left(4+7 x+2 x^2\right)^2} \, dx","Integrate[(1 + 4*x + 3*x^2)/(4 + 7*x + 2*x^2)^2,x]","\frac{-3 x-2}{2 \left(2 x^2+7 x+4\right)}","-\frac{3 x+2}{2 \left(2 x^2+7 x+4\right)}",1,"(-2 - 3*x)/(2*(4 + 7*x + 2*x^2))","A",1
176,1,39,39,0.0272988,"\int \frac{1+x+x^2}{\left(3+2 x+x^2\right)^2} \, dx","Integrate[(1 + x + x^2)/(3 + 2*x + x^2)^2,x]","\frac{1-x}{4 \left(x^2+2 x+3\right)}+\frac{3 \tan ^{-1}\left(\frac{x+1}{\sqrt{2}}\right)}{4 \sqrt{2}}","\frac{1-x}{4 \left(x^2+2 x+3\right)}+\frac{3 \tan ^{-1}\left(\frac{x+1}{\sqrt{2}}\right)}{4 \sqrt{2}}",1,"(1 - x)/(4*(3 + 2*x + x^2)) + (3*ArcTan[(1 + x)/Sqrt[2]])/(4*Sqrt[2])","A",1
177,1,11,11,0.0062094,"\int \frac{-1+2 x+5 x^2}{\left(1+x+x^2\right)^4} \, dx","Integrate[(-1 + 2*x + 5*x^2)/(1 + x + x^2)^4,x]","-\frac{x}{\left(x^2+x+1\right)^3}","-\frac{x}{\left(x^2+x+1\right)^3}",1,"-(x/(1 + x + x^2)^3)","A",1
178,1,344,267,0.8610372,"\int \left(a+b x+c x^2\right)^{5/2} \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^(5/2)*(A + C*x^2),x]","\frac{-\frac{1120 A \left(b^2-4 a c\right) \left(16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left(b^2-4 a c\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)\right)}{c^{5/2}}+57344 A (b+2 c x) (a+x (b+c x))^{5/2}+\frac{7 C \left(9 b^2-4 a c\right) \left(256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left(b^2-4 a c\right) \left(16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left(b^2-4 a c\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)\right)\right)}{c^{9/2}}-\frac{55296 b C (a+x (b+c x))^{7/2}}{c}+86016 C x (a+x (b+c x))^{7/2}}{688128 c}","-\frac{5 \left(b^2-4 a c\right)^3 \left(-4 a c C+32 A c^2+9 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{32768 c^{11/2}}+\frac{5 \left(b^2-4 a c\right)^2 (b+2 c x) \sqrt{a+b x+c x^2} \left(-4 a c C+32 A c^2+9 b^2 C\right)}{16384 c^5}-\frac{5 \left(b^2-4 a c\right) (b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-4 a c C+32 A c^2+9 b^2 C\right)}{6144 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{5/2} \left(-4 a c C+32 A c^2+9 b^2 C\right)}{384 c^3}-\frac{9 b C \left(a+b x+c x^2\right)^{7/2}}{112 c^2}+\frac{C x \left(a+b x+c x^2\right)^{7/2}}{8 c}",1,"(57344*A*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - (55296*b*C*(a + x*(b + c*x))^(7/2))/c + 86016*C*x*(a + x*(b + c*x))^(7/2) - (1120*A*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 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) + (7*(9*b^2 - 4*a*c)*C*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 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^(9/2))/(688128*c)","A",1
179,1,267,212,0.5808161,"\int \left(a+b x+c x^2\right)^{3/2} \left(A+C x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^(3/2)*(A + C*x^2),x]","\frac{\frac{360 A \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^{3/2}}+1920 A (b+2 c x) (a+x (b+c x))^{3/2}+\frac{C \left(5 \left(7 b^2-4 a c\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 b (a+x (b+c x))^{5/2}\right)}{c}+2560 C x (a+x (b+c x))^{5/2}}{15360 c}","\frac{\left(b^2-4 a c\right)^2 \left(-4 a c C+24 A c^2+7 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{1024 c^{9/2}}-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} \left(-4 a c C+24 A c^2+7 b^2 C\right)}{512 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-4 a c C+24 A c^2+7 b^2 C\right)}{192 c^3}-\frac{7 b C \left(a+b x+c x^2\right)^{5/2}}{60 c^2}+\frac{C x \left(a+b x+c x^2\right)^{5/2}}{6 c}",1,"(1920*A*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 2560*C*x*(a + x*(b + c*x))^(5/2) + (360*A*(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^(3/2) + (C*(-1792*b*(a + x*(b + c*x))^(5/2) + 5*(7*b^2 - 4*a*c)*((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)/(15360*c)","A",1
180,1,144,157,0.2081919,"\int \sqrt{a+b x+c x^2} \left(A+C x^2\right) \, dx","Integrate[Sqrt[a + b*x + c*x^2]*(A + C*x^2),x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left(C \left(b \left(8 c^2 x^2-52 a c\right)+24 c^2 x \left(a+2 c x^2\right)+15 b^3-10 b^2 c x\right)+48 A c^2 (b+2 c x)\right)-3 \left(b^2-4 a c\right) \left(-4 a c C+16 A c^2+5 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{384 c^{7/2}}","-\frac{\left(b^2-4 a c\right) \left(-4 a c C+16 A c^2+5 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left(-4 a c C+16 A c^2+5 b^2 C\right)}{64 c^3}-\frac{5 b C \left(a+b x+c x^2\right)^{3/2}}{24 c^2}+\frac{C x \left(a+b x+c x^2\right)^{3/2}}{4 c}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(48*A*c^2*(b + 2*c*x) + C*(15*b^3 - 10*b^2*c*x + 24*c^2*x*(a + 2*c*x^2) + b*(-52*a*c + 8*c^2*x^2))) - 3*(b^2 - 4*a*c)*(16*A*c^2 + 5*b^2*C - 4*a*c*C)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(384*c^(7/2))","A",1
181,1,86,104,0.1456445,"\int \frac{A+C x^2}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(A + C*x^2)/Sqrt[a + b*x + c*x^2],x]","\frac{\left(-4 a c C+8 A c^2+3 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{8 c^{5/2}}+\frac{C (2 c x-3 b) \sqrt{a+x (b+c x)}}{4 c^2}","\frac{\left(-4 a c C+8 A c^2+3 b^2 C\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{8 c^{5/2}}-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c}",1,"(C*(-3*b + 2*c*x)*Sqrt[a + x*(b + c*x)])/(4*c^2) + ((8*A*c^2 + 3*b^2*C - 4*a*c*C)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(5/2))","A",1
182,1,104,98,0.7275404,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(a C (b-2 c x)+A c (b+2 c x)+b^2 C x\right)}{\sqrt{a+x (b+c x)}}-C \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)}{c^{3/2} \left(4 a c-b^2\right)}","\frac{C \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2}}-\frac{2 \left(x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}",1,"((2*Sqrt[c]*(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x)))/Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*C*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(c^(3/2)*(-b^2 + 4*a*c))","A",1
183,1,107,114,0.8943334,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(5/2),x]","\frac{2 C \left(8 a^2 b+4 a x \left(3 b^2+3 b c x+2 c^2 x^2\right)+b^2 x^2 (3 b+2 c x)\right)-2 A (b+2 c x) \left(-4 c \left(3 a+2 c x^2\right)+b^2-8 b c x\right)}{3 \left(b^2-4 a c\right)^2 (a+x (b+c x))^{3/2}}","\frac{2 (b+2 c x) \left(4 a C+8 A c+\frac{b^2 C}{c}\right)}{3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}-\frac{2 \left(x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)\right)}{3 c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}",1,"(-2*A*(b + 2*c*x)*(b^2 - 8*b*c*x - 4*c*(3*a + 2*c*x^2)) + 2*C*(8*a^2*b + b^2*x^2*(3*b + 2*c*x) + 4*a*x*(3*b^2 + 3*b*c*x + 2*c^2*x^2)))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))","A",1
184,1,148,167,1.9331349,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^{7/2}} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(7/2),x]","\frac{2 \left(\left(b^2-4 a c\right) (b+2 c x) (a+x (b+c x)) \left(4 a c C+16 A c^2+3 b^2 C\right)-8 c (b+2 c x) (a+x (b+c x))^2 \left(4 a c C+16 A c^2+3 b^2 C\right)-3 \left(b^2-4 a c\right)^2 \left(a C (b-2 c x)+A c (b+2 c x)+b^2 C x\right)\right)}{15 c \left(b^2-4 a c\right)^3 (a+x (b+c x))^{5/2}}","-\frac{16 (b+2 c x) \left(4 a c C+16 A c^2+3 b^2 C\right)}{15 \left(b^2-4 a c\right)^3 \sqrt{a+b x+c x^2}}-\frac{2 \left(x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)\right)}{5 c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{5/2}}+\frac{2 (b+2 c x) \left(4 a C+16 A c+\frac{3 b^2 C}{c}\right)}{15 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{3/2}}",1,"(2*((b^2 - 4*a*c)*(16*A*c^2 + 3*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b + c*x)) - 8*c*(16*A*c^2 + 3*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b + c*x))^2 - 3*(b^2 - 4*a*c)^2*(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))))/(15*c*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))","A",1
185,1,199,220,1.7377106,"\int \frac{A+C x^2}{\left(a+b x+c x^2\right)^{9/2}} \, dx","Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(9/2),x]","\frac{2 \left(3 \left(b^2-4 a c\right)^2 (b+2 c x) (a+x (b+c x)) \left(4 a c C+24 A c^2+5 b^2 C\right)-16 c \left(b^2-4 a c\right) (b+2 c x) (a+x (b+c x))^2 \left(4 a c C+24 A c^2+5 b^2 C\right)+128 c^2 (b+2 c x) (a+x (b+c x))^3 \left(4 a c C+24 A c^2+5 b^2 C\right)-15 \left(b^2-4 a c\right)^3 \left(a C (b-2 c x)+A c (b+2 c x)+b^2 C x\right)\right)}{105 c \left(b^2-4 a c\right)^4 (a+x (b+c x))^{7/2}}","\frac{256 c (b+2 c x) \left(4 a c C+24 A c^2+5 b^2 C\right)}{105 \left(b^2-4 a c\right)^4 \sqrt{a+b x+c x^2}}-\frac{32 (b+2 c x) \left(4 a c C+24 A c^2+5 b^2 C\right)}{105 \left(b^2-4 a c\right)^3 \left(a+b x+c x^2\right)^{3/2}}-\frac{2 \left(x \left(C \left(b^2-2 a c\right)+2 A c^2\right)+b c \left(\frac{a C}{c}+A\right)\right)}{7 c \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{7/2}}+\frac{2 (b+2 c x) \left(4 a C+24 A c+\frac{5 b^2 C}{c}\right)}{35 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)^{5/2}}",1,"(2*(3*(b^2 - 4*a*c)^2*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b + c*x)) - 16*c*(b^2 - 4*a*c)*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b + c*x))^2 + 128*c^2*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b + c*x))^3 - 15*(b^2 - 4*a*c)^3*(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))))/(105*c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))^(7/2))","A",1
186,1,1093,930,2.418003,"\int (g+h x)^3 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^3*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left(-3465 f h^3 b^6+210 c h^2 (63 f g+21 e h+11 f h x) b^5-84 c h \left(-260 a f h^2+35 c (6 e g+2 d h+e h x) h+c f \left(210 g^2+105 h x g+22 h^2 x^2\right)\right) b^4+16 c^2 \left(c \left(f \left(525 g^3+735 h x g^2+441 h^2 x^2 g+99 h^3 x^3\right)+7 h \left(5 d h (45 g+7 h x)+3 e \left(75 g^2+35 h x g+7 h^2 x^2\right)\right)\right)-42 a h^2 (35 e h+3 f (35 g+6 h x))\right) b^3-16 c^2 \left(2163 a^2 f h^3-2 a c \left(7 h (345 e g+115 d h+56 e h x)+3 f \left(805 g^2+392 h x g+81 h^2 x^2\right)\right) h+2 c^2 \left(7 d h \left(180 g^2+75 h x g+14 h^2 x^2\right)+21 e \left(20 g^3+25 h x g^2+14 h^2 x^2 g+3 h^3 x^3\right)+f x \left(175 g^3+294 h x g^2+189 h^2 x^2 g+44 h^3 x^3\right)\right)\right) b^2+32 c^3 \left(4 \left(21 d \left(10 g^3+10 h x g^2+5 h^2 x^2 g+h^3 x^3\right)+x \left(7 e \left(10 g^3+15 h x g^2+9 h^2 x^2 g+2 h^3 x^3\right)+f x \left(35 g^3+63 h x g^2+42 h^2 x^2 g+10 h^3 x^3\right)\right)\right) c^2-2 a \left(f \left(455 g^3+609 h x g^2+357 h^2 x^2 g+79 h^3 x^3\right)+7 h \left(d h (195 g+29 h x)+e \left(195 g^2+87 h x g+17 h^2 x^2\right)\right)\right) c+a^2 h^2 (2373 f g+791 e h+397 f h x)\right) b+64 c^3 \left(4 x \left(21 d \left(10 g^3+20 h x g^2+15 h^2 x^2 g+4 h^3 x^3\right)+x \left(7 e \left(20 g^3+45 h x g^2+36 h^2 x^2 g+10 h^3 x^3\right)+3 f x \left(35 g^3+84 h x g^2+70 h^2 x^2 g+20 h^3 x^3\right)\right)\right) c^3+2 a \left(7 d h \left(120 g^2+45 h x g+8 h^2 x^2\right)+7 e \left(40 g^3+45 h x g^2+24 h^2 x^2 g+5 h^3 x^3\right)+3 f x \left(35 g^3+56 h x g^2+35 h^2 x^2 g+8 h^3 x^3\right)\right) c^2-a^2 h \left(7 h (96 e g+32 d h+15 e h x)+f \left(672 g^2+315 h x g+64 h^2 x^2\right)\right) c+128 a^3 f h^3\right)\right)+105 \left(b^2-4 a c\right) \left(33 f h^3 b^5-6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3+8 c^2 h \left(7 \left(3 f g^2+3 e h g+d h^2\right) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right) b-256 c^5 d g^3+64 c^4 g \left(a f g^2+2 b (e g+3 d h) g+3 a h (e g+d h)\right)-16 c^3 \left(5 g \left(f g^2+3 h (e g+d h)\right) b^2+6 a h \left(3 f g^2+h (3 e g+d h)\right) b+2 a^2 h^2 (3 f g+e h)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{215040 c^{13/2}}","\frac{f \left(c x^2+b x+a\right)^{3/2} (g+h x)^4}{7 c h}-\frac{(6 c f g-14 c e h+11 b f h) \left(c x^2+b x+a\right)^{3/2} (g+h x)^3}{84 c^2 h}+\frac{\left(-4 \left(3 f g^2-7 h (e g+2 d h)\right) c^2-2 h (8 b f g+21 b e h+16 a f h) c+33 b^2 f h^2\right) \left(c x^2+b x+a\right)^{3/2} (g+h x)^2}{280 c^3 h}+\frac{\left(-128 g^2 \left(3 f g^2-7 h (e g+12 d h)\right) c^4-16 h \left(16 a h \left(15 f g^2+7 h (3 e g+d h)\right)+b g \left(17 f g^2+21 h (19 e g+25 d h)\right)\right) c^3+8 h^2 \left(\left(537 f g^2+245 h (3 e g+d h)\right) b^2+343 a h (3 f g+e h) b+128 a^2 f h^2\right) c^2-42 b^2 h^3 (78 a f h+35 b (3 f g+e h)) c-6 h \left(16 g \left(3 f g^2-7 h (e g+7 d h)\right) c^3+8 h \left(a h (41 f g+35 e h)+b \left(5 f g^2+7 h (9 e g+7 d h)\right)\right) c^2-6 b h^2 (59 b f g+49 b e h+74 a f h) c+231 b^3 f h^3\right) x c+1155 b^4 f h^4\right) \left(c x^2+b x+a\right)^{3/2}}{13440 c^5 h}-\frac{\left(b^2-4 a c\right) \left(-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left(7 \left(3 f g^2+3 e h g+d h^2\right) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right) b+256 c^5 d g^3-64 c^4 g \left(2 b g (e g+3 d h)+a \left(f g^2+3 h (e g+d h)\right)\right)+16 c^3 \left(5 g \left(f g^2+3 h (e g+d h)\right) b^2+6 a h \left(3 f g^2+h (3 e g+d h)\right) b+2 a^2 h^2 (3 f g+e h)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{2048 c^{13/2}}+\frac{\left(-33 f h^3 b^5+6 c h^2 (20 a f h+7 b (3 f g+e h)) b^3-8 c^2 h \left(7 \left(3 f g^2+3 e h g+d h^2\right) b^2+14 a h (3 f g+e h) b+10 a^2 f h^2\right) b+256 c^5 d g^3-64 c^4 g \left(2 b g (e g+3 d h)+a \left(f g^2+3 h (e g+d h)\right)\right)+16 c^3 \left(5 g \left(f g^2+3 h (e g+d h)\right) b^2+6 a h \left(3 f g^2+h (3 e g+d h)\right) b+2 a^2 h^2 (3 f g+e h)\right)\right) (b+2 c x) \sqrt{c x^2+b x+a}}{1024 c^6}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^6*f*h^3 + 210*b^5*c*h^2*(63*f*g + 21*e*h + 11*f*h*x) - 84*b^4*c*h*(-260*a*f*h^2 + 35*c*h*(6*e*g + 2*d*h + e*h*x) + c*f*(210*g^2 + 105*g*h*x + 22*h^2*x^2)) - 16*b^2*c^2*(2163*a^2*f*h^3 - 2*a*c*h*(7*h*(345*e*g + 115*d*h + 56*e*h*x) + 3*f*(805*g^2 + 392*g*h*x + 81*h^2*x^2)) + 2*c^2*(7*d*h*(180*g^2 + 75*g*h*x + 14*h^2*x^2) + 21*e*(20*g^3 + 25*g^2*h*x + 14*g*h^2*x^2 + 3*h^3*x^3) + f*x*(175*g^3 + 294*g^2*h*x + 189*g*h^2*x^2 + 44*h^3*x^3))) + 16*b^3*c^2*(-42*a*h^2*(35*e*h + 3*f*(35*g + 6*h*x)) + c*(f*(525*g^3 + 735*g^2*h*x + 441*g*h^2*x^2 + 99*h^3*x^3) + 7*h*(5*d*h*(45*g + 7*h*x) + 3*e*(75*g^2 + 35*g*h*x + 7*h^2*x^2)))) + 32*b*c^3*(a^2*h^2*(2373*f*g + 791*e*h + 397*f*h*x) - 2*a*c*(f*(455*g^3 + 609*g^2*h*x + 357*g*h^2*x^2 + 79*h^3*x^3) + 7*h*(d*h*(195*g + 29*h*x) + e*(195*g^2 + 87*g*h*x + 17*h^2*x^2))) + 4*c^2*(21*d*(10*g^3 + 10*g^2*h*x + 5*g*h^2*x^2 + h^3*x^3) + x*(7*e*(10*g^3 + 15*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3) + f*x*(35*g^3 + 63*g^2*h*x + 42*g*h^2*x^2 + 10*h^3*x^3)))) + 64*c^3*(128*a^3*f*h^3 - a^2*c*h*(7*h*(96*e*g + 32*d*h + 15*e*h*x) + f*(672*g^2 + 315*g*h*x + 64*h^2*x^2)) + 2*a*c^2*(7*d*h*(120*g^2 + 45*g*h*x + 8*h^2*x^2) + 7*e*(40*g^3 + 45*g^2*h*x + 24*g*h^2*x^2 + 5*h^3*x^3) + 3*f*x*(35*g^3 + 56*g^2*h*x + 35*g*h^2*x^2 + 8*h^3*x^3)) + 4*c^3*x*(21*d*(10*g^3 + 20*g^2*h*x + 15*g*h^2*x^2 + 4*h^3*x^3) + x*(7*e*(20*g^3 + 45*g^2*h*x + 36*g*h^2*x^2 + 10*h^3*x^3) + 3*f*x*(35*g^3 + 84*g^2*h*x + 70*g*h^2*x^2 + 20*h^3*x^3))))) + 105*(b^2 - 4*a*c)*(-256*c^5*d*g^3 + 33*b^5*f*h^3 + 64*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 2*b*g*(e*g + 3*d*h)) - 6*b^3*c*h^2*(20*a*f*h + 7*b*(3*f*g + e*h)) + 8*b*c^2*h*(10*a^2*f*h^2 + 14*a*b*h*(3*f*g + e*h) + 7*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) - 16*c^3*(2*a^2*h^2*(3*f*g + e*h) + 5*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 6*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(215040*c^(13/2))","A",1
187,1,436,584,0.9658388,"\int (g+h x)^2 \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^2*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]","\frac{\frac{3 h \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) \left(8 c^2 \left(2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left(h (d h+2 e g)+f g^2\right)\right)-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left(a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right)+21 b^4 f h^2+128 c^4 d g^2\right)}{512 c^{9/2}}-\frac{(a+x (b+c x))^{3/2} \left(8 c^2 h (a h (16 e h+32 f g+15 f h x)+b h (25 d h+50 e g+21 e h x)+b f g (7 g+6 h x))-14 b c h^2 (14 a f h+b (10 e h+20 f g+9 f h x))+105 b^3 f h^3+16 c^3 \left(f g^2 (4 g+3 h x)-h (5 d h (8 g+3 h x)+2 e g (4 g+3 h x))\right)\right)}{160 c^3}-\frac{3 (g+h x)^2 (a+x (b+c x))^{3/2} (3 b f h-4 c e h+2 c f g)}{10 c}+f (g+h x)^3 (a+x (b+c x))^{3/2}}{6 c h}","-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(8 c^2 \left(2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left(d h^2+2 e g h+f g^2\right)\right)-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+2 b g (2 d h+e g)\right)+21 b^4 f h^2+128 c^4 d g^2\right)}{1024 c^{11/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left(8 c^2 \left(2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left(d h^2+2 e g h+f g^2\right)\right)-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+2 b g (2 d h+e g)\right)+21 b^4 f h^2+128 c^4 d g^2\right)}{512 c^5}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(-6 c h x \left(-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left(f g^2-h (5 d h+2 e g)\right)\right)+8 c^2 h \left(16 a h (e h+2 f g)+b \left(25 h (d h+2 e g)+7 f g^2\right)\right)-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 g \left(f g^2-2 h (5 d h+e g)\right)\right)}{960 c^4 h}-\frac{(g+h x)^2 \left(a+b x+c x^2\right)^{3/2} (3 b f h-4 c e h+2 c f g)}{20 c^2 h}+\frac{f (g+h x)^3 \left(a+b x+c x^2\right)^{3/2}}{6 c h}",1,"((-3*(2*c*f*g - 4*c*e*h + 3*b*f*h)*(g + h*x)^2*(a + x*(b + c*x))^(3/2))/(10*c) + f*(g + h*x)^3*(a + x*(b + c*x))^(3/2) - ((a + x*(b + c*x))^(3/2)*(105*b^3*f*h^3 - 14*b*c*h^2*(14*a*f*h + b*(20*f*g + 10*e*h + 9*f*h*x)) + 8*c^2*h*(b*f*g*(7*g + 6*h*x) + b*h*(50*e*g + 25*d*h + 21*e*h*x) + a*h*(32*f*g + 16*e*h + 15*f*h*x)) + 16*c^3*(f*g^2*(4*g + 3*h*x) - h*(2*e*g*(4*g + 3*h*x) + 5*d*h*(8*g + 3*h*x)))))/(160*c^3) + (3*h*(128*c^4*d*g^2 + 21*b^4*f*h^2 - 28*b^2*c*h*(2*b*f*g + b*e*h + 2*a*f*h) - 32*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 2*b*g*(e*g + 2*d*h)) + 8*c^2*(2*a^2*f*h^2 + 6*a*b*h*(2*f*g + e*h) + 5*b^2*(f*g^2 + h*(2*e*g + d*h))))*(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)])]))/(512*c^(9/2)))/(6*c*h)","A",1
188,1,258,322,0.475879,"\int (g+h x) \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]","\frac{\frac{(a+x (b+c x))^{3/2} \left(-2 c h (16 a f h+b (25 e h+25 f g+21 f h x))+35 b^2 f h^2+c^2 (20 h (4 d h+4 e g+3 e h x)-12 f g (4 g+3 h x))\right)}{48 c^2}-\frac{5 h \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) \left(8 c^2 (a e h+a f g+2 b d h+2 b e g)-2 b c (6 a f h+5 b (e h+f g))+7 b^3 f h-32 c^3 d g\right)}{256 c^{7/2}}+f (g+h x)^2 (a+x (b+c x))^{3/2}}{5 c h}","\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left(-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right)}{128 c^4}+\frac{\left(a+b x+c x^2\right)^{3/2} \left(-2 c h (16 a f h+25 b (e h+f g))+35 b^2 f h^2-6 c h x (7 b f h-10 c e h+6 c f g)-16 c^2 \left(3 f g^2-5 h (d h+e g)\right)\right)}{240 c^3 h}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-8 c^2 (a e h+a f g+2 b d h+2 b e g)+2 b c (6 a f h+5 b (e h+f g))-7 b^3 f h+32 c^3 d g\right)}{256 c^{9/2}}+\frac{f (g+h x)^2 \left(a+b x+c x^2\right)^{3/2}}{5 c h}",1,"(f*(g + h*x)^2*(a + x*(b + c*x))^(3/2) + ((a + x*(b + c*x))^(3/2)*(35*b^2*f*h^2 + c^2*(-12*f*g*(4*g + 3*h*x) + 20*h*(4*e*g + 4*d*h + 3*e*h*x)) - 2*c*h*(16*a*f*h + b*(25*f*g + 25*e*h + 21*f*h*x))))/(48*c^2) - (5*h*(-32*c^3*d*g + 7*b^3*f*h + 8*c^2*(2*b*e*g + a*f*g + 2*b*d*h + a*e*h) - 2*b*c*(6*a*f*h + 5*b*(f*g + e*h)))*(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)])]))/(256*c^(7/2)))/(5*c*h)","A",1
189,1,173,175,0.2983928,"\int \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]","\frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left(4 b c \left(2 c \left(6 d+2 e x+f x^2\right)-13 a f\right)+8 c^2 \left(a (8 e+3 f x)+2 c x \left(6 d+4 e x+3 f x^2\right)\right)+15 b^3 f-2 b^2 c (12 e+5 f x)\right)-3 \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) \left(-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right)}{384 c^{7/2}}","-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right)}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left(-4 a c f+5 b^2 f-8 b c e+16 c^2 d\right)}{64 c^3}+\frac{\left(a+b x+c x^2\right)^{3/2} (8 c e-5 b f)}{24 c^2}+\frac{f x \left(a+b x+c x^2\right)^{3/2}}{4 c}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^3*f - 2*b^2*c*(12*e + 5*f*x) + 4*b*c*(-13*a*f + 2*c*(6*d + 2*e*x + f*x^2)) + 8*c^2*(a*(8*e + 3*f*x) + 2*c*x*(6*d + 4*e*x + 3*f*x^2))) - 3*(b^2 - 4*a*c)*(16*c^2*d + 5*b^2*f - 4*c*(2*b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(384*c^(7/2))","A",1
190,1,331,321,0.7862612,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{g+h x} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x),x]","\frac{2 \sqrt{c} \left(h \sqrt{a+x (b+c x)} \left(2 c h (4 a f h+b (3 e h-3 f g+f h x))-3 b^2 f h^2+4 c^2 \left(3 h (2 d h-2 e g+e h x)+f \left(6 g^2-3 g h x+2 h^2 x^2\right)\right)\right)-24 c^2 \sqrt{h (a h-b g)+c g^2} \left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)\right)-3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-8 c^2 h \left(a h (e h-f g)+b h (d h-e g)+b f g^2\right)+2 b c h^2 (2 a f h+b e h-b f g)-b^3 f h^3+16 c^3 g \left(h (d h-e g)+f g^2\right)\right)}{48 c^{5/2} h^4}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c h (2 c g-b h) (b f g-2 c d h)-\left(-4 c h (b g-a h)-b^2 h^2+8 c^2 g^2\right) (b f h-2 c e h+2 c f g)\right)}{16 c^{5/2} h^4}-\frac{\sqrt{a+b x+c x^2} (4 c h (b f g-2 c d h)+2 c h x (b f h-2 c e h+2 c f g)-(4 c g-b h) (b f h-2 c e h+2 c f g))}{8 c^2 h^3}+\frac{\sqrt{a h^2-b g h+c g^2} \left(d h^2-e g h+f g^2\right) \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right)}{h^4}+\frac{f \left(a+b x+c x^2\right)^{3/2}}{3 c h}",1,"(-3*(-(b^3*f*h^3) + 2*b*c*h^2*(-(b*f*g) + b*e*h + 2*a*f*h) + 16*c^3*g*(f*g^2 + h*(-(e*g) + d*h)) - 8*c^2*h*(b*f*g^2 + b*h*(-(e*g) + d*h) + a*h*(-(f*g) + e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(h*Sqrt[a + x*(b + c*x)]*(-3*b^2*f*h^2 + 2*c*h*(4*a*f*h + b*(-3*f*g + 3*e*h + f*h*x)) + 4*c^2*(3*h*(-2*e*g + 2*d*h + e*h*x) + f*(6*g^2 - 3*g*h*x + 2*h^2*x^2))) - 24*c^2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])]))/(48*c^(5/2)*h^4)","A",1
191,1,486,459,1.5609033,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^2} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^2,x]","\frac{-\frac{\frac{\left(h (a h-b g)+c g^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(4 c h (a f h+b e h-2 b f g)-b^2 f h^2+8 c^2 \left(h (d h-2 e g)+3 f g^2\right)\right)}{\sqrt{c}}+2 h \sqrt{a+x (b+c x)} \left(c h (2 a h (2 e h-4 f g+f h x)+4 b h (d h-2 e g)+b f g (13 g-2 h x))+b f h^2 (a h-b g)+c^2 \left(4 h (d h (h x-g)+e g (2 g-h x))+6 f g^2 (h x-2 g)\right)\right)+4 c \sqrt{h (a h-b g)+c g^2} \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(2 c \left(g h (d h-2 e g)+3 f g^3\right)-h \left(2 a h (e h-2 f g)+b h (d h-3 e g)+5 b f g^2\right)\right)}{4 h^3 \left(h (b g-a h)-c g^2\right)}-\frac{(a+x (b+c x))^{3/2} \left(f h (a h-b g)+2 c h (d h-e g)+3 c f g^2\right)}{(g+h x) \left(h (a h-b g)+c g^2\right)}+\frac{f (a+x (b+c x))^{3/2}}{g+h x}}{2 c h}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c h (-a f h-b e h+2 b f g)+b^2 f h^2-8 c^2 \left(3 f g^2-h (2 e g-d h)\right)\right)}{8 c^{3/2} h^4}-\frac{\sqrt{a+b x+c x^2} \left(2 c h^2 x \left(-a f h+b f g-2 c d h+2 c e g-\frac{3 c f g^2}{h}\right)+c h \left(4 a h (2 f g-e h)-b \left(4 d h^2-8 e g h+13 f g^2\right)\right)+b f h^2 (b g-a h)+4 c^2 g \left(3 f g^2-h (2 e g-d h)\right)\right)}{4 c h^3 \left(a h^2-b g h+c g^2\right)}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(f g^2-h (e g-d h)\right)}{h (g+h x) \left(a h^2-b g h+c g^2\right)}-\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(h \left(2 a h (2 f g-e h)-b \left(d h^2-3 e g h+5 f g^2\right)\right)+2 c g \left(3 f g^2-h (2 e g-d h)\right)\right)}{2 h^4 \sqrt{a h^2-b g h+c g^2}}",1,"((f*(a + x*(b + c*x))^(3/2))/(g + h*x) - ((3*c*f*g^2 + f*h*(-(b*g) + a*h) + 2*c*h*(-(e*g) + d*h))*(a + x*(b + c*x))^(3/2))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)) - (2*h*Sqrt[a + x*(b + c*x)]*(b*f*h^2*(-(b*g) + a*h) + c*h*(4*b*h*(-2*e*g + d*h) + b*f*g*(13*g - 2*h*x) + 2*a*h*(-4*f*g + 2*e*h + f*h*x)) + c^2*(6*f*g^2*(-2*g + h*x) + 4*h*(e*g*(2*g - h*x) + d*h*(-g + h*x)))) + ((c*g^2 + h*(-(b*g) + a*h))*(-(b^2*f*h^2) + 4*c*h*(-2*b*f*g + b*e*h + a*f*h) + 8*c^2*(3*f*g^2 + h*(-2*e*g + d*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 4*c*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*(2*c*(3*f*g^3 + g*h*(-2*e*g + d*h)) - h*(5*b*f*g^2 + b*h*(-3*e*g + d*h) + 2*a*h*(-2*f*g + e*h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(4*h^3*(-(c*g^2) + h*(b*g - a*h))))/(2*c*h)","A",1
192,1,645,448,3.5836701,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^3} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^3,x]","\frac{-\frac{\frac{2 c \sqrt{a+x (b+c x)} \left(h^2 \left(-4 a^2 f h^2-4 a b h (e h-4 f g)+b^2 \left(d h^2+3 e g h-11 f g^2\right)\right)+c h \left(b \left(h (d h (h x-g)+e g (3 h x-7 g))+f g^2 (23 g-7 h x)\right)-2 a h (h (d h-3 e g+2 e h x)+f g (9 g-4 h x))\right)-2 c^2 \left(g h \left(d h^2 x+e g (h x-2 g)\right)+3 f g^3 (2 g-h x)\right)\right)}{h^2}+\frac{c \sqrt{h (a h-b g)+c g^2} \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(h^2 \left(8 a^2 f h^2+4 a b h (e h-6 f g)+b^2 \left(15 f g^2-h (d h+3 e g)\right)\right)+4 c h \left(a h \left(d h^2-3 e g h+9 f g^2\right)+b g^2 (3 e h-10 f g)\right)+8 c^2 g^3 (3 f g-e h)\right)+4 \sqrt{c} \left(h (a h-b g)+c g^2\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) (-b f h-2 c e h+6 c f g)}{h^3}-\frac{2 c (a+x (b+c x))^{3/2} \left(-4 a h^2 (e h-2 f g)+b h \left(h (d h+3 e g)-7 f g^2\right)-2 c g h (d h+e g)+6 c f g^3\right)}{g+h x}}{8 \left(h (a h-b g)+c g^2\right)^2}-\frac{(a+x (b+c x))^{3/2} \left(2 f h (a h-b g)+c h (d h-e g)+3 c f g^2\right)}{2 (g+h x)^2 \left(h (a h-b g)+c g^2\right)}+\frac{f (a+x (b+c x))^{3/2}}{(g+h x)^2}}{c h}","\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(h^2 \left(8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left(15 f g^2-h (d h+3 e g)\right)\right)-4 c h \left(b g^2 (10 f g-3 e h)-a h \left(d h^2-3 e g h+9 f g^2\right)\right)+8 c^2 g^3 (3 f g-e h)\right)}{8 h^4 \left(a h^2-b g h+c g^2\right)^{3/2}}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(f g^2-h (e g-d h)\right)}{2 h (g+h x)^2 \left(a h^2-b g h+c g^2\right)}+\frac{\sqrt{a+b x+c x^2} \left(-2 h x \left(-2 a f h+2 b f g-c d h+c e g-\frac{3 c f g^2}{h}\right)+4 a h (3 f g-e h)-b \left(-d h^2-3 e g h+11 f g^2\right)+\frac{4 c g^2 (3 f g-e h)}{h}\right)}{4 h^2 (g+h x) \left(a h^2-b g h+c g^2\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (-b f h-2 c e h+6 c f g)}{2 \sqrt{c} h^4}",1,"((f*(a + x*(b + c*x))^(3/2))/(g + h*x)^2 - ((3*c*f*g^2 + 2*f*h*(-(b*g) + a*h) + c*h*(-(e*g) + d*h))*(a + x*(b + c*x))^(3/2))/(2*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) - ((-2*c*(6*c*f*g^3 - 2*c*g*h*(e*g + d*h) - 4*a*h^2*(-2*f*g + e*h) + b*h*(-7*f*g^2 + h*(3*e*g + d*h)))*(a + x*(b + c*x))^(3/2))/(g + h*x) + (2*c*Sqrt[a + x*(b + c*x)]*(h^2*(-4*a^2*f*h^2 - 4*a*b*h*(-4*f*g + e*h) + b^2*(-11*f*g^2 + 3*e*g*h + d*h^2)) - 2*c^2*(3*f*g^3*(2*g - h*x) + g*h*(d*h^2*x + e*g*(-2*g + h*x))) + c*h*(-2*a*h*(f*g*(9*g - 4*h*x) + h*(-3*e*g + d*h + 2*e*h*x)) + b*(f*g^2*(23*g - 7*h*x) + h*(d*h*(-g + h*x) + e*g*(-7*g + 3*h*x))))))/h^2 + (4*Sqrt[c]*(6*c*f*g - 2*c*e*h - b*f*h)*(c*g^2 + h*(-(b*g) + a*h))^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + c*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*(8*c^2*g^3*(3*f*g - e*h) + 4*c*h*(b*g^2*(-10*f*g + 3*e*h) + a*h*(9*f*g^2 - 3*e*g*h + d*h^2)) + h^2*(8*a^2*f*h^2 + 4*a*b*h*(-6*f*g + e*h) + b^2*(15*f*g^2 - h*(3*e*g + d*h))))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/h^3)/(8*(c*g^2 + h*(-(b*g) + a*h))^2))/(c*h)","A",1
193,1,439,603,1.9250768,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^4} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]","\frac{\frac{\left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{8 \left(h (a h-b g)+c g^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a h+b (g-h x)+2 c g x)}{4 (g+h x)^2 \left(h (a h-b g)+c g^2\right)}\right) \left(2 a h^2 (e h-2 f g)-b h \left(h (d h+e g)-3 f g^2\right)+c \left(2 d g h^2-2 f g^3\right)\right)}{2 \left(h (a h-b g)+c g^2\right)}-\frac{h (a+x (b+c x))^{3/2} \left(h (d h-e g)+f g^2\right)}{3 (g+h x)^3 \left(h (a h-b g)+c g^2\right)}+\frac{f \left(\frac{(2 c g-b h) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{2 \sqrt{h (a h-b g)+c g^2}}-\frac{h \sqrt{a+x (b+c x)}}{g+h x}+\sqrt{c} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)\right)}{h^2}}{h^2}","-\frac{\sqrt{a+b x+c x^2} \left(h x \left(h^2 \left(8 a^2 f h^2-2 a b h (10 f g-e h)+b^2 \left(11 f g^2-h (d h+e g)\right)\right)+2 c g h \left(2 a h (6 f g-e h)-b \left(12 f g^2-h (2 d h+e g)\right)\right)+4 c^2 \left(3 f g^4-d g^2 h^2\right)\right)+h^2 \left(4 a^2 e h^3-2 a b h \left(d h^2+2 e g h+3 f g^2\right)+b^2 g \left(d h^2+e g h+5 f g^2\right)\right)-2 c g h \left(-2 a d h^3-6 a f g^2 h+b d g h^2+7 b f g^3\right)+8 c^2 f g^5\right)}{8 h^3 (g+h x)^2 \left(a h^2-b g h+c g^2\right)^2}-\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(2 c h^2 \left(4 a^2 h^2 (4 f g-e h)-2 a b h \left(-d h^2-e g h+15 f g^2\right)+b^2 \left(d g h^2+15 f g^3\right)\right)-b h^3 \left(8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left(d h^2+e g h+5 f g^2\right)\right)-8 c^2 g h \left(a d h^3-5 a f g^2 h+5 b f g^3\right)+16 c^3 f g^5\right)}{16 h^4 \left(a h^2-b g h+c g^2\right)^{5/2}}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(f g^2-h (e g-d h)\right)}{3 h (g+h x)^3 \left(a h^2-b g h+c g^2\right)}+\frac{\sqrt{c} f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{h^4}",1,"(-1/3*(h*(f*g^2 + h*(-(e*g) + d*h))*(a + x*(b + c*x))^(3/2))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^3) + ((2*a*h^2*(-2*f*g + e*h) + c*(-2*f*g^3 + 2*d*g*h^2) - b*h*(-3*f*g^2 + h*(e*g + d*h)))*((Sqrt[a + x*(b + c*x)]*(-2*a*h + 2*c*g*x + b*(g - h*x)))/(4*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(8*(c*g^2 + h*(-(b*g) + a*h))^(3/2))))/(2*(c*g^2 + h*(-(b*g) + a*h))) + (f*(-((h*Sqrt[a + x*(b + c*x)])/(g + h*x)) + Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + ((2*c*g - b*h)*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)])))/h^2)/h^2","A",1
194,1,447,497,3.9014148,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^5} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^5,x]","\frac{\frac{\frac{3}{2} c h \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{8 \left(h (a h-b g)+c g^2\right)^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a h+b (g-h x)+2 c g x)}{4 (g+h x)^2 \left(h (a h-b g)+c g^2\right)}\right) \left(16 a^2 f h^2-4 c \left(a h (d h-5 e g)+a f g^2+2 b g (2 d h+e g)\right)-8 a b h (e h+2 f g)+b^2 \left(h (5 d h+3 e g)+5 f g^2\right)+16 c^2 d g^2\right)+\frac{c (a+x (b+c x))^{3/2} \left(-8 a h^2 (e h-2 f g)+b h \left(h (5 d h+3 e g)-11 f g^2\right)+2 c g h (e g-5 d h)+6 c f g^3\right)}{(g+h x)^3}}{24 \left(h (a h-b g)+c g^2\right)^2}+\frac{(a+x (b+c x))^{3/2} \left(4 f h (a h-b g)+c h (e g-d h)+3 c f g^2\right)}{4 (g+h x)^4 \left(h (a h-b g)+c g^2\right)}-\frac{f (a+x (b+c x))^{3/2}}{(g+h x)^4}}{c h}","\frac{\sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left(16 a^2 f h^2-4 c \left(a \left(d h^2-5 e g h+f g^2\right)+2 b g (2 d h+e g)\right)-8 a b h (e h+2 f g)+b^2 \left(5 d h^2+3 e g h+5 f g^2\right)+16 c^2 d g^2\right)}{64 (g+h x)^2 \left(a h^2-b g h+c g^2\right)^3}-\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(16 a^2 f h^2-4 c \left(a \left(d h^2-5 e g h+f g^2\right)+2 b g (2 d h+e g)\right)-8 a b h (e h+2 f g)+b^2 \left(5 d h^2+3 e g h+5 f g^2\right)+16 c^2 d g^2\right)}{128 \left(a h^2-b g h+c g^2\right)^{7/2}}+\frac{\left(a+b x+c x^2\right)^{3/2} \left(h \left(8 a h (2 f g-e h)-b \left(-5 d h^2-3 e g h+11 f g^2\right)\right)+2 c g \left(h (e g-5 d h)+3 f g^2\right)\right)}{24 h (g+h x)^3 \left(a h^2-b g h+c g^2\right)^2}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(f g^2-h (e g-d h)\right)}{4 h (g+h x)^4 \left(a h^2-b g h+c g^2\right)}",1,"(-((f*(a + x*(b + c*x))^(3/2))/(g + h*x)^4) + ((3*c*f*g^2 + 4*f*h*(-(b*g) + a*h) + c*h*(e*g - d*h))*(a + x*(b + c*x))^(3/2))/(4*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^4) + ((c*(6*c*f*g^3 + 2*c*g*h*(e*g - 5*d*h) - 8*a*h^2*(-2*f*g + e*h) + b*h*(-11*f*g^2 + h*(3*e*g + 5*d*h)))*(a + x*(b + c*x))^(3/2))/(g + h*x)^3 + (3*c*h*(16*c^2*d*g^2 + 16*a^2*f*h^2 - 8*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 + a*h*(-5*e*g + d*h) + 2*b*g*(e*g + 2*d*h)) + b^2*(5*f*g^2 + h*(3*e*g + 5*d*h)))*((Sqrt[a + x*(b + c*x)]*(-2*a*h + 2*c*g*x + b*(g - h*x)))/(4*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(8*(c*g^2 + h*(-(b*g) + a*h))^(3/2))))/2)/(24*(c*g^2 + h*(-(b*g) + a*h))^2))/(c*h)","A",1
195,1,1128,824,6.3327539,"\int \frac{\sqrt{a+b x+c x^2} \left(d+e x+f x^2\right)}{(g+h x)^6} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^6,x]","\frac{\sqrt{a+x (b+c x)} \left(-\frac{\left(\frac{1}{2} h (3 b f g+4 c d h-10 a f h)-\frac{1}{2} g (6 c f g+4 c e h-7 b f h)\right) \left(c x^2+b x+a\right)^{3/2}}{5 \left(c g^2-b h g+a h^2\right) (g+h x)^5}-\frac{-\frac{\left(2 c g \left(3 c f g^2-5 f h (b g-a h)+2 c h (e g-d h)\right)-c h \left(3 b f g^2-b h (3 e g+7 d h)+10 h (c d g-a f g+a e h)\right)\right) \left(c x^2+b x+a\right)^{3/2}}{4 \left(c g^2-b h g+a h^2\right) (g+h x)^4}-\frac{\frac{\left(c^2 g \left(6 c f g^3+2 c h (2 e g-7 d h) g+10 a h^2 (2 f g-e h)-b h \left(13 f g^2-h (3 e g+7 d h)\right)\right)-\frac{1}{2} c h \left(5 h \left(3 f g^2+h (3 e g+7 d h)\right) b^2+2 \left(3 c f g^3-c h (18 e g+47 d h) g-5 a h^2 (6 f g+5 e h)\right) b+16 h \left(5 c^2 d g^2+5 a^2 f h^2-a c \left(2 f g^2-h (7 e g-2 d h)\right)\right)\right)\right) \left(c x^2+b x+a\right)^{3/2}}{3 \left(c g^2-b h g+a h^2\right) (g+h x)^3}-\frac{\left(b \left(g \left(6 c f g^3+2 c h (2 e g-7 d h) g+10 a h^2 (2 f g-e h)-b h \left(13 f g^2-h (3 e g+7 d h)\right)\right) c^2+\frac{1}{2} h \left(5 h \left(3 f g^2+h (3 e g+7 d h)\right) b^2+2 \left(3 c f g^3-c h (18 e g+47 d h) g-5 a h^2 (6 f g+5 e h)\right) b+16 h \left(5 c^2 d g^2+5 a^2 f h^2-a c \left(2 f g^2-h (7 e g-2 d h)\right)\right)\right) c\right)-2 \left(a h \left(6 c f g^3+2 c h (2 e g-7 d h) g+10 a h^2 (2 f g-e h)-b h \left(13 f g^2-h (3 e g+7 d h)\right)\right) c^2+\frac{1}{2} g \left(5 h \left(3 f g^2+h (3 e g+7 d h)\right) b^2+2 \left(3 c f g^3-c h (18 e g+47 d h) g-5 a h^2 (6 f g+5 e h)\right) b+16 h \left(5 c^2 d g^2+5 a^2 f h^2-a c \left(2 f g^2-h (7 e g-2 d h)\right)\right)\right) c^2\right)\right) \left(\frac{\sqrt{c x^2+b x+a} (b g-2 a h+(2 c g-b h) x)}{4 \left(c g^2-b h g+a h^2\right) (g+h x)^2}+\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-b g+2 a h-(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{2 \sqrt{c g^2-b h g+a h^2} \left(4 c g^2-4 b h g+4 a h^2\right)}\right)}{2 \left(c g^2-b h g+a h^2\right)}}{4 \left(c g^2-b h g+a h^2\right)}}{5 \left(c g^2-b h g+a h^2\right)}\right)}{2 c h \sqrt{c x^2+b x+a}}-\frac{f \left(c x^2+b x+a\right) \sqrt{a+x (b+c x)}}{2 c h (g+h x)^5}","\frac{\left(4 c^2 \left(3 f g^2+h (2 e g-27 d h)\right) g^2-5 h^2 \left(\left(3 f g^2+3 e h g+7 d h^2\right) b^2-2 a h (6 f g+5 e h) b+16 a^2 f h^2\right)-2 c h \left(b g \left(16 f g^2-21 e h g-54 d h^2\right)-2 a h \left(18 f g^2-33 e h g+8 d h^2\right)\right)\right) \left(c x^2+b x+a\right)^{3/2}}{240 h \left(c g^2-b h g+a h^2\right)^3 (g+h x)^3}+\frac{\left(2 c g \left(3 f g^2+h (2 e g-7 d h)\right)+h \left(10 a h (2 f g-e h)-b \left(13 f g^2-3 e h g-7 d h^2\right)\right)\right) \left(c x^2+b x+a\right)^{3/2}}{40 h \left(c g^2-b h g+a h^2\right)^2 (g+h x)^4}-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{3/2}}{5 h \left(c g^2-b h g+a h^2\right) (g+h x)^5}+\frac{\left(32 c^3 d g^3-8 c^2 \left(2 b g (e g+3 d h)+a \left(f g^2-6 e h g+3 d h^2\right)\right) g-b h \left(\left(3 f g^2+3 e h g+7 d h^2\right) b^2-2 a h (6 f g+5 e h) b+16 a^2 f h^2\right)+2 c \left(g \left(5 f g^2+6 e h g+15 d h^2\right) b^2-6 a h \left(3 f g^2+3 e h g-d h^2\right) b+4 a^2 h^2 (6 f g-e h)\right)\right) (b g-2 a h+(2 c g-b h) x) \sqrt{c x^2+b x+a}}{128 \left(c g^2-b h g+a h^2\right)^4 (g+h x)^2}-\frac{\left(b^2-4 a c\right) \left(32 c^3 d g^3-8 c^2 \left(2 b g (e g+3 d h)+a \left(f g^2-6 e h g+3 d h^2\right)\right) g-b h \left(\left(3 f g^2+3 e h g+7 d h^2\right) b^2-2 a h (6 f g+5 e h) b+16 a^2 f h^2\right)+2 c \left(g \left(5 f g^2+6 e h g+15 d h^2\right) b^2-6 a h \left(3 f g^2+3 e h g-d h^2\right) b+4 a^2 h^2 (6 f g-e h)\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{256 \left(c g^2-b h g+a h^2\right)^{9/2}}",1,"-1/2*(f*(a + b*x + c*x^2)*Sqrt[a + x*(b + c*x)])/(c*h*(g + h*x)^5) + (Sqrt[a + x*(b + c*x)]*(-1/5*(((h*(3*b*f*g + 4*c*d*h - 10*a*f*h))/2 - (g*(6*c*f*g + 4*c*e*h - 7*b*f*h))/2)*(a + b*x + c*x^2)^(3/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - (-1/4*((2*c*g*(3*c*f*g^2 - 5*f*h*(b*g - a*h) + 2*c*h*(e*g - d*h)) - c*h*(3*b*f*g^2 - b*h*(3*e*g + 7*d*h) + 10*h*(c*d*g - a*f*g + a*e*h)))*(a + b*x + c*x^2)^(3/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (((c^2*g*(6*c*f*g^3 + 2*c*g*h*(2*e*g - 7*d*h) + 10*a*h^2*(2*f*g - e*h) - b*h*(13*f*g^2 - h*(3*e*g + 7*d*h))) - (c*h*(5*b^2*h*(3*f*g^2 + h*(3*e*g + 7*d*h)) + 2*b*(3*c*f*g^3 - c*g*h*(18*e*g + 47*d*h) - 5*a*h^2*(6*f*g + 5*e*h)) + 16*h*(5*c^2*d*g^2 + 5*a^2*f*h^2 - a*c*(2*f*g^2 - h*(7*e*g - 2*d*h)))))/2)*(a + b*x + c*x^2)^(3/2))/(3*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) - ((-2*(a*c^2*h*(6*c*f*g^3 + 2*c*g*h*(2*e*g - 7*d*h) + 10*a*h^2*(2*f*g - e*h) - b*h*(13*f*g^2 - h*(3*e*g + 7*d*h))) + (c^2*g*(5*b^2*h*(3*f*g^2 + h*(3*e*g + 7*d*h)) + 2*b*(3*c*f*g^3 - c*g*h*(18*e*g + 47*d*h) - 5*a*h^2*(6*f*g + 5*e*h)) + 16*h*(5*c^2*d*g^2 + 5*a^2*f*h^2 - a*c*(2*f*g^2 - h*(7*e*g - 2*d*h)))))/2) + b*(c^2*g*(6*c*f*g^3 + 2*c*g*h*(2*e*g - 7*d*h) + 10*a*h^2*(2*f*g - e*h) - b*h*(13*f*g^2 - h*(3*e*g + 7*d*h))) + (c*h*(5*b^2*h*(3*f*g^2 + h*(3*e*g + 7*d*h)) + 2*b*(3*c*f*g^3 - c*g*h*(18*e*g + 47*d*h) - 5*a*h^2*(6*f*g + 5*e*h)) + 16*h*(5*c^2*d*g^2 + 5*a^2*f*h^2 - a*c*(2*f*g^2 - h*(7*e*g - 2*d*h)))))/2))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(2*(c*g^2 - b*g*h + a*h^2)))/(4*(c*g^2 - b*g*h + a*h^2)))/(5*(c*g^2 - b*g*h + a*h^2))))/(2*c*h*Sqrt[a + b*x + c*x^2])","A",1
196,1,721,1169,2.7065029,"\int (g+h x)^3 \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^3*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\frac{(a+x (b+c x))^{5/2} \left(4 c^2 h^2 \left(512 a^2 f h^2+2 a b h (837 e h+2511 f g+935 f h x)+b^2 (27 h (56 d h+168 e g+55 e h x)+f g (3106 g+1595 h x))\right)-66 b^2 c h^3 (114 a f h+b (63 e h+189 f g+65 f h x))-16 c^3 h \left(a h (9 h (32 d h+96 e g+35 e h x)+f g (544 g+305 h x))+b \left(9 h (4 d h (49 g+15 h x)+e g (141 g+70 h x))+f g^2 (13 g+15 h x)\right)\right)+3003 b^4 f h^4-32 c^4 \left(5 f g^3 (6 g+5 h x)-9 g h (4 d h (32 g+15 h x)+e g (6 g+5 h x))\right)\right)}{8960 c^4}+\frac{3 h \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) \left(32 c^3 \left(3 a^2 h^2 (e h+3 f g)+12 a b h \left(h (d h+3 e g)+3 f g^2\right)+14 b^2 g \left(3 h (d h+e g)+f g^2\right)\right)-48 b c^2 h \left(5 a^2 f h^2+9 a b h (e h+3 f g)+6 b^2 \left(d h^2+3 e g h+3 f g^2\right)\right)+22 b^3 c h^2 (20 a f h+9 b (e h+3 f g))-256 c^4 g \left(3 a h (d h+e g)+a f g^2+3 b g (3 d h+e g)\right)-143 b^5 f h^3+1536 c^5 d g^3\right)}{65536 c^{13/2}}+\frac{(g+h x)^2 (a+x (b+c x))^{5/2} \left(-2 c h (64 a f h+99 b e h+24 b f g)+143 b^2 f h^2-12 c^2 \left(5 f g^2-3 h (8 d h+3 e g)\right)\right)}{224 c^2}-\frac{(g+h x)^3 (a+x (b+c x))^{5/2} (13 b f h+2 c (5 f g-9 e h))}{16 c}+f (g+h x)^4 (a+x (b+c x))^{5/2}}{9 c h}","\frac{f \left(c x^2+b x+a\right)^{5/2} (g+h x)^4}{9 c h}-\frac{(10 c f g-18 c e h+13 b f h) \left(c x^2+b x+a\right)^{5/2} (g+h x)^3}{144 c^2 h}+\frac{\left(-12 \left(5 f g^2-3 h (3 e g+8 d h)\right) c^2-2 h (24 b f g+99 b e h+64 a f h) c+143 b^2 f h^2\right) \left(c x^2+b x+a\right)^{5/2} (g+h x)^2}{2016 c^3 h}+\frac{\left(-192 g^2 \left(5 f g^2-3 h (3 e g+64 d h)\right) c^4-16 h \left(32 a h \left(17 f g^2+9 h (3 e g+d h)\right)+b g \left(13 f g^2+9 h (141 e g+196 d h)\right)\right) c^3+8 h^2 \left(\left(1553 f g^2+756 h (3 e g+d h)\right) b^2+837 a h (3 f g+e h) b+256 a^2 f h^2\right) c^2-198 b^2 h^3 (38 a f h+21 b (3 f g+e h)) c-10 h \left(16 g \left(5 f g^2-9 h (e g+12 d h)\right) c^3+8 h \left(a h (61 f g+63 e h)+3 b \left(f g^2+6 h (7 e g+6 d h)\right)\right) c^2-22 b h^2 (29 b f g+27 b e h+34 a f h) c+429 b^3 f h^3\right) x c+3003 b^4 f h^4\right) \left(c x^2+b x+a\right)^{5/2}}{80640 c^5 h}+\frac{\left(-143 f h^3 b^5+22 c h^2 (20 a f h+9 b (3 f g+e h)) b^3-48 c^2 h \left(6 \left(3 f g^2+3 e h g+d h^2\right) b^2+9 a h (3 f g+e h) b+5 a^2 f h^2\right) b+1536 c^5 d g^3-256 c^4 g \left(3 b g (e g+3 d h)+a \left(f g^2+3 h (e g+d h)\right)\right)+32 c^3 \left(14 g \left(f g^2+3 h (e g+d h)\right) b^2+12 a h \left(3 f g^2+h (3 e g+d h)\right) b+3 a^2 h^2 (3 f g+e h)\right)\right) (b+2 c x) \left(c x^2+b x+a\right)^{3/2}}{12288 c^6}+\frac{\left(b^2-4 a c\right)^2 \left(-143 f h^3 b^5+22 c h^2 (20 a f h+9 b (3 f g+e h)) b^3-48 c^2 h \left(6 \left(3 f g^2+3 e h g+d h^2\right) b^2+9 a h (3 f g+e h) b+5 a^2 f h^2\right) b+1536 c^5 d g^3-256 c^4 g \left(3 b g (e g+3 d h)+a \left(f g^2+3 h (e g+d h)\right)\right)+32 c^3 \left(14 g \left(f g^2+3 h (e g+d h)\right) b^2+12 a h \left(3 f g^2+h (3 e g+d h)\right) b+3 a^2 h^2 (3 f g+e h)\right)\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{65536 c^{15/2}}-\frac{\left(b^2-4 a c\right) \left(-143 f h^3 b^5+22 c h^2 (20 a f h+9 b (3 f g+e h)) b^3-48 c^2 h \left(6 \left(3 f g^2+3 e h g+d h^2\right) b^2+9 a h (3 f g+e h) b+5 a^2 f h^2\right) b+1536 c^5 d g^3-256 c^4 g \left(3 b g (e g+3 d h)+a \left(f g^2+3 h (e g+d h)\right)\right)+32 c^3 \left(14 g \left(f g^2+3 h (e g+d h)\right) b^2+12 a h \left(3 f g^2+h (3 e g+d h)\right) b+3 a^2 h^2 (3 f g+e h)\right)\right) (b+2 c x) \sqrt{c x^2+b x+a}}{32768 c^7}",1,"(((143*b^2*f*h^2 - 2*c*h*(24*b*f*g + 99*b*e*h + 64*a*f*h) - 12*c^2*(5*f*g^2 - 3*h*(3*e*g + 8*d*h)))*(g + h*x)^2*(a + x*(b + c*x))^(5/2))/(224*c^2) - ((13*b*f*h + 2*c*(5*f*g - 9*e*h))*(g + h*x)^3*(a + x*(b + c*x))^(5/2))/(16*c) + f*(g + h*x)^4*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(3003*b^4*f*h^4 - 66*b^2*c*h^3*(114*a*f*h + b*(189*f*g + 63*e*h + 65*f*h*x)) - 32*c^4*(5*f*g^3*(6*g + 5*h*x) - 9*g*h*(e*g*(6*g + 5*h*x) + 4*d*h*(32*g + 15*h*x))) + 4*c^2*h^2*(512*a^2*f*h^2 + 2*a*b*h*(2511*f*g + 837*e*h + 935*f*h*x) + b^2*(f*g*(3106*g + 1595*h*x) + 27*h*(168*e*g + 56*d*h + 55*e*h*x))) - 16*c^3*h*(a*h*(f*g*(544*g + 305*h*x) + 9*h*(96*e*g + 32*d*h + 35*e*h*x)) + b*(f*g^2*(13*g + 15*h*x) + 9*h*(4*d*h*(49*g + 15*h*x) + e*g*(141*g + 70*h*x))))))/(8960*c^4) + (3*h*(1536*c^5*d*g^3 - 143*b^5*f*h^3 - 256*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + 3*b*g*(e*g + 3*d*h)) + 22*b^3*c*h^2*(20*a*f*h + 9*b*(3*f*g + e*h)) - 48*b*c^2*h*(5*a^2*f*h^2 + 9*a*b*h*(3*f*g + e*h) + 6*b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 32*c^3*(3*a^2*h^2*(3*f*g + e*h) + 14*b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 12*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*(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)])]))/(65536*c^(13/2)))/(9*c*h)","A",1
197,1,468,753,1.5984494,"\int (g+h x)^2 \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^2*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\frac{h \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) \left(16 c^2 \left(3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left(h (d h+2 e g)+f g^2\right)\right)-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left(a h (d h+2 e g)+a f g^2+3 b g (2 d h+e g)\right)+99 b^4 f h^2+768 c^4 d g^2\right)}{12288 c^{11/2}}-\frac{(a+x (b+c x))^{5/2} \left(8 c^2 h (3 a h (32 e h+64 f g+35 f h x)+4 b h (49 d h+98 e g+45 e h x)+b f g (31 g+30 h x))-18 b c h^2 (62 a f h+b (56 e h+112 f g+55 f h x))+693 b^3 f h^3+16 c^3 \left(5 f g^2 (6 g+5 h x)-4 h (7 d h (12 g+5 h x)+2 e g (6 g+5 h x))\right)\right)}{1680 c^3}-\frac{(g+h x)^2 (a+x (b+c x))^{5/2} (11 b f h+2 c (5 f g-8 e h))}{14 c}+f (g+h x)^3 (a+x (b+c x))^{5/2}}{8 c h}","\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(16 c^2 \left(3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left(d h^2+2 e g h+f g^2\right)\right)-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+3 b g (2 d h+e g)\right)+99 b^4 f h^2+768 c^4 d g^2\right)}{32768 c^{13/2}}-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} \left(16 c^2 \left(3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left(d h^2+2 e g h+f g^2\right)\right)-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+3 b g (2 d h+e g)\right)+99 b^4 f h^2+768 c^4 d g^2\right)}{16384 c^6}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(16 c^2 \left(3 a^2 f h^2+12 a b h (e h+2 f g)+14 b^2 \left(d h^2+2 e g h+f g^2\right)\right)-72 b^2 c h (3 a f h+2 b e h+4 b f g)-128 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+3 b g (2 d h+e g)\right)+99 b^4 f h^2+768 c^4 d g^2\right)}{6144 c^5}-\frac{\left(a+b x+c x^2\right)^{5/2} \left(-10 c h x \left(-12 c h (7 a f h+2 b (6 e h+f g))+99 b^2 f h^2-8 c^2 \left(5 f g^2-4 h (7 d h+2 e g)\right)\right)+8 c^2 h \left(96 a h (e h+2 f g)+b \left(196 h (d h+2 e g)+31 f g^2\right)\right)-36 b c h^2 (31 a f h+28 b (e h+2 f g))+693 b^3 f h^3+96 c^3 g \left(5 f g^2-8 h (7 d h+e g)\right)\right)}{13440 c^4 h}-\frac{(g+h x)^2 \left(a+b x+c x^2\right)^{5/2} (11 b f h-16 c e h+10 c f g)}{112 c^2 h}+\frac{f (g+h x)^3 \left(a+b x+c x^2\right)^{5/2}}{8 c h}",1,"(-1/14*((11*b*f*h + 2*c*(5*f*g - 8*e*h))*(g + h*x)^2*(a + x*(b + c*x))^(5/2))/c + f*(g + h*x)^3*(a + x*(b + c*x))^(5/2) - ((a + x*(b + c*x))^(5/2)*(693*b^3*f*h^3 + 8*c^2*h*(b*f*g*(31*g + 30*h*x) + 4*b*h*(98*e*g + 49*d*h + 45*e*h*x) + 3*a*h*(64*f*g + 32*e*h + 35*f*h*x)) - 18*b*c*h^2*(62*a*f*h + b*(112*f*g + 56*e*h + 55*f*h*x)) + 16*c^3*(5*f*g^2*(6*g + 5*h*x) - 4*h*(2*e*g*(6*g + 5*h*x) + 7*d*h*(12*g + 5*h*x)))))/(1680*c^3) + (h*(768*c^4*d*g^2 + 99*b^4*f*h^2 - 72*b^2*c*h*(4*b*f*g + 2*b*e*h + 3*a*f*h) - 128*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + 3*b*g*(e*g + 2*d*h)) + 16*c^2*(3*a^2*f*h^2 + 12*a*b*h*(2*f*g + e*h) + 14*b^2*(f*g^2 + h*(2*e*g + d*h))))*(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)])]))/(12288*c^(11/2)))/(8*c*h)","A",1
198,1,285,418,0.7847645,"\int (g+h x) \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\frac{(a+x (b+c x))^{5/2} \left(-2 c h (24 a f h+b (49 e h+49 f g+45 f h x))+63 b^2 f h^2-4 c^2 (5 f g (6 g+5 h x)-7 h (6 d h+6 e g+5 e h x))\right)}{120 c^2}-\frac{7 h \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) \left(8 c^2 (a e h+a f g+3 b d h+3 b e g)-2 b c (6 a f h+7 b (e h+f g))+9 b^3 f h-48 c^3 d g\right)}{6144 c^{9/2}}+f (g+h x)^2 (a+x (b+c x))^{5/2}}{7 c h}","\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right)}{384 c^4}+\frac{\left(a+b x+c x^2\right)^{5/2} \left(-2 c h (24 a f h+49 b (e h+f g))+63 b^2 f h^2-10 c h x (9 b f h-14 c e h+10 c f g)-24 c^2 \left(5 f g^2-7 h (d h+e g)\right)\right)}{840 c^3 h}+\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right)}{2048 c^{11/2}}-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} \left(-8 c^2 (a e h+a f g+3 b d h+3 b e g)+2 b c (6 a f h+7 b (e h+f g))-9 b^3 f h+48 c^3 d g\right)}{1024 c^5}+\frac{f (g+h x)^2 \left(a+b x+c x^2\right)^{5/2}}{7 c h}",1,"(f*(g + h*x)^2*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(63*b^2*f*h^2 - 4*c^2*(5*f*g*(6*g + 5*h*x) - 7*h*(6*e*g + 6*d*h + 5*e*h*x)) - 2*c*h*(24*a*f*h + b*(49*f*g + 49*e*h + 45*f*h*x))))/(120*c^2) - (7*h*(-48*c^3*d*g + 9*b^3*f*h + 8*c^2*(3*b*e*g + a*f*g + 3*b*d*h + a*e*h) - 2*b*c*(6*a*f*h + 7*b*(f*g + e*h)))*(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)])]))/(6144*c^(9/2)))/(7*c*h)","A",1
199,1,392,236,0.69103,"\int \left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right) \, dx","Integrate[(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2),x]","\frac{\frac{360 d \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^{3/2}}-60 b e \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)+\frac{f \left(5 \left(7 b^2-4 a c\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 b (a+x (b+c x))^{5/2}\right)}{c}+1920 d (b+2 c x) (a+x (b+c x))^{3/2}+3072 e (a+x (b+c x))^{5/2}+2560 f x (a+x (b+c x))^{5/2}}{15360 c}","\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right)}{1024 c^{9/2}}-\frac{\left(b^2-4 a c\right) (b+2 c x) \sqrt{a+b x+c x^2} \left(-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right)}{512 c^4}+\frac{(b+2 c x) \left(a+b x+c x^2\right)^{3/2} \left(-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right)}{192 c^3}+\frac{\left(a+b x+c x^2\right)^{5/2} (12 c e-7 b f)}{60 c^2}+\frac{f x \left(a+b x+c x^2\right)^{5/2}}{6 c}",1,"(1920*d*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 3072*e*(a + x*(b + c*x))^(5/2) + 2560*f*x*(a + x*(b + c*x))^(5/2) + (360*(b^2 - 4*a*c)*d*(-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^(3/2) - 60*b*e*((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)) + (f*(-1792*b*(a + x*(b + c*x))^(5/2) + 5*(7*b^2 - 4*a*c)*((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)/(15360*c)","A",1
200,1,635,660,2.2099481,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{g+h x} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x),x]","-\frac{(a+x (b+c x))^{3/2} \left(3 b^2 f h^2+6 b c h (f (g+h x)-e h)-4 c^2 (h (4 d h-4 e g+3 e h x)+f g (4 g-3 h x))\right)}{48 c^2 h^3}+\frac{-\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(\left(2 c h (b g-a h)+\frac{b^2 h^2}{2}-4 c^2 g^2\right) \left(8 c h (2 c g-b h) (b f g-2 c d h)-\left(4 c h (3 a h-2 b g)-3 b^2 h^2+16 c^2 g^2\right) (b f h-2 c e h+2 c f g)\right)+2 c h (2 c g-b h) \left(8 c h (b g-2 a h) (b f g-2 c d h)-g \left(-4 a c h-3 b^2 h+8 b c g\right) (b f h-2 c e h+2 c f g)\right)\right)+\sqrt{c} h \sqrt{a+x (b+c x)} \left(6 b^2 c h^3 (-2 a f h-b e h+b f (g+h x))-16 c^3 h \left(a h (h (-8 d h+8 e g-3 e h x)+f g (3 h x-8 g))+2 b (5 g-h x) \left(h (d h-e g)+f g^2\right)\right)+4 b c^2 h^2 \left(6 a e h^2-6 a f h (g+h x)+b h (4 d h-4 e g-3 e h x)+b f g (4 g+3 h x)\right)+3 b^4 f h^4+64 c^4 g (2 g-h x) \left(h (d h-e g)+f g^2\right)\right)-128 c^{7/2} \left(h (a h-b g)+c g^2\right)^{3/2} \left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{128 c^{7/2} h^6}+\frac{f (a+x (b+c x))^{5/2}}{5 c h}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(4 c h (2 c g-b h) \left(8 c h (b g-2 a h) (b f g-2 c d h)-g \left(-4 a c h-3 b^2 h+8 b c g\right) (b f h-2 c e h+2 c f g)\right)-2 \left(-2 c h (b g-a h)-\frac{b^2 h^2}{2}+4 c^2 g^2\right) \left(8 c h (2 c g-b h) (b f g-2 c d h)-\left(-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right) (b f h-2 c e h+2 c f g)\right)\right)}{256 c^{7/2} h^6}+\frac{\sqrt{a+b x+c x^2} \left(2 c h x \left(8 c h (2 c g-b h) (b f g-2 c d h)-\left(-4 c h (2 b g-3 a h)-3 b^2 h^2+16 c^2 g^2\right) (b f h-2 c e h+2 c f g)\right)+6 b^2 c h^3 (-2 a f h-b e h+b f g)-32 c^3 h (5 b g-4 a h) \left(f g^2-h (e g-d h)\right)-8 b c^2 h^2 \left(3 a h (f g-e h)-2 b \left(d h^2-e g h+f g^2\right)\right)+3 b^4 f h^4+128 c^4 g^2 \left(f g^2-h (e g-d h)\right)\right)}{128 c^3 h^5}-\frac{\left(a+b x+c x^2\right)^{3/2} (8 c h (b f g-2 c d h)+6 c h x (b f h-2 c e h+2 c f g)-(8 c g-3 b h) (b f h-2 c e h+2 c f g))}{48 c^2 h^3}+\frac{\left(a h^2-b g h+c g^2\right)^{3/2} \left(f g^2-h (e g-d h)\right) \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right)}{h^6}+\frac{f \left(a+b x+c x^2\right)^{5/2}}{5 c h}",1,"(f*(a + x*(b + c*x))^(5/2))/(5*c*h) - ((a + x*(b + c*x))^(3/2)*(3*b^2*f*h^2 + 6*b*c*h*(-(e*h) + f*(g + h*x)) - 4*c^2*(f*g*(4*g - 3*h*x) + h*(-4*e*g + 4*d*h + 3*e*h*x))))/(48*c^2*h^3) + (Sqrt[c]*h*Sqrt[a + x*(b + c*x)]*(3*b^4*f*h^4 + 64*c^4*g*(f*g^2 + h*(-(e*g) + d*h))*(2*g - h*x) + 6*b^2*c*h^3*(-(b*e*h) - 2*a*f*h + b*f*(g + h*x)) + 4*b*c^2*h^2*(6*a*e*h^2 - 6*a*f*h*(g + h*x) + b*f*g*(4*g + 3*h*x) + b*h*(-4*e*g + 4*d*h - 3*e*h*x)) - 16*c^3*h*(2*b*(f*g^2 + h*(-(e*g) + d*h))*(5*g - h*x) + a*h*(f*g*(-8*g + 3*h*x) + h*(8*e*g - 8*d*h - 3*e*h*x)))) - (2*c*h*(2*c*g - b*h)*(8*c*h*(b*g - 2*a*h)*(b*f*g - 2*c*d*h) - g*(8*b*c*g - 3*b^2*h - 4*a*c*h)*(2*c*f*g - 2*c*e*h + b*f*h)) + (-4*c^2*g^2 + (b^2*h^2)/2 + 2*c*h*(b*g - a*h))*(8*c*h*(2*c*g - b*h)*(b*f*g - 2*c*d*h) - (2*c*f*g - 2*c*e*h + b*f*h)*(16*c^2*g^2 - 3*b^2*h^2 + 4*c*h*(-2*b*g + 3*a*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 128*c^(7/2)*(c*g^2 + h*(-(b*g) + a*h))^(3/2)*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(128*c^(7/2)*h^6)","A",1
201,1,756,754,4.1575453,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^2} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^2,x]","-\frac{\frac{\frac{-2 c h \sqrt{a+x (b+c x)} \left(h (a h-b g)+c g^2\right) \left(-4 c^2 h (a h (8 e h-16 f g+3 f h x)+2 b (h (9 d h-14 e g+e h x)+f g (19 g-2 h x)))+b c h^2 (b (-4 e h+8 f g+3 f h x)-6 a f h)+\frac{3}{2} b^3 f h^3+16 c^3 (2 g-h x) \left(h (3 d h-4 e g)+5 f g^2\right)\right)+\sqrt{c} \left(h (a h-b g)+c g^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(\frac{1}{2} \left(4 c h (a h-b g)-b^2 h^2+8 c^2 g^2\right) \left(4 c h (3 a f h+2 b e h-4 b f g)-3 b^2 f h^2+16 c^2 \left(h (3 d h-4 e g)+5 f g^2\right)\right)+2 c h (2 c g-b h) \left(4 a c h (5 f g-4 e h)+3 b^2 f g h-8 b c \left(h (3 d h-4 e g)+5 f g^2\right)\right)\right)+32 c^3 \left(h (a h-b g)+c g^2\right)^{3/2} \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(h \left(-2 a h (e h-2 f g)+b h (5 e g-3 d h)-7 b f g^2\right)+2 c \left(g h (3 d h-4 e g)+5 f g^3\right)\right)}{16 c^2 h^5}+\frac{(a+x (b+c x))^{3/2} \left(c h (2 a h (4 e h-8 f g+3 f h x)+8 b h (3 d h-4 e g)+b f g (43 g-6 h x))+3 b f h^2 (a h-b g)+c^2 \left(8 h (3 d h (h x-g)+e g (4 g-3 h x))+10 f g^2 (3 h x-4 g)\right)\right)}{6 h^2}}{h (b g-a h)-c g^2}+\frac{(a+x (b+c x))^{5/2} \left(f h (a h-b g)+4 c h (d h-e g)+5 c f g^2\right)}{(g+h x) \left(h (a h-b g)+c g^2\right)}-\frac{f (a+x (b+c x))^{5/2}}{g+h x}}{4 c h}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(48 c^2 h^2 \left(a^2 f h^2-2 a b h (2 f g-e h)+b^2 \left(d h^2-2 e g h+3 f g^2\right)\right)+8 b^2 c h^3 (-3 a f h-b e h+2 b f g)+192 c^3 h \left(a h \left(d h^2-2 e g h+3 f g^2\right)-b g \left(2 d h^2-3 e g h+4 f g^2\right)\right)+3 b^4 f h^4+128 c^4 g^2 \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{128 c^{5/2} h^6}-\frac{\sqrt{a+b x+c x^2} \left(2 c h x \left(4 c h (-3 a f h-2 b e h+4 b f g)+3 b^2 f h^2-16 c^2 \left(5 f g^2-h (4 e g-3 d h)\right)\right)+16 c^2 h \left(4 a h (2 f g-e h)-b \left(9 d h^2-14 e g h+19 f g^2\right)\right)+4 b c h^2 (-3 a f h-2 b e h+4 b f g)+3 b^3 f h^3+64 c^3 g \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{64 c^2 h^5}-\frac{\left(a+b x+c x^2\right)^{3/2} \left(6 c h^2 x \left(-a f h+b f g-4 c d h+4 c e g-\frac{5 c f g^2}{h}\right)+c h \left(8 a h (2 f g-e h)-b \left(43 f g^2-8 h (4 e g-3 d h)\right)\right)+3 b f h^2 (b g-a h)+8 c^2 g \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{24 c h^3 \left(a h^2-b g h+c g^2\right)}-\frac{\left(a+b x+c x^2\right)^{5/2} \left(f g^2-h (e g-d h)\right)}{h (g+h x) \left(a h^2-b g h+c g^2\right)}-\frac{\sqrt{a h^2-b g h+c g^2} \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(h \left(2 a h (2 f g-e h)-b \left(3 d h^2-5 e g h+7 f g^2\right)\right)+2 c g \left(5 f g^2-h (4 e g-3 d h)\right)\right)}{2 h^6}",1,"-1/4*(-((f*(a + x*(b + c*x))^(5/2))/(g + h*x)) + ((5*c*f*g^2 + f*h*(-(b*g) + a*h) + 4*c*h*(-(e*g) + d*h))*(a + x*(b + c*x))^(5/2))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)) + (((a + x*(b + c*x))^(3/2)*(3*b*f*h^2*(-(b*g) + a*h) + c*h*(8*b*h*(-4*e*g + 3*d*h) + b*f*g*(43*g - 6*h*x) + 2*a*h*(-8*f*g + 4*e*h + 3*f*h*x)) + c^2*(10*f*g^2*(-4*g + 3*h*x) + 8*h*(e*g*(4*g - 3*h*x) + 3*d*h*(-g + h*x)))))/(6*h^2) + (-2*c*h*(c*g^2 + h*(-(b*g) + a*h))*Sqrt[a + x*(b + c*x)]*((3*b^3*f*h^3)/2 + 16*c^3*(5*f*g^2 + h*(-4*e*g + 3*d*h))*(2*g - h*x) + b*c*h^2*(-6*a*f*h + b*(8*f*g - 4*e*h + 3*f*h*x)) - 4*c^2*h*(a*h*(-16*f*g + 8*e*h + 3*f*h*x) + 2*b*(f*g*(19*g - 2*h*x) + h*(-14*e*g + 9*d*h + e*h*x)))) + Sqrt[c]*(c*g^2 + h*(-(b*g) + a*h))*(2*c*h*(2*c*g - b*h)*(3*b^2*f*g*h + 4*a*c*h*(5*f*g - 4*e*h) - 8*b*c*(5*f*g^2 + h*(-4*e*g + 3*d*h))) + ((8*c^2*g^2 - b^2*h^2 + 4*c*h*(-(b*g) + a*h))*(-3*b^2*f*h^2 + 4*c*h*(-4*b*f*g + 2*b*e*h + 3*a*f*h) + 16*c^2*(5*f*g^2 + h*(-4*e*g + 3*d*h))))/2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 32*c^3*(c*g^2 + h*(-(b*g) + a*h))^(3/2)*(2*c*(5*f*g^3 + g*h*(-4*e*g + 3*d*h)) + h*(-7*b*f*g^2 + b*h*(5*e*g - 3*d*h) - 2*a*h*(-2*f*g + e*h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(16*c^2*h^5))/(-(c*g^2) + h*(b*g - a*h)))/(c*h)","A",1
202,1,4162,824,6.2740344,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^3} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^3,x]","\text{Result too large to show}","-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{5/2}}{2 h \left(c g^2-b h g+a h^2\right) (g+h x)^2}-\frac{\left(4 c g \left(-\frac{10 f g^2}{h}+6 e g-3 d h\right)-4 a h (7 f g-3 e h)+b \left(31 f g^2-3 h (5 e g-d h)\right)+2 h \left(-\frac{5 c f g^2}{h}+3 c e g+2 b f g-3 c d h-2 a f h\right) x\right) \left(c x^2+b x+a\right)^{3/2}}{12 h^2 \left(c g^2-b h g+a h^2\right) (g+h x)}-\frac{\left(-8 g^2 \left(10 f g^2-3 h (2 e g-d h)\right) c^3-2 h \left(2 a h \left(19 f g^2-9 e h g+3 d h^2\right)-3 b g \left(22 f g^2-12 e h g+5 d h^2\right)\right) c^2-h^2 \left(\left(53 f g^2-6 h (4 e g-d h)\right) b^2-18 a h (3 f g-e h) b+8 a^2 f h^2\right) c+2 h \left(2 g \left(10 f g^2-3 h (2 e g-d h)\right) c^2+h \left(2 a h (7 f g-3 e h)-3 b \left(6 f g^2-3 e h g+d h^2\right)\right) c+b f h^2 (b g-a h)\right) x c+b^2 f h^3 (b g-a h)\right) \sqrt{c x^2+b x+a}}{8 c h^5 \left(c g^2-b h g+a h^2\right)}-\frac{\left(16 g \left(10 f g^2-3 h (2 e g-d h)\right) c^3+24 h \left(a h (3 f g-e h)-b \left(6 f g^2-3 e h g+d h^2\right)\right) c^2+6 b h^2 (3 b f g-b e h-2 a f h) c+b^3 f h^3\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{16 c^{3/2} h^6}+\frac{\left(8 c^2 \left(10 f g^2-3 h (2 e g-d h)\right) g^2+4 c h \left(a h \left(19 f g^2-9 e h g+3 d h^2\right)-b g \left(28 f g^2-15 e h g+6 d h^2\right)\right)+h^2 \left(\left(35 f g^2-3 h (5 e g-d h)\right) b^2-4 a h (10 f g-3 e h) b+8 a^2 f h^2\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{8 h^6 \sqrt{c g^2-b h g+a h^2}}",1,"(f*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(3*c*h*(g + h*x)^2) - ((a + x*(b + c*x))^(3/2)*(-1/2*(((h*(5*b*f*g - 6*c*d*h - 4*a*f*h))/2 - (g*(10*c*f*g - 6*c*e*h + b*f*h))/2)*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - (((-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2)*(a + b*x + c*x^2)^(5/2))/((-(c*g^2) + b*g*h - a*h^2)*(g + h*x)) + (((4*c*(4*c*g - (3*b*h)/2)*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2) - 12*c^2*h*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2)*x)*(a + b*x + c*x^2)^(3/2))/(12*c*h^2) - (((2*c*h*(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) - (2*c*g - (b*h)/2)*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) + c*h*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2))*x)*Sqrt[a + b*x + c*x^2])/(2*c*h^2) - (((2*c*h*(2*c*g - b*h)*(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) + (-4*c^2*g^2 + (b^2*h^2)/2 - c*h*(-2*b*g + 2*a*h))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*h) - (4*Sqrt[c*g^2 - b*g*h + a*h^2]*(-(g*(2*c*h*(2*c*g - b*h)*(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) + (-4*c^2*g^2 + (b^2*h^2)/2 - c*h*(-2*b*g + 2*a*h))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)))) + h*(2*c*h*(b*g - 2*a*h)*(-4*c*(2*a*c*g*h + b*g*(-4*c*g + (3*b*h)/2))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(b*g - 2*a*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2)) + (2*a*c*g*h + b*g*(-2*c*g + (b*h)/2))*(-4*c*(-8*c^2*g^2 + (3*b^2*h^2)/2 - c*h*(-4*b*g + 6*a*h))*(-3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) + (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2) + 4*c*h*(2*c*g - b*h)*(-3*a*c*h*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c^2*g*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (3*b*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2 + (5*b*(3*c*g*(5*c*f*g^2 - 2*f*h*(b*g - a*h) - 3*c*h*(e*g - d*h)) - (3*c*h*(5*b*f*g^2 - b*h*(5*e*g - d*h) + 4*h*(c*d*g - a*f*g + a*e*h)))/2))/2))))*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(h*(4*c*g^2 - 4*b*g*h + 4*a*h^2)))/(4*c*h^2))/(8*c*h^2))/(-(c*g^2) + b*g*h - a*h^2))/(2*(c*g^2 - b*g*h + a*h^2))))/(3*c*h*(a + b*x + c*x^2)^(3/2))","B",1
203,1,7806,833,6.492431,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^4} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4,x]","\text{Result too large to show}","-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{5/2}}{3 h \left(c g^2-b h g+a h^2\right) (g+h x)^3}-\frac{\left(2 c g \left(-\frac{10 f g^2}{h}+4 e g-d h\right)-6 a h (3 f g-e h)+b \left(17 f g^2-h (5 e g+d h)\right)+2 h \left(-\frac{5 c f g^2}{h}+2 c e g+3 b f g-2 c d h-3 a f h\right) x\right) \left(c x^2+b x+a\right)^{3/2}}{12 h^2 \left(c g^2-b h g+a h^2\right) (g+h x)^2}-\frac{\left(8 c^2 \left(10 f g^2-h (4 e g-d h)\right) g^2-2 c h \left(3 b g \left(18 f g^2-6 e h g+d h^2\right)-2 a h \left(23 f g^2-8 e h g+2 d h^2\right)\right)+h^2 \left(\left(29 f g^2-h (5 e g+d h)\right) b^2-6 a h (7 f g-e h) b+12 a^2 f h^2\right)+2 h \left(2 g \left(10 f g^2-h (4 e g-d h)\right) c^2+h \left(6 a h (3 f g-e h)-b \left(22 f g^2-7 e h g+d h^2\right)\right) c+3 b f h^2 (b g-a h)\right) x\right) \sqrt{c x^2+b x+a}}{8 h^5 \left(c g^2-b h g+a h^2\right) (g+h x)}+\frac{\left(8 \left(10 f g^2-h (4 e g-d h)\right) c^2-12 h (4 b f g-b e h-a f h) c+3 b^2 f h^2\right) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{8 \sqrt{c} h^6}-\frac{\left(16 c^3 \left(10 f g^2-h (4 e g-d h)\right) g^3-24 c^2 h \left(b g \left(14 f g^2-5 e h g+d h^2\right)-a h \left(11 f g^2-4 e h g+d h^2\right)\right) g-b h^3 \left(\left(35 f g^2-5 e h g-d h^2\right) b^2-6 a h (10 f g-e h) b+24 a^2 f h^2\right)+6 c h^2 \left(g \left(35 f g^2-10 e h g+d h^2\right) b^2-2 a h \left(25 f g^2-7 e h g+d h^2\right) b+4 a^2 h^2 (4 f g-e h)\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{16 h^6 \left(c g^2-b h g+a h^2\right)^{3/2}}",1,"Result too large to show","B",1
204,1,46895,1097,6.6217301,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^5} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]","\text{Result too large to show}","-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{5/2}}{4 h \left(c g^2-b h g+a h^2\right) (g+h x)^4}-\frac{\left(16 c^2 (5 f g-e h) g^4-4 c h \left(b g \left(31 f g^2-5 e h g+3 d h^2\right)-a h \left(25 f g^2-5 e h g+9 d h^2\right)\right) g-h^2 \left(-g \left(35 f g^2+5 e h g+3 d h^2\right) b^2+4 a h \left(7 f g^2+7 e h g+3 d h^2\right) b+16 a^2 h^2 (f g-2 e h)\right)+3 h \left(8 c^2 \left(5 f g^2-h (e g+d h)\right) g^2+h^2 \left(\left(29 f g^2-5 e h g-3 d h^2\right) b^2-8 a h (6 f g-e h) b+16 a^2 f h^2\right)-4 c h \left(2 b g \left(9 f g^2-2 e h g-d h^2\right)-a h \left(17 f g^2-5 e h g+d h^2\right)\right)\right) x\right) \left(c x^2+b x+a\right)^{3/2}}{96 h^3 \left(c g^2-b h g+a h^2\right)^2 (g+h x)^3}+\frac{\left(64 c^3 (5 f g-e h) g^4-16 c^2 h (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h^2 \left(2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left(173 f g^2-25 e h g-3 d h^2\right)\right)-b h^3 \left(\left(35 f g^2+5 e h g+3 d h^2\right) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right)+2 c h \left(16 c^2 (5 f g-e h) g^3-4 c h \left(6 b g^2 (6 f g-e h)-a h \left(35 f g^2-h (7 e g-3 d h)\right)\right)+h^2 \left(\left(61 f g^2-h (5 e g+3 d h)\right) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right)\right) x\right) \sqrt{c x^2+b x+a}}{64 h^5 \left(c g^2-b h g+a h^2\right)^2 (g+h x)}-\frac{\sqrt{c} (10 c f g-2 c e h-3 b f h) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{2 h^6}+\frac{\left(128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left(-5 g^2 (14 f g-e h) b^3+3 a h \left(55 f g^2-5 e h g-d h^2\right) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right)-48 c^2 h^2 \left(-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left(25 f g^2-5 e h g+d h^2\right)\right)+b^2 h^4 \left(\left(35 f g^2+5 e h g+3 d h^2\right) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{128 h^6 \left(c g^2-b h g+a h^2\right)^{5/2}}",1,"Result too large to show","B",1
205,1,1111,1226,6.2955559,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^6} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]","\frac{f (a+x (b+c x))^{3/2} \left(\frac{-\frac{(b h-2 c g) \left(c x^2+b x+a\right)^{3/2}}{2 \left(c g^2-b h g+a h^2\right) (g+h x)^2}-\frac{\frac{\left(\frac{1}{2} h \left(h b^2+2 c g b-8 a c h\right)-c g (2 c g-b h)\right) \left(c x^2+b x+a\right)^{3/2}}{\left(-c g^2+b h g-a h^2\right) (g+h x)}+\frac{\frac{\left(h \left(4 c^2 g^2-b^2 h^2-4 c h (b g-2 a h)\right) x c^2-\left(2 c g-\frac{b h}{2}\right) \left(4 c^2 g^2-b^2 h^2-4 c h (b g-2 a h)\right) c+\frac{1}{2} h \left(-h^2 b^3-10 c g h b^2+4 c \left(2 c g^2+3 a h^2\right) b+8 a c^2 g h\right) c\right) \sqrt{c x^2+b x+a}}{2 c h^2}-\frac{-\frac{16 \left(c g^2-h (b g-a h)\right)^2 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right) c^{5/2}}{h}-\frac{4 \sqrt{c g^2-b h g+a h^2} \left(16 c^3 g \left(c g^2-h (b g-a h)\right)^2-c h \left(c g^2-b h g+a h^2\right) \left(-h^2 b^3-6 c g h b^2+8 c^2 g^2 b+12 a c h^2 b-8 a c^2 g h\right)\right) \tanh ^{-1}\left(\frac{-b g+2 a h-(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{h \left(4 c g^2-4 b h g+4 a h^2\right)}}{4 c h^2}}{-c g^2+b h g-a h^2}}{2 \left(c g^2-b h g+a h^2\right)}}{2 h}-\frac{\left(c x^2+b x+a\right)^{3/2}}{3 h (g+h x)^3}\right)}{h^2 \left(c x^2+b x+a\right)^{3/2}}-\frac{(a+x (b+c x))^{3/2} \left(\frac{\left(g h (2 f g-e h)-h \left(f g^2-d h^2\right)\right) \left(c x^2+b x+a\right)^{5/2}}{5 \left(c g^2-b h g+a h^2\right) (g+h x)^5}-\frac{\left(b \left(g h (2 f g-e h)+h \left(f g^2-d h^2\right)\right)-2 \left(a (2 f g-e h) h^2+c g \left(f g^2-d h^2\right)\right)\right) \left(\frac{(b g-2 a h+(2 c g-b h) x) \left(c x^2+b x+a\right)^{3/2}}{8 \left(c g^2-b h g+a h^2\right) (g+h x)^4}-\frac{3 \left(b^2-4 a c\right) \left(\frac{\sqrt{c x^2+b x+a} (b g-2 a h+(2 c g-b h) x)}{4 \left(c g^2-b h g+a h^2\right) (g+h x)^2}+\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-b g+2 a h-(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{2 \sqrt{c g^2-b h g+a h^2} \left(4 c g^2-4 b h g+4 a h^2\right)}\right)}{16 \left(c g^2-b h g+a h^2\right)}\right)}{2 \left(c g^2-b h g+a h^2\right)}\right)}{h^2 \left(c x^2+b x+a\right)^{3/2}}","-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{5/2}}{5 h \left(c g^2-b h g+a h^2\right) (g+h x)^5}-\frac{\left(16 c^2 f g^5-2 c h \left(13 b f g^3-10 a f h g^2+3 b d h^2 g-6 a d h^3\right) g-h^2 \left(-g \left(7 f g^2+3 h (e g+d h)\right) b^2+2 a h \left(f g^2+3 h (2 e g+d h)\right) b+4 a^2 h^2 (2 f g-3 e h)\right)+h \left(4 \left(7 f g^4-3 d g^2 h^2\right) c^2+2 g h \left(2 a h (14 f g-3 e h)-b \left(28 f g^2-3 e h g-6 d h^2\right)\right) c+h^2 \left(\left(25 f g^2-3 h (e g+d h)\right) b^2-2 a h (22 f g-3 e h) b+16 a^2 f h^2\right)\right) x\right) \left(c x^2+b x+a\right)^{3/2}}{48 h^3 \left(c g^2-b h g+a h^2\right)^2 (g+h x)^4}-\frac{\left(128 c^4 f g^7-32 c^3 f h (11 b g-10 a h) g^5+8 c^2 h^2 \left(38 b^2 f g^4-a b h \left(65 f g^2+3 d h^2\right) g+2 a^2 h^2 \left(13 f g^2+3 d h^2\right)\right) g-2 c h^3 \left(\left(35 f g^4-3 d g^2 h^2\right) b^3-2 a g^2 h (34 f g+3 e h) b^2+4 a^2 h^2 \left(5 f g^2+6 e h g+3 d h^2\right) b+8 a^3 h^3 (2 f g-3 e h)\right)-b h^4 (b g-2 a h) \left(\left(7 f g^2+3 h (e g+d h)\right) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right)+h \left(128 c f \left(c g^2-h (b g-a h)\right)^3+(2 c g-b h) \left(32 c^3 f g^5-8 c^2 h \left(10 b f g^3-11 a f h g^2+3 a d h^3\right) g+2 c h^2 \left(\left(29 f g^3+3 d h^2 g\right) b^2-6 a h \left(11 f g^2-e h g-d h^2\right) b+4 a^2 h^2 (10 f g-3 e h)\right)-b h^3 \left(\left(7 f g^2+3 h (e g+d h)\right) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right)\right)\right) x\right) \sqrt{c x^2+b x+a}}{128 h^5 \left(c g^2-b h g+a h^2\right)^3 (g+h x)^2}+\frac{c^{3/2} f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right)}{h^6}-\frac{\left(256 c^5 f g^7-896 c^4 f h (b g-a h) g^5+32 c^3 h^2 \left(35 b^2 f g^4-70 a b f h g^3+a^2 h^2 \left(35 f g^2-3 d h^2\right)\right) g-16 c^2 h^3 \left(35 b^3 f g^4-3 a b^2 h \left(35 f g^2+d h^2\right) g-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left(35 f g^2-e h g-d h^2\right)\right)+b^3 h^5 \left(\left(7 f g^2+3 h (e g+d h)\right) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right)-2 b c h^4 \left(-\left(\left(35 f g^3-3 d g h^2\right) b^3\right)+4 a h \left(35 f g^2+3 h (e g+d h)\right) b^2-24 a^2 h^2 (8 f g+e h) b+96 a^3 f h^3\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{256 h^6 \left(c g^2-b h g+a h^2\right)^{7/2}}",1,"-(((a + x*(b + c*x))^(3/2)*(((g*h*(2*f*g - e*h) - h*(f*g^2 - d*h^2))*(a + b*x + c*x^2)^(5/2))/(5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - ((-2*(a*h^2*(2*f*g - e*h) + c*g*(f*g^2 - d*h^2)) + b*(g*h*(2*f*g - e*h) + h*(f*g^2 - d*h^2)))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2*(c*g^2 - b*g*h + a*h^2))))/(h^2*(a + b*x + c*x^2)^(3/2))) + (f*(a + x*(b + c*x))^(3/2)*(-1/3*(a + b*x + c*x^2)^(3/2)/(h*(g + h*x)^3) + (-1/2*((-2*c*g + b*h)*(a + b*x + c*x^2)^(3/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - (((-(c*g*(2*c*g - b*h)) + (h*(2*b*c*g + b^2*h - 8*a*c*h))/2)*(a + b*x + c*x^2)^(3/2))/((-(c*g^2) + b*g*h - a*h^2)*(g + h*x)) + (((-(c*(2*c*g - (b*h)/2)*(4*c^2*g^2 - b^2*h^2 - 4*c*h*(b*g - 2*a*h))) + (c*h*(-10*b^2*c*g*h + 8*a*c^2*g*h - b^3*h^2 + 4*b*c*(2*c*g^2 + 3*a*h^2)))/2 + c^2*h*(4*c^2*g^2 - b^2*h^2 - 4*c*h*(b*g - 2*a*h))*x)*Sqrt[a + b*x + c*x^2])/(2*c*h^2) - ((-16*c^(5/2)*(c*g^2 - h*(b*g - a*h))^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h - (4*Sqrt[c*g^2 - b*g*h + a*h^2]*(-(c*h*(c*g^2 - b*g*h + a*h^2)*(8*b*c^2*g^2 - 6*b^2*c*g*h - 8*a*c^2*g*h - b^3*h^2 + 12*a*b*c*h^2)) + 16*c^3*g*(c*g^2 - h*(b*g - a*h))^2)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(h*(4*c*g^2 - 4*b*g*h + 4*a*h^2)))/(4*c*h^2))/(-(c*g^2) + b*g*h - a*h^2))/(2*(c*g^2 - b*g*h + a*h^2)))/(2*h)))/(h^2*(a + b*x + c*x^2)^(3/2))","A",1
206,1,766,657,6.2432761,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^7} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]","\frac{(a+x (b+c x))^{3/2} \left(-\frac{\frac{\left(a+b x+c x^2\right)^{5/2} \left(\frac{1}{2} c h \left(12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right)-c g \left(-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right)\right)}{5 (g+h x)^5 \left(a h^2-b g h+c g^2\right)}-\frac{\left(\frac{\left(a+b x+c x^2\right)^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left(a h^2-b g h+c g^2\right)}-\frac{3 \left(b^2-4 a c\right) \left(\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a h-x (2 c g-b h)-b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right)}{2 \sqrt{a h^2-b g h+c g^2} \left(4 a h^2-4 b g h+4 c g^2\right)}+\frac{\sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left(a h^2-b g h+c g^2\right)}\right)}{16 \left(a h^2-b g h+c g^2\right)}\right) \left(b \left(-c g \left(-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right)-\frac{1}{2} c h \left(12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right)\right)-2 \left(-\frac{1}{2} c^2 g \left(12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right)-a c h \left(-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right)\right)\right)}{2 \left(a h^2-b g h+c g^2\right)}}{6 \left(a h^2-b g h+c g^2\right)}-\frac{\left(a+b x+c x^2\right)^{5/2} \left(\frac{1}{2} h (-12 a f h+5 b f g+2 c d h)-\frac{1}{2} g (2 c (e h+5 f g)-7 b f h)\right)}{6 (g+h x)^6 \left(a h^2-b g h+c g^2\right)}\right)}{c h \left(a+b x+c x^2\right)^{3/2}}-\frac{f \left(a+b x+c x^2\right) (a+x (b+c x))^{3/2}}{c h (g+h x)^6}","\frac{\left(a+b x+c x^2\right)^{3/2} (-2 a h+x (2 c g-b h)+b g) \left(24 a^2 f h^2-4 c \left(a \left(d h^2-7 e g h+f g^2\right)+3 b g (2 d h+e g)\right)-12 a b h (e h+2 f g)+b^2 \left(7 d h^2+5 e g h+7 f g^2\right)+24 c^2 d g^2\right)}{192 (g+h x)^4 \left(a h^2-b g h+c g^2\right)^3}-\frac{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left(24 a^2 f h^2-4 c \left(a \left(d h^2-7 e g h+f g^2\right)+3 b g (2 d h+e g)\right)-12 a b h (e h+2 f g)+b^2 \left(7 d h^2+5 e g h+7 f g^2\right)+24 c^2 d g^2\right)}{512 (g+h x)^2 \left(a h^2-b g h+c g^2\right)^4}+\frac{\left(b^2-4 a c\right)^2 \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(24 a^2 f h^2-4 c \left(a \left(d h^2-7 e g h+f g^2\right)+3 b g (2 d h+e g)\right)-12 a b h (e h+2 f g)+b^2 \left(7 d h^2+5 e g h+7 f g^2\right)+24 c^2 d g^2\right)}{1024 \left(a h^2-b g h+c g^2\right)^{9/2}}+\frac{\left(a+b x+c x^2\right)^{5/2} \left(h \left(12 a h (2 f g-e h)-b \left(-7 d h^2-5 e g h+17 f g^2\right)\right)+2 c g \left(h (e g-7 d h)+5 f g^2\right)\right)}{60 h (g+h x)^5 \left(a h^2-b g h+c g^2\right)^2}-\frac{\left(a+b x+c x^2\right)^{5/2} \left(f g^2-h (e g-d h)\right)}{6 h (g+h x)^6 \left(a h^2-b g h+c g^2\right)}",1,"-((f*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(c*h*(g + h*x)^6)) + ((a + x*(b + c*x))^(3/2)*(-1/6*(((h*(5*b*f*g + 2*c*d*h - 12*a*f*h))/2 - (g*(-7*b*f*h + 2*c*(5*f*g + e*h)))/2)*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^6) - (((-(c*g*(5*c*f*g^2 - 6*f*h*(b*g - a*h) + c*h*(e*g - d*h))) + (c*h*(5*b*f*g^2 - b*h*(5*e*g + 7*d*h) + 12*h*(c*d*g - a*f*g + a*e*h)))/2)*(a + b*x + c*x^2)^(5/2))/(5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - ((-2*(-(a*c*h*(5*c*f*g^2 - 6*f*h*(b*g - a*h) + c*h*(e*g - d*h))) - (c^2*g*(5*b*f*g^2 - b*h*(5*e*g + 7*d*h) + 12*h*(c*d*g - a*f*g + a*e*h)))/2) + b*(-(c*g*(5*c*f*g^2 - 6*f*h*(b*g - a*h) + c*h*(e*g - d*h))) - (c*h*(5*b*f*g^2 - b*h*(5*e*g + 7*d*h) + 12*h*(c*d*g - a*f*g + a*e*h)))/2))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2*(c*g^2 - b*g*h + a*h^2)))/(6*(c*g^2 - b*g*h + a*h^2))))/(c*h*(a + b*x + c*x^2)^(3/2))","A",1
207,1,1221,1062,6.4186033,"\int \frac{\left(a+b x+c x^2\right)^{3/2} \left(d+e x+f x^2\right)}{(g+h x)^8} \, dx","Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]","\frac{(a+x (b+c x))^{3/2} \left(-\frac{\left(\frac{1}{2} h (5 b f g+4 c d h-14 a f h)-\frac{1}{2} g (10 c f g+4 c e h-9 b f h)\right) \left(c x^2+b x+a\right)^{5/2}}{7 \left(c g^2-b h g+a h^2\right) (g+h x)^7}-\frac{-\frac{\left(2 c g \left(5 c f g^2-7 f h (b g-a h)+2 c h (e g-d h)\right)-c h \left(5 b f g^2-b h (5 e g+9 d h)+14 h (c d g-a f g+a e h)\right)\right) \left(c x^2+b x+a\right)^{5/2}}{6 \left(c g^2-b h g+a h^2\right) (g+h x)^6}-\frac{\frac{\left(c^2 g \left(2 c \left(5 f g^3+h (2 e g-9 d h) g\right)-h \left(19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right)\right)-\frac{1}{2} c h \left(7 h \left(5 f g^2+h (5 e g+9 d h)\right) b^2+2 \left(5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right) b+24 h \left(7 c^2 d g^2+7 a^2 f h^2-a c \left(2 f g^2-h (9 e g-2 d h)\right)\right)\right)\right) \left(c x^2+b x+a\right)^{5/2}}{5 \left(c g^2-b h g+a h^2\right) (g+h x)^5}-\frac{\left(b \left(g \left(2 c \left(5 f g^3+h (2 e g-9 d h) g\right)-h \left(19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right)\right) c^2+\frac{1}{2} h \left(7 h \left(5 f g^2+h (5 e g+9 d h)\right) b^2+2 \left(5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right) b+24 h \left(7 c^2 d g^2+7 a^2 f h^2-a c \left(2 f g^2-h (9 e g-2 d h)\right)\right)\right) c\right)-2 \left(a h \left(2 c \left(5 f g^3+h (2 e g-9 d h) g\right)-h \left(19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right)\right) c^2+\frac{1}{2} g \left(7 h \left(5 f g^2+h (5 e g+9 d h)\right) b^2+2 \left(5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right) b+24 h \left(7 c^2 d g^2+7 a^2 f h^2-a c \left(2 f g^2-h (9 e g-2 d h)\right)\right)\right) c^2\right)\right) \left(\frac{(b g-2 a h+(2 c g-b h) x) \left(c x^2+b x+a\right)^{3/2}}{8 \left(c g^2-b h g+a h^2\right) (g+h x)^4}-\frac{3 \left(b^2-4 a c\right) \left(\frac{\sqrt{c x^2+b x+a} (b g-2 a h+(2 c g-b h) x)}{4 \left(c g^2-b h g+a h^2\right) (g+h x)^2}+\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{-b g+2 a h-(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{2 \sqrt{c g^2-b h g+a h^2} \left(4 c g^2-4 b h g+4 a h^2\right)}\right)}{16 \left(c g^2-b h g+a h^2\right)}\right)}{2 \left(c g^2-b h g+a h^2\right)}}{6 \left(c g^2-b h g+a h^2\right)}}{7 \left(c g^2-b h g+a h^2\right)}\right)}{2 c h \left(c x^2+b x+a\right)^{3/2}}-\frac{f \left(c x^2+b x+a\right) (a+x (b+c x))^{3/2}}{2 c h (g+h x)^7}","\frac{\left(4 c^2 \left(5 f g^2+h (2 e g-51 d h)\right) g^2-7 h^2 \left(\left(5 f g^2+5 e h g+9 d h^2\right) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right)-2 c h \left(3 b g \left(8 f g^2-15 e h g-34 d h^2\right)-2 a h \left(26 f g^2-61 e h g+12 d h^2\right)\right)\right) \left(c x^2+b x+a\right)^{5/2}}{840 h \left(c g^2-b h g+a h^2\right)^3 (g+h x)^5}+\frac{\left(2 c g \left(5 f g^2+h (2 e g-9 d h)\right)+h \left(14 a h (2 f g-e h)-b \left(19 f g^2-5 e h g-9 d h^2\right)\right)\right) \left(c x^2+b x+a\right)^{5/2}}{84 h \left(c g^2-b h g+a h^2\right)^2 (g+h x)^6}-\frac{\left(f g^2-h (e g-d h)\right) \left(c x^2+b x+a\right)^{5/2}}{7 h \left(c g^2-b h g+a h^2\right) (g+h x)^7}+\frac{\left(48 c^3 d g^3-8 c^2 \left(3 b g (e g+3 d h)+a \left(f g^2-8 e h g+3 d h^2\right)\right) g-b h \left(\left(5 f g^2+5 e h g+9 d h^2\right) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right)+2 c \left(g \left(7 f g^2+10 e h g+21 d h^2\right) b^2-2 a h \left(13 f g^2+13 e h g-3 d h^2\right) b+4 a^2 h^2 (8 f g-e h)\right)\right) (b g-2 a h+(2 c g-b h) x) \left(c x^2+b x+a\right)^{3/2}}{384 \left(c g^2-b h g+a h^2\right)^4 (g+h x)^4}-\frac{\left(b^2-4 a c\right) \left(48 c^3 d g^3-8 c^2 \left(3 b g (e g+3 d h)+a \left(f g^2-8 e h g+3 d h^2\right)\right) g-b h \left(\left(5 f g^2+5 e h g+9 d h^2\right) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right)+2 c \left(g \left(7 f g^2+10 e h g+21 d h^2\right) b^2-2 a h \left(13 f g^2+13 e h g-3 d h^2\right) b+4 a^2 h^2 (8 f g-e h)\right)\right) (b g-2 a h+(2 c g-b h) x) \sqrt{c x^2+b x+a}}{1024 \left(c g^2-b h g+a h^2\right)^5 (g+h x)^2}+\frac{\left(b^2-4 a c\right)^2 \left(48 c^3 d g^3-8 c^2 \left(3 b g (e g+3 d h)+a \left(f g^2-8 e h g+3 d h^2\right)\right) g-b h \left(\left(5 f g^2+5 e h g+9 d h^2\right) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right)+2 c \left(g \left(7 f g^2+10 e h g+21 d h^2\right) b^2-2 a h \left(13 f g^2+13 e h g-3 d h^2\right) b+4 a^2 h^2 (8 f g-e h)\right)\right) \tanh ^{-1}\left(\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right)}{2048 \left(c g^2-b h g+a h^2\right)^{11/2}}",1,"-1/2*(f*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(c*h*(g + h*x)^7) + ((a + x*(b + c*x))^(3/2)*(-1/7*(((h*(5*b*f*g + 4*c*d*h - 14*a*f*h))/2 - (g*(10*c*f*g + 4*c*e*h - 9*b*f*h))/2)*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^7) - (-1/6*((2*c*g*(5*c*f*g^2 - 7*f*h*(b*g - a*h) + 2*c*h*(e*g - d*h)) - c*h*(5*b*f*g^2 - b*h*(5*e*g + 9*d*h) + 14*h*(c*d*g - a*f*g + a*e*h)))*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*h^2)*(g + h*x)^6) - (((c^2*g*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g - e*h))) - (c*h*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c*f*g^3 - c*g*h*(40*e*g + 93*d*h) - 7*a*h^2*(10*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2 - h*(9*e*g - 2*d*h)))))/2)*(a + b*x + c*x^2)^(5/2))/(5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - ((-2*(a*c^2*h*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g - e*h))) + (c^2*g*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c*f*g^3 - c*g*h*(40*e*g + 93*d*h) - 7*a*h^2*(10*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2 - h*(9*e*g - 2*d*h)))))/2) + b*(c^2*g*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g - e*h))) + (c*h*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c*f*g^3 - c*g*h*(40*e*g + 93*d*h) - 7*a*h^2*(10*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2 - h*(9*e*g - 2*d*h)))))/2))*(((b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2*(c*g^2 - b*g*h + a*h^2)))/(6*(c*g^2 - b*g*h + a*h^2)))/(7*(c*g^2 - b*g*h + a*h^2))))/(2*c*h*(a + b*x + c*x^2)^(3/2))","A",1
208,1,70,143,0.0499655,"\int (1+2 x)^3 \sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^3*Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(2488320 x^6+6462720 x^5+7491456 x^4+5497776 x^3+3280872 x^2+1493894 x-543069\right)-4341365 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{3265920}","\frac{2}{21} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^4+\frac{67}{378} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^3+\frac{17}{105} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^2-\frac{(26982 x+75295) \left(3 x^2-x+2\right)^{3/2}}{68040}+\frac{5393 (1-6 x) \sqrt{3 x^2-x+2}}{15552}+\frac{124039 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{31104 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(-543069 + 1493894*x + 3280872*x^2 + 5497776*x^3 + 7491456*x^4 + 6462720*x^5 + 2488320*x^6) - 4341365*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/3265920","A",1
209,1,65,118,0.0375512,"\int (1+2 x)^2 \sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^2*Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(17280 x^5+35712 x^4+33552 x^3+22344 x^2+14638 x+5607\right)-27025 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{38880}","\frac{1}{9} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^3+\frac{1}{5} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^2+\frac{1}{810} (306 x+25) \left(3 x^2-x+2\right)^{3/2}+\frac{235 (1-6 x) \sqrt{3 x^2-x+2}}{1296}+\frac{5405 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{2592 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(5607 + 14638*x + 22344*x^2 + 33552*x^3 + 35712*x^4 + 17280*x^5) - 27025*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/38880","A",1
210,1,60,93,0.0284859,"\int (1+2 x) \sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)*Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(20736 x^4+31536 x^3+24072 x^2+17374 x+15471\right)-2185 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{77760}","\frac{2}{15} \left(3 x^2-x+2\right)^{3/2} (2 x+1)^2+\frac{(738 x+745) \left(3 x^2-x+2\right)^{3/2}}{1620}+\frac{19 (1-6 x) \sqrt{3 x^2-x+2}}{2592}+\frac{437 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{5184 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(15471 + 17374*x + 24072*x^2 + 31536*x^3 + 20736*x^4) - 2185*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/77760","A",1
211,1,86,101,0.0538521,"\int \frac{\sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right)}{1+2 x} \, dx","Integrate[(Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2))/(1 + 2*x),x]","\frac{1}{432} \left(6 \sqrt{3 x^2-x+2} \left(48 x^2+14 x+45\right)-54 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+43 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)\right)","\frac{2}{9} \left(3 x^2-x+2\right)^{3/2}+\frac{1}{72} (30 x+13) \sqrt{3 x^2-x+2}-\frac{1}{8} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)-\frac{43 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{144 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(45 + 14*x + 48*x^2) + 43*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 54*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/432","A",1
212,1,92,108,0.0863927,"\int \frac{\sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right)}{(1+2 x)^2} \, dx","Integrate[(Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]","\frac{\sqrt{3 x^2-x+2} \left(12 x^2-2 x-7\right)}{24 x+12}+\frac{17 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{8 \sqrt{13}}+\frac{11 \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{6 \sqrt{3}}","-\frac{\left(3 x^2-x+2\right)^{3/2}}{13 (2 x+1)}-\frac{1}{156} (67-96 x) \sqrt{3 x^2-x+2}+\frac{17 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{8 \sqrt{13}}-\frac{11 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{6 \sqrt{3}}",1,"(Sqrt[2 - x + 3*x^2]*(-7 - 2*x + 12*x^2))/(12 + 24*x) + (11*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(6*Sqrt[3]) + (17*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(8*Sqrt[13])","A",1
213,1,93,115,0.085498,"\int \frac{\sqrt{2-x+3 x^2} \left(1+3 x+4 x^2\right)}{(1+2 x)^3} \, dx","Integrate[(Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2))/(1 + 2*x)^3,x]","\frac{\frac{78 \sqrt{3 x^2-x+2} \left(208 x^2+268 x+69\right)}{(2 x+1)^2}-2409 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)-3718 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{8112}","-\frac{\left(3 x^2-x+2\right)^{3/2}}{26 (2 x+1)^2}+\frac{11 (10 x+7) \sqrt{3 x^2-x+2}}{104 (2 x+1)}-\frac{803 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{208 \sqrt{13}}+\frac{11 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{8 \sqrt{3}}",1,"((78*Sqrt[2 - x + 3*x^2]*(69 + 268*x + 208*x^2))/(1 + 2*x)^2 - 3718*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 2409*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/8112","A",1
214,1,80,158,0.0502214,"\int (1+2 x)^3 \left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^3*(2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(238878720 x^8+510105600 x^7+635765760 x^6+711210240 x^5+649452672 x^4+421626672 x^3+201289704 x^2+84014278 x+12499587\right)-144398485 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{134369280}","\frac{2}{27} \left(3 x^2-x+2\right)^{5/2} (2 x+1)^4+\frac{913}{486} x^2 \left(3 x^2-x+2\right)^{5/2}-\frac{11 (283-5850 x) \left(3 x^2-x+2\right)^{5/2}}{58320}+\frac{54593 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{559872}+\frac{1255639 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}+\frac{77}{81} x^3 \left(3 x^2-x+2\right)^{5/2}+\frac{28879697 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{8957952 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(12499587 + 84014278*x + 201289704*x^2 + 421626672*x^3 + 649452672*x^4 + 711210240*x^5 + 635765760*x^6 + 510105600*x^7 + 238878720*x^8) - 144398485*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/134369280","A",1
215,1,75,141,0.0455872,"\int (1+2 x)^2 \left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(5806080 x^7+9262080 x^6+10656000 x^5+12173952 x^4+10119792 x^3+5694024 x^2+2735918 x+1517367\right)-1684865 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{5806080}","\frac{1}{12} \left(3 x^2-x+2\right)^{5/2} (2 x+1)^3+\frac{8}{63} \left(3 x^2-x+2\right)^{5/2} (2 x+1)^2+\frac{13 (50 x+29) \left(3 x^2-x+2\right)^{5/2}}{2520}+\frac{91 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{3456}+\frac{2093 (1-6 x) \sqrt{3 x^2-x+2}}{27648}+\frac{48139 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{55296 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(1517367 + 2735918*x + 5694024*x^2 + 10119792*x^3 + 12173952*x^4 + 10656000*x^5 + 9262080*x^6 + 5806080*x^7) - 1684865*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/5806080","A",1
216,1,70,116,0.0365799,"\int (1+2 x) \left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)*(2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(497664 x^6+518400 x^5+653184 x^4+744336 x^3+531384 x^2+275410 x+203337\right)+262913 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{870912}","\frac{2}{21} (2 x+1)^2 \left(3 x^2-x+2\right)^{5/2}+\frac{1}{378} (102 x+109) \left(3 x^2-x+2\right)^{5/2}-\frac{71 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{2592}-\frac{1633 (1-6 x) \sqrt{3 x^2-x+2}}{20736}-\frac{37559 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{41472 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(203337 + 275410*x + 531384*x^2 + 744336*x^3 + 653184*x^4 + 518400*x^5 + 497664*x^6) + 262913*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/870912","A",1
217,1,96,124,0.0615363,"\int \frac{\left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right)}{1+2 x} \, dx","Integrate[((2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2))/(1 + 2*x),x]","\frac{-14040 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+6 \sqrt{3 x^2-x+2} \left(6912 x^4-1008 x^3+9624 x^2+1058 x+7977\right)-11015 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{34560}","\frac{2}{15} \left(3 x^2-x+2\right)^{5/2}+\frac{1}{144} (30 x+7) \left(3 x^2-x+2\right)^{3/2}+\frac{(402 x+869) \sqrt{3 x^2-x+2}}{1152}-\frac{13}{32} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{2203 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{2304 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(7977 + 1058*x + 9624*x^2 - 1008*x^3 + 6912*x^4) - 11015*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 14040*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/34560","A",1
218,1,103,131,0.0916591,"\int \frac{\left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right)}{(1+2 x)^2} \, dx","Integrate[((2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]","\frac{900 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{6 \sqrt{3 x^2-x+2} \left(288 x^4-96 x^3+564 x^2-332 x-493\right)}{2 x+1}+2327 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{1152}","-\frac{\left(3 x^2-x+2\right)^{5/2}}{13 (2 x+1)}-\frac{1}{104} (23-38 x) \left(3 x^2-x+2\right)^{3/2}-\frac{1}{192} (349-294 x) \sqrt{3 x^2-x+2}+\frac{25}{32} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)-\frac{2327 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{384 \sqrt{3}}",1,"((6*Sqrt[2 - x + 3*x^2]*(-493 - 332*x + 564*x^2 - 96*x^3 + 288*x^4))/(1 + 2*x) + 2327*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] + 900*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/1152","A",1
219,1,103,138,0.1068523,"\int \frac{\left(2-x+3 x^2\right)^{3/2} \left(1+3 x+4 x^2\right)}{(1+2 x)^3} \, dx","Integrate[((2 - x + 3*x^2)^(3/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^3,x]","\frac{-10377 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{156 \sqrt{3 x^2-x+2} \left(96 x^4-68 x^3+390 x^2+627 x+182\right)}{(2 x+1)^2}-19747 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{7488}","-\frac{\left(3 x^2-x+2\right)^{5/2}}{26 (2 x+1)^2}+\frac{(122 x+151) \left(3 x^2-x+2\right)^{3/2}}{312 (2 x+1)}+\frac{1}{624} (1858-771 x) \sqrt{3 x^2-x+2}-\frac{1153 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{64 \sqrt{13}}+\frac{1519 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{192 \sqrt{3}}",1,"((156*Sqrt[2 - x + 3*x^2]*(182 + 627*x + 390*x^2 - 68*x^3 + 96*x^4))/(1 + 2*x)^2 - 19747*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 10377*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/7488","A",1
220,1,90,189,0.0623085,"\int (1+2 x)^3 \left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^3*(2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(120394874880 x^{10}+207681159168 x^9+308846297088 x^8+419978151936 x^7+415908006912 x^6+347247744768 x^5+263636134272 x^4+161269204752 x^3+72088585464 x^2+26646633218 x+9173509857\right)-23838377255 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{27590492160}","\frac{2}{33} \left(3 x^2-x+2\right)^{7/2} (2 x+1)^4+\frac{29}{330} \left(3 x^2-x+2\right)^{7/2} (2 x+1)^3+\frac{133 \left(3 x^2-x+2\right)^{7/2} (2 x+1)^2}{1485}-\frac{(26353-21350 x) \left(3 x^2-x+2\right)^{7/2}}{498960}+\frac{5089 (1-6 x) \left(3 x^2-x+2\right)^{5/2}}{155520}+\frac{117047 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{1492992}+\frac{2692081 (1-6 x) \sqrt{3 x^2-x+2}}{11943936}+\frac{61917863 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{23887872 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(9173509857 + 26646633218*x + 72088585464*x^2 + 161269204752*x^3 + 263636134272*x^4 + 347247744768*x^5 + 415908006912*x^6 + 419978151936*x^7 + 308846297088*x^8 + 207681159168*x^9 + 120394874880*x^10) - 23838377255*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/27590492160","A",1
221,1,85,164,0.0560236,"\int (1+2 x)^2 \left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)^2*(2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(2257403904 x^9+2675441664 x^8+4427716608 x^7+5671627776 x^6+4996802304 x^5+4171579776 x^4+3096104976 x^3+1693765752 x^2+692659234 x+387182961\right)+124772585 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{940584960}","\frac{1}{15} (2 x+1)^3 \left(3 x^2-x+2\right)^{7/2}+\frac{37}{405} (2 x+1)^2 \left(3 x^2-x+2\right)^{7/2}+\frac{(3430 x+2731) \left(3 x^2-x+2\right)^{7/2}}{17010}-\frac{293 (1-6 x) \left(3 x^2-x+2\right)^{5/2}}{58320}-\frac{6739 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{559872}-\frac{154997 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}-\frac{3564931 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{8957952 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(387182961 + 692659234*x + 1693765752*x^2 + 3096104976*x^3 + 4171579776*x^4 + 4996802304*x^5 + 5671627776*x^6 + 4427716608*x^7 + 2675441664*x^8 + 2257403904*x^9) + 124772585*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/940584960","A",1
222,1,80,139,0.0479097,"\int (1+2 x) \left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right) \, dx","Integrate[(1 + 2*x)*(2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2),x]","\frac{6 \sqrt{3 x^2-x+2} \left(47775744 x^8+30357504 x^7+79377408 x^6+80034048 x^5+66969216 x^4+58946544 x^3+41031048 x^2+19860062 x+10960335\right)+27071575 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{35831808}","\frac{2}{27} (2 x+1)^2 \left(3 x^2-x+2\right)^{7/2}+\frac{1}{648} (122 x+137) \left(3 x^2-x+2\right)^{7/2}-\frac{445 (1-6 x) \left(3 x^2-x+2\right)^{5/2}}{15552}-\frac{51175 (1-6 x) \left(3 x^2-x+2\right)^{3/2}}{746496}-\frac{1177025 (1-6 x) \sqrt{3 x^2-x+2}}{5971968}-\frac{27071575 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{11943936 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(10960335 + 19860062*x + 41031048*x^2 + 58946544*x^3 + 66969216*x^4 + 80034048*x^5 + 79377408*x^6 + 30357504*x^7 + 47775744*x^8) + 27071575*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/35831808","A",1
223,1,106,147,0.0750522,"\int \frac{\left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right)}{1+2 x} \, dx","Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x),x]","\frac{-22997520 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+6 \sqrt{3 x^2-x+2} \left(7464960 x^6-3836160 x^5+15700608 x^4-3646512 x^3+12466776 x^2-2120998 x+11665053\right)-33058235 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{17418240}","\frac{2}{21} \left(3 x^2-x+2\right)^{7/2}+\frac{(150 x+29) \left(3 x^2-x+2\right)^{5/2}}{1080}+\frac{(2154 x+2449) \left(3 x^2-x+2\right)^{3/2}}{10368}+\frac{(221999-17850 x) \sqrt{3 x^2-x+2}}{82944}-\frac{169}{128} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{944521 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{165888 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(11665053 - 2120998*x + 12466776*x^2 - 3646512*x^3 + 15700608*x^4 - 3836160*x^5 + 7464960*x^6) - 33058235*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 22997520*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/17418240","A",1
224,1,113,154,0.1159858,"\int \frac{\left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right)}{(1+2 x)^2} \, dx","Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]","\frac{694980 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{6 \sqrt{3 x^2-x+2} \left(103680 x^6-65664 x^5+251424 x^4-115680 x^3+310660 x^2-322972 x-364257\right)}{2 x+1}+1578115 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{207360}","-\frac{\left(3 x^2-x+2\right)^{7/2}}{13 (2 x+1)}-\frac{11 (37-60 x) \left(3 x^2-x+2\right)^{5/2}}{2340}-\frac{11}{864} (67-78 x) \left(3 x^2-x+2\right)^{3/2}-\frac{11 (4727-3090 x) \sqrt{3 x^2-x+2}}{6912}+\frac{429}{128} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)-\frac{315623 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{13824 \sqrt{3}}",1,"((6*Sqrt[2 - x + 3*x^2]*(-364257 - 322972*x + 310660*x^2 - 115680*x^3 + 251424*x^4 - 65664*x^5 + 103680*x^6))/(1 + 2*x) + 1578115*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] + 694980*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/207360","A",1
225,1,113,161,0.1144537,"\int \frac{\left(2-x+3 x^2\right)^{5/2} \left(1+3 x+4 x^2\right)}{(1+2 x)^3} \, dx","Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^3,x]","\frac{-293580 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{6 \sqrt{3 x^2-x+2} \left(27648 x^6-22464 x^5+83616 x^4-76200 x^3+256564 x^2+464446 x+142057\right)}{(2 x+1)^2}-592115 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{46080}","-\frac{\left(3 x^2-x+2\right)^{7/2}}{26 (2 x+1)^2}+\frac{(134 x+257) \left(3 x^2-x+2\right)^{5/2}}{520 (2 x+1)}+\frac{1}{832} (1227-838 x) \left(3 x^2-x+2\right)^{3/2}+\frac{(21317-10470 x) \sqrt{3 x^2-x+2}}{1536}-\frac{1631}{256} \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)+\frac{118423 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{3072 \sqrt{3}}",1,"((6*Sqrt[2 - x + 3*x^2]*(142057 + 464446*x + 256564*x^2 - 76200*x^3 + 83616*x^4 - 22464*x^5 + 27648*x^6))/(1 + 2*x)^2 - 592115*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]] - 293580*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/46080","A",1
226,1,588,693,1.2505821,"\int \frac{(g+h x)^3 \left(d+e x+f x^2\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((g + h*x)^3*(d + e*x + f*x^2))/Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} \left(4 c^2 h \left(256 a^2 f h^2+2 a b h (275 e h+825 f g+161 f h x)+b^2 \left(25 h (12 d h+36 e g+7 e h x)+3 f \left(300 g^2+175 g h x+42 h^2 x^2\right)\right)\right)-210 b^2 c h^2 (14 a f h+5 b e h+3 b f (5 g+h x))-16 c^3 \left(a h \left(5 h (16 d h+48 e g+9 e h x)+f \left(240 g^2+135 g h x+32 h^2 x^2\right)\right)+b \left(5 h \left(2 d h (27 g+5 h x)+e \left(54 g^2+30 g h x+7 h^2 x^2\right)\right)+3 f \left(30 g^3+50 g^2 h x+35 g h^2 x^2+9 h^3 x^3\right)\right)\right)+945 b^4 f h^3+32 c^4 \left(10 d h \left(18 g^2+9 g h x+2 h^2 x^2\right)+15 e \left(4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right)+3 f x \left(10 g^3+20 g^2 h x+15 g h^2 x^2+4 h^3 x^3\right)\right)\right)}{1920 c^5}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(96 c^3 \left(a^2 h^2 (e h+3 f g)+2 a b h \left(h (d h+3 e g)+3 f g^2\right)+b^2 g \left(3 h (d h+e g)+f g^2\right)\right)-80 b c^2 h \left(3 a^2 f h^2+3 a b h (e h+3 f g)+b^2 \left(d h^2+3 e g h+3 f g^2\right)\right)+70 b^3 c h^2 (4 a f h+b e h+3 b f g)-128 c^4 g \left(3 a h (d h+e g)+a f g^2+b g (3 d h+e g)\right)-63 b^5 f h^3+256 c^5 d g^3\right)}{256 c^{11/2}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(96 c^3 \left(a^2 h^2 (e h+3 f g)+2 a b h \left(h (d h+3 e g)+3 f g^2\right)+b^2 g \left(3 h (d h+e g)+f g^2\right)\right)-80 b c^2 h \left(3 a^2 f h^2+3 a b h (e h+3 f g)+b^2 \left(d h^2+3 e g h+3 f g^2\right)\right)+70 b^3 c h^2 (4 a f h+b e h+3 b f g)-128 c^4 g \left(a \left(3 h (d h+e g)+f g^2\right)+b g (3 d h+e g)\right)-63 b^5 f h^3+256 c^5 d g^3\right)}{256 c^{11/2}}+\frac{\sqrt{a+b x+c x^2} \left(8 c^2 h^2 \left(128 a^2 f h^2+275 a b h (e h+3 f g)+3 b^2 \left(50 h (d h+3 e g)+129 f g^2\right)\right)-2 c h x \left(8 c^2 h \left(a h (45 e h+71 f g)+b \left(50 d h^2+80 e g h+21 f g^2\right)\right)-14 b c h^2 (46 a f h+25 b e h+39 b f g)+315 b^3 f h^3+16 c^3 g \left(3 f g^2-5 h (10 d h+3 e g)\right)\right)-210 b^2 c h^3 (14 a f h+5 b (e h+3 f g))-16 c^3 h \left(16 a h \left(5 h (d h+3 e g)+13 f g^2\right)+b g \left(5 h (54 d h+47 e g)+39 f g^2\right)\right)+945 b^4 f h^4-64 c^4 g^2 \left(3 f g^2-5 h (16 d h+3 e g)\right)\right)}{1920 c^5 h}+\frac{(g+h x)^2 \sqrt{a+b x+c x^2} \left(-2 c h (32 a f h+35 b e h+24 b f g)+63 b^2 f h^2-\left(c^2 \left(12 f g^2-20 h (4 d h+3 e g)\right)\right)\right)}{240 c^3 h}-\frac{(g+h x)^3 \sqrt{a+b x+c x^2} (9 b f h+2 c (f g-5 e h))}{40 c^2 h}+\frac{f (g+h x)^4 \sqrt{a+b x+c x^2}}{5 c h}",1,"(Sqrt[a + x*(b + c*x)]*(945*b^4*f*h^3 - 210*b^2*c*h^2*(5*b*e*h + 14*a*f*h + 3*b*f*(5*g + h*x)) + 32*c^4*(10*d*h*(18*g^2 + 9*g*h*x + 2*h^2*x^2) + 15*e*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3) + 3*f*x*(10*g^3 + 20*g^2*h*x + 15*g*h^2*x^2 + 4*h^3*x^3)) + 4*c^2*h*(256*a^2*f*h^2 + 2*a*b*h*(825*f*g + 275*e*h + 161*f*h*x) + b^2*(25*h*(36*e*g + 12*d*h + 7*e*h*x) + 3*f*(300*g^2 + 175*g*h*x + 42*h^2*x^2))) - 16*c^3*(a*h*(5*h*(48*e*g + 16*d*h + 9*e*h*x) + f*(240*g^2 + 135*g*h*x + 32*h^2*x^2)) + b*(3*f*(30*g^3 + 50*g^2*h*x + 35*g*h^2*x^2 + 9*h^3*x^3) + 5*h*(2*d*h*(27*g + 5*h*x) + e*(54*g^2 + 30*g*h*x + 7*h^2*x^2))))))/(1920*c^5) + ((256*c^5*d*g^3 - 63*b^5*f*h^3 + 70*b^3*c*h^2*(3*b*f*g + b*e*h + 4*a*f*h) - 128*c^4*g*(a*f*g^2 + 3*a*h*(e*g + d*h) + b*g*(e*g + 3*d*h)) - 80*b*c^2*h*(3*a^2*f*h^2 + 3*a*b*h*(3*f*g + e*h) + b^2*(3*f*g^2 + 3*e*g*h + d*h^2)) + 96*c^3*(a^2*h^2*(3*f*g + e*h) + b^2*g*(f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(3*f*g^2 + h*(3*e*g + d*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(11/2))","A",1
227,1,343,420,0.6494528,"\int \frac{(g+h x)^2 \left(d+e x+f x^2\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((g + h*x)^2*(d + e*x + f*x^2))/Sqrt[a + b*x + c*x^2],x]","\frac{3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(48 c^2 \left(a^2 f h^2+2 a b h (e h+2 f g)+b^2 \left(h (d h+2 e g)+f g^2\right)\right)-40 b^2 c h (3 a f h+b e h+2 b f g)-64 c^3 \left(a h (d h+2 e g)+a f g^2+b g (2 d h+e g)\right)+35 b^4 f h^2+128 c^4 d g^2\right)+2 \sqrt{c} \sqrt{a+x (b+c x)} \left(-8 c^2 \left(a h (16 e h+32 f g+9 f h x)+2 b h (9 d h+18 e g+5 e h x)+b f \left(18 g^2+20 g h x+7 h^2 x^2\right)\right)+10 b c h (22 a f h+b (12 e h+24 f g+7 f h x))-105 b^3 f h^2+16 c^3 \left(6 d h (4 g+h x)+4 e \left(3 g^2+3 g h x+h^2 x^2\right)+f x \left(6 g^2+8 g h x+3 h^2 x^2\right)\right)\right)}{384 c^{9/2}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(48 c^2 \left(a^2 f h^2+2 a b h (e h+2 f g)+b^2 \left(d h^2+2 e g h+f g^2\right)\right)-40 b^2 c h (3 a f h+b e h+2 b f g)-64 c^3 \left(a \left(d h^2+2 e g h+f g^2\right)+b g (2 d h+e g)\right)+35 b^4 f h^2+128 c^4 d g^2\right)}{128 c^{9/2}}-\frac{\sqrt{a+b x+c x^2} \left(-2 c h x \left(-4 c h (9 a f h+10 b e h+6 b f g)+35 b^2 f h^2-8 c^2 \left(f g^2-2 h (3 d h+2 e g)\right)\right)+8 c^2 h \left(16 a h (e h+2 f g)+b \left(18 h (d h+2 e g)+11 f g^2\right)\right)-20 b c h^2 (11 a f h+6 b (e h+2 f g))+105 b^3 f h^3+32 c^3 g \left(f g^2-4 h (3 d h+e g)\right)\right)}{192 c^4 h}-\frac{(g+h x)^2 \sqrt{a+b x+c x^2} (7 b f h-8 c e h+2 c f g)}{24 c^2 h}+\frac{f (g+h x)^3 \sqrt{a+b x+c x^2}}{4 c h}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^3*f*h^2 + 10*b*c*h*(22*a*f*h + b*(24*f*g + 12*e*h + 7*f*h*x)) + 16*c^3*(6*d*h*(4*g + h*x) + 4*e*(3*g^2 + 3*g*h*x + h^2*x^2) + f*x*(6*g^2 + 8*g*h*x + 3*h^2*x^2)) - 8*c^2*(2*b*h*(18*e*g + 9*d*h + 5*e*h*x) + a*h*(32*f*g + 16*e*h + 9*f*h*x) + b*f*(18*g^2 + 20*g*h*x + 7*h^2*x^2))) + 3*(128*c^4*d*g^2 + 35*b^4*f*h^2 - 40*b^2*c*h*(2*b*f*g + b*e*h + 3*a*f*h) - 64*c^3*(a*f*g^2 + a*h*(2*e*g + d*h) + b*g*(e*g + 2*d*h)) + 48*c^2*(a^2*f*h^2 + 2*a*b*h*(2*f*g + e*h) + b^2*(f*g^2 + h*(2*e*g + d*h))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(384*c^(9/2))","A",1
228,1,215,223,0.2341874,"\int \frac{(g+h x) \left(d+e x+f x^2\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((g + h*x)*(d + e*x + f*x^2))/Sqrt[a + b*x + c*x^2],x]","\frac{-\frac{3 h \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(8 c^2 (a e h+a f g+b d h+b e g)-6 b c (2 a f h+b e h+b f g)+5 b^3 f h-16 c^3 d g\right)}{16 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} \left(-2 c h (8 a f h+b (9 e h+9 f g+5 f h x))+15 b^2 f h^2-4 c^2 (f g (2 g+h x)-3 h (2 d h+2 e g+e h x))\right)}{8 c^2}+f (g+h x)^2 \sqrt{a+x (b+c x)}}{3 c h}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-8 c^2 (a e h+a f g+b d h+b e g)+6 b c (2 a f h+b e h+b f g)-5 b^3 f h+16 c^3 d g\right)}{16 c^{7/2}}+\frac{\sqrt{a+b x+c x^2} \left(-2 c h (8 a f h+9 b (e h+f g))+15 b^2 f h^2-2 c h x (5 b f h-6 c e h+2 c f g)-8 c^2 \left(f g^2-3 h (d h+e g)\right)\right)}{24 c^3 h}+\frac{f (g+h x)^2 \sqrt{a+b x+c x^2}}{3 c h}",1,"(f*(g + h*x)^2*Sqrt[a + x*(b + c*x)] + (Sqrt[a + x*(b + c*x)]*(15*b^2*f*h^2 - 4*c^2*(f*g*(2*g + h*x) - 3*h*(2*e*g + 2*d*h + e*h*x)) - 2*c*h*(8*a*f*h + b*(9*f*g + 9*e*h + 5*f*h*x))))/(8*c^2) - (3*h*(-16*c^3*d*g + 5*b^3*f*h + 8*c^2*(b*e*g + a*f*g + b*d*h + a*e*h) - 6*b*c*(b*f*g + b*e*h + 2*a*f*h))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(16*c^(5/2)))/(3*c*h)","A",1
229,1,96,116,0.1519132,"\int \frac{d+e x+f x^2}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/Sqrt[a + b*x + c*x^2],x]","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-4 c (a f+b e)+3 b^2 f+8 c^2 d\right)}{8 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} (-3 b f+4 c e+2 c f x)}{4 c^2}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c (a f+b e)+3 b^2 f+8 c^2 d\right)}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c e-3 b f)}{4 c^2}+\frac{f x \sqrt{a+b x+c x^2}}{2 c}",1,"((4*c*e - 3*b*f + 2*c*f*x)*Sqrt[a + x*(b + c*x)])/(4*c^2) + ((8*c^2*d + 3*b^2*f - 4*c*(b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(5/2))","A",1
230,1,172,179,0.2812151,"\int \frac{d+e x+f x^2}{(g+h x) \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)*Sqrt[a + b*x + c*x^2]),x]","-\frac{\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) (b f h-2 c e h+2 c f g)}{c^{3/2}}+\frac{2 \left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{\sqrt{h (a h-b g)+c g^2}}-\frac{2 f h \sqrt{a+x (b+c x)}}{c}}{2 h^2}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (b f h-2 c e h+2 c f g)}{2 c^{3/2} h^2}+\frac{\left(f g^2-h (e g-d h)\right) \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right)}{h^2 \sqrt{a h^2-b g h+c g^2}}+\frac{f \sqrt{a+b x+c x^2}}{c h}",1,"-1/2*((-2*f*h*Sqrt[a + x*(b + c*x)])/c + ((2*c*f*g - 2*c*e*h + b*f*h)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (2*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/Sqrt[c*g^2 + h*(-(b*g) + a*h)])/h^2","A",1
231,1,227,241,0.3476139,"\int \frac{d+e x+f x^2}{(g+h x)^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^2*Sqrt[a + b*x + c*x^2]),x]","\frac{-\frac{h \sqrt{a+x (b+c x)} \left(h (d h-e g)+f g^2\right)}{(g+h x) \left(h (a h-b g)+c g^2\right)}+\frac{\tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(h \left(-2 a h (e h-2 f g)+b h (d h+e g)-3 b f g^2\right)+2 c \left(f g^3-d g h^2\right)\right)}{2 \left(h (a h-b g)+c g^2\right)^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}}{h^2}","-\frac{\sqrt{a+b x+c x^2} \left(f g^2-h (e g-d h)\right)}{h (g+h x) \left(a h^2-b g h+c g^2\right)}-\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(h \left(2 a h (2 f g-e h)-b \left(-d h^2-e g h+3 f g^2\right)\right)+2 c \left(f g^3-d g h^2\right)\right)}{2 h^2 \left(a h^2-b g h+c g^2\right)^{3/2}}+\frac{f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c} h^2}",1,"(-((h*(f*g^2 + h*(-(e*g) + d*h))*Sqrt[a + x*(b + c*x)])/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x))) + (f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + ((2*c*(f*g^3 - d*g*h^2) + h*(-3*b*f*g^2 + b*h*(e*g + d*h) - 2*a*h*(-2*f*g + e*h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(2*(c*g^2 + h*(-(b*g) + a*h))^(3/2)))/h^2","A",1
232,1,367,336,1.0998254,"\int \frac{d+e x+f x^2}{(g+h x)^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^3*Sqrt[a + b*x + c*x^2]),x]","\frac{-\frac{c h \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(8 a^2 f h^2-4 c \left(a h (d h-3 e g)+a f g^2+b g (2 d h+e g)\right)-4 a b h (e h+2 f g)+b^2 \left(h (3 d h+e g)+3 f g^2\right)+8 c^2 d g^2\right)}{8 \left(h (a h-b g)+c g^2\right)^{5/2}}+\frac{\sqrt{a+x (b+c x)} \left(2 f h (a h-b g)+c h (e g-d h)+c f g^2\right)}{2 (g+h x)^2 \left(h (a h-b g)+c g^2\right)}+\frac{c \sqrt{a+x (b+c x)} \left(h \left(-4 a h (e h-2 f g)+b h (3 d h+e g)-5 b f g^2\right)+2 c \left(g h (e g-3 d h)+f g^3\right)\right)}{4 (g+h x) \left(h (a h-b g)+c g^2\right)^2}-\frac{f \sqrt{a+x (b+c x)}}{(g+h x)^2}}{c h}","\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(8 a^2 f h^2-4 c \left(a \left(d h^2-3 e g h+f g^2\right)+b g (2 d h+e g)\right)-4 a b h (e h+2 f g)+b^2 \left(3 d h^2+e g h+3 f g^2\right)+8 c^2 d g^2\right)}{8 \left(a h^2-b g h+c g^2\right)^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left(f g^2-h (e g-d h)\right)}{2 h (g+h x)^2 \left(a h^2-b g h+c g^2\right)}+\frac{\sqrt{a+b x+c x^2} \left(h \left(4 a h (2 f g-e h)-b \left(-3 d h^2-e g h+5 f g^2\right)\right)+2 c g \left(h (e g-3 d h)+f g^2\right)\right)}{4 h (g+h x) \left(a h^2-b g h+c g^2\right)^2}",1,"(-((f*Sqrt[a + x*(b + c*x)])/(g + h*x)^2) + ((c*f*g^2 + 2*f*h*(-(b*g) + a*h) + c*h*(e*g - d*h))*Sqrt[a + x*(b + c*x)])/(2*(c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) + (c*(2*c*(f*g^3 + g*h*(e*g - 3*d*h)) + h*(-5*b*f*g^2 + b*h*(e*g + 3*d*h) - 4*a*h*(-2*f*g + e*h)))*Sqrt[a + x*(b + c*x)])/(4*(c*g^2 + h*(-(b*g) + a*h))^2*(g + h*x)) - (c*h*(8*c^2*d*g^2 + 8*a^2*f*h^2 - 4*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 + a*h*(-3*e*g + d*h) + b*g*(e*g + 2*d*h)) + b^2*(3*f*g^2 + h*(e*g + 3*d*h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(8*(c*g^2 + h*(-(b*g) + a*h))^(5/2)))/(c*h)","A",1
233,1,715,504,1.5945652,"\int \frac{(g+h x)^3 \left(d+e x+f x^2\right)}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)^3*(d + e*x + f*x^2))/(a + b*x + c*x^2)^(3/2),x]","\frac{3 \left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(24 c^2 h \left(a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right)-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 g \left(3 h (d h+e g)+f g^2\right)\right)-2 \sqrt{c} \left(4 b^2 c \left(-115 a^2 f h^3+a c h \left(3 h (6 d h+18 e g+31 e h x)+f \left(54 g^2+279 g h x-43 h^2 x^2\right)\right)+c^2 x \left(3 h \left(2 d h (h x-6 g)+e \left(-12 g^2+6 g h x+h^2 x^2\right)\right)+f \left(-12 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right)\right)\right)-8 b c^2 \left(-a^2 h^2 (39 e h+117 f g+61 f h x)+a c \left(3 h \left(2 d h (3 g+5 h x)+e \left(6 g^2+30 g h x-5 h^2 x^2\right)\right)+f \left(6 g^3+90 g^2 h x-45 g h^2 x^2-7 h^3 x^3\right)\right)-6 c^2 g^2 (-d g+3 d h x+e g x)\right)-16 c^2 \left(-16 a^3 f h^3+a^2 c h \left(3 h (4 d h+3 e (4 g+h x))+f \left(36 g^2+27 g h x-8 h^2 x^2\right)\right)+a c^2 \left(6 d h \left(-3 g^2-3 g h x+h^2 x^2\right)-3 e \left(2 g^3+6 g^2 h x-6 g h^2 x^2-h^3 x^3\right)+f x \left(-6 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right)\right)+6 c^3 d g^3 x\right)+5 b^4 h^2 (21 a f h+c x (-18 e h-54 f g+7 f h x))-2 b^3 c h \left(5 a h (9 e h+27 f g+53 f h x)+c x \left(3 h (-12 d h-36 e g+5 e h x)+f \left(-108 g^2+45 g h x+7 h^2 x^2\right)\right)\right)+105 b^5 f h^3 x\right)}{48 c^{9/2} \left(4 a c-b^2\right) \sqrt{a+x (b+c x)}}","\frac{h \sqrt{a+b x+c x^2} \left(8 c^2 \left(32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left(9 h (d h+3 e g)+20 f g^2\right)\right)+2 c h x \left(-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right)-10 b^2 c h (46 a f h+9 b (e h+3 f g))-16 c^3 \left(4 a \left(3 d h^2+9 e g h+7 f g^2\right)+3 b g (3 d h+2 e g)\right)+105 b^4 f h^2+192 c^4 d g^2\right)}{24 c^4 \left(b^2-4 a c\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(24 c^2 h \left(a h (e h+3 f g)+b \left(d h^2+3 e g h+3 f g^2\right)\right)-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 g \left(3 h (d h+e g)+f g^2\right)\right)}{16 c^{9/2}}+\frac{2 (g+h x)^3 \left(c \left(2 a e-b \left(\frac{a f}{c}+d\right)\right)-x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{h (g+h x)^2 \sqrt{a+b x+c x^2} \left(-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right)}{3 c^2 \left(b^2-4 a c\right)}",1,"(-2*Sqrt[c]*(105*b^5*f*h^3*x + 5*b^4*h^2*(21*a*f*h + c*x*(-54*f*g - 18*e*h + 7*f*h*x)) - 2*b^3*c*h*(5*a*h*(27*f*g + 9*e*h + 53*f*h*x) + c*x*(3*h*(-36*e*g - 12*d*h + 5*e*h*x) + f*(-108*g^2 + 45*g*h*x + 7*h^2*x^2))) - 16*c^2*(-16*a^3*f*h^3 + 6*c^3*d*g^3*x + a*c^2*(6*d*h*(-3*g^2 - 3*g*h*x + h^2*x^2) - 3*e*(2*g^3 + 6*g^2*h*x - 6*g*h^2*x^2 - h^3*x^3) + f*x*(-6*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3)) + a^2*c*h*(f*(36*g^2 + 27*g*h*x - 8*h^2*x^2) + 3*h*(4*d*h + 3*e*(4*g + h*x)))) - 8*b*c^2*(-6*c^2*g^2*(-(d*g) + e*g*x + 3*d*h*x) - a^2*h^2*(117*f*g + 39*e*h + 61*f*h*x) + a*c*(f*(6*g^3 + 90*g^2*h*x - 45*g*h^2*x^2 - 7*h^3*x^3) + 3*h*(2*d*h*(3*g + 5*h*x) + e*(6*g^2 + 30*g*h*x - 5*h^2*x^2)))) + 4*b^2*c*(-115*a^2*f*h^3 + a*c*h*(3*h*(18*e*g + 6*d*h + 31*e*h*x) + f*(54*g^2 + 279*g*h*x - 43*h^2*x^2)) + c^2*x*(f*(-12*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3) + 3*h*(2*d*h*(-6*g + h*x) + e*(-12*g^2 + 6*g*h*x + h^2*x^2))))) + 3*(b^2 - 4*a*c)*(35*b^3*f*h^3 - 30*b*c*h^2*(3*b*f*g + b*e*h + 2*a*f*h) - 16*c^3*g*(f*g^2 + 3*h*(e*g + d*h)) + 24*c^2*h*(3*b*f*g^2 + b*h*(3*e*g + d*h) + a*h*(3*f*g + e*h)))*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(48*c^(9/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])","A",1
234,1,412,289,0.8267508,"\int \frac{(g+h x)^2 \left(d+e x+f x^2\right)}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)^2*(d + e*x + f*x^2))/(a + b*x + c*x^2)^(3/2),x]","\frac{2 \sqrt{c} \left(4 b c \left(-13 a^2 f h^2+a c \left(2 h (d h+2 e g+5 e h x)+f \left(2 g^2+20 g h x-5 h^2 x^2\right)\right)+2 c^2 g (d (g-2 h x)-e g x)\right)+8 c^2 \left(a^2 h (4 e h+8 f g+3 f h x)+a c \left(-2 d h (2 g+h x)-2 e \left(g^2+2 g h x-h^2 x^2\right)+f x \left(-2 g^2+4 g h x+h^2 x^2\right)\right)+2 c^2 d g^2 x\right)+b^3 h (15 a f h+c x (-12 e h-24 f g+5 f h x))-2 b^2 c \left(a h (6 e h+12 f g+31 f h x)+c x \left(2 h (-2 d h-4 e g+e h x)+f \left(-4 g^2+4 g h x+h^2 x^2\right)\right)\right)+15 b^4 f h^2 x\right)-\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-12 c h (a f h+b e h+2 b f g)+15 b^2 f h^2+8 c^2 \left(h (d h+2 e g)+f g^2\right)\right)}{8 c^{7/2} \left(4 a c-b^2\right) \sqrt{a+x (b+c x)}}","\frac{2 (g+h x)^2 \left(c \left(2 a e-b \left(\frac{a f}{c}+d\right)\right)-x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-12 c h (a f h+b e h+2 b f g)+15 b^2 f h^2+8 c^2 \left(h (d h+2 e g)+f g^2\right)\right)}{8 c^{7/2}}+\frac{h \sqrt{a+b x+c x^2} \left(2 c h x \left(-12 a c f+5 b^2 f-4 b c e+8 c^2 d\right)-8 c^2 (4 a e h+8 a f g+b d h+2 b e g)+4 b c (13 a f h+3 b e h+6 b f g)-15 b^3 f h+32 c^3 d g\right)}{4 c^3 \left(b^2-4 a c\right)}",1,"(2*Sqrt[c]*(15*b^4*f*h^2*x + b^3*h*(15*a*f*h + c*x*(-24*f*g - 12*e*h + 5*f*h*x)) + 4*b*c*(-13*a^2*f*h^2 + 2*c^2*g*(-(e*g*x) + d*(g - 2*h*x)) + a*c*(2*h*(2*e*g + d*h + 5*e*h*x) + f*(2*g^2 + 20*g*h*x - 5*h^2*x^2))) - 2*b^2*c*(a*h*(12*f*g + 6*e*h + 31*f*h*x) + c*x*(2*h*(-4*e*g - 2*d*h + e*h*x) + f*(-4*g^2 + 4*g*h*x + h^2*x^2))) + 8*c^2*(2*c^2*d*g^2*x + a^2*h*(8*f*g + 4*e*h + 3*f*h*x) + a*c*(-2*d*h*(2*g + h*x) - 2*e*(g^2 + 2*g*h*x - h^2*x^2) + f*x*(-2*g^2 + 4*g*h*x + h^2*x^2)))) - (b^2 - 4*a*c)*(15*b^2*f*h^2 - 12*c*h*(2*b*f*g + b*e*h + a*f*h) + 8*c^2*(f*g^2 + h*(2*e*g + d*h)))*Sqrt[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(7/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])","A",1
235,1,205,186,0.734427,"\int \frac{(g+h x) \left(d+e x+f x^2\right)}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[((g + h*x)*(d + e*x + f*x^2))/(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(4 c \left(2 a^2 f h-a c (d h+e (g+h x)+f x (g-h x))+c^2 d g x\right)+b^2 (c x (2 e h+2 f g-f h x)-3 a f h)+2 b c (a e h+a f (g+5 h x)+c d (g-h x)-c e g x)-3 b^3 f h x\right)}{\sqrt{a+x (b+c x)}}+\left(b^2-4 a c\right) \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right) (3 b f h-2 c (e h+f g))}{2 c^{5/2} \left(4 a c-b^2\right)}","\frac{2 (g+h x) \left(c \left(2 a e-b \left(\frac{a f}{c}+d\right)\right)-x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{h \sqrt{a+b x+c x^2} \left(-8 a c f+3 b^2 f-2 b c e+4 c^2 d\right)}{c^2 \left(b^2-4 a c\right)}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) (3 b f h-2 c (e h+f g))}{2 c^{5/2}}",1,"((2*Sqrt[c]*(-3*b^3*f*h*x + 2*b*c*(a*e*h - c*e*g*x + c*d*(g - h*x) + a*f*(g + 5*h*x)) + b^2*(-3*a*f*h + c*x*(2*f*g + 2*e*h - f*h*x)) + 4*c*(2*a^2*f*h + c^2*d*g*x - a*c*(d*h + f*x*(g - h*x) + e*(g + h*x)))))/Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*(3*b*f*h - 2*c*(f*g + e*h))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(2*c^(5/2)*(-b^2 + 4*a*c))","A",1
236,1,113,111,0.2950309,"\int \frac{d+e x+f x^2}{\left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/(a + b*x + c*x^2)^(3/2),x]","\frac{\frac{2 \sqrt{c} \left(a b f-2 a c (e+f x)+b^2 f x+b c (d-e x)+2 c^2 d x\right)}{\sqrt{a+x (b+c x)}}-f \left(b^2-4 a c\right) \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)}{c^{3/2} \left(4 a c-b^2\right)}","\frac{2 \left(c \left(2 a e-b \left(\frac{a f}{c}+d\right)\right)-x \left(-2 a c f+b^2 f-b c e+2 c^2 d\right)\right)}{c \left(b^2-4 a c\right) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{3/2}}",1,"((2*Sqrt[c]*(a*b*f + 2*c^2*d*x + b^2*f*x + b*c*(d - e*x) - 2*a*c*(e + f*x)))/Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*f*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(c^(3/2)*(-b^2 + 4*a*c))","A",1
237,1,271,225,0.4929789,"\int \frac{d+e x+f x^2}{(g+h x) \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)*(a + b*x + c*x^2)^(3/2)),x]","\frac{\frac{-b^2 \left(a f h^2+2 c d h^2+c f g (g-2 h x)\right)-2 b c h (-a e h+a f (g+h x)+c (-d g+d h x+e g x))+4 c^2 (a h (d h-e g+e h x)+a f g (g-h x)+c d g h x)+b^3 f g h}{\left(b^2-4 a c\right) \sqrt{a+x (b+c x)} \left(h (b g-a h)-c g^2\right)}-\frac{c h \left(h (d h-e g)+f g^2\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right)}{\left(h (a h-b g)+c g^2\right)^{3/2}}-\frac{f}{\sqrt{a+x (b+c x)}}}{c h}","\frac{2 \left(-x \left(-c (-2 a e h+2 a f g+b d h+b e g)+b f (b g-a h)+2 c^2 d g\right)-b (a e h+a f g+c d g)+2 a (a f h-c d h+c e g)+b^2 d h\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a h^2-b g h+c g^2\right)}+\frac{\left(f g^2-h (e g-d h)\right) \tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right)}{\left(a h^2-b g h+c g^2\right)^{3/2}}",1,"(-(f/Sqrt[a + x*(b + c*x)]) + (b^3*f*g*h - b^2*(2*c*d*h^2 + a*f*h^2 + c*f*g*(g - 2*h*x)) - 2*b*c*h*(-(a*e*h) + a*f*(g + h*x) + c*(-(d*g) + e*g*x + d*h*x)) + 4*c^2*(c*d*g*h*x + a*f*g*(g - h*x) + a*h*(-(e*g) + d*h + e*h*x)))/((b^2 - 4*a*c)*(-(c*g^2) + h*(b*g - a*h))*Sqrt[a + x*(b + c*x)]) - (c*h*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(c*g^2 + h*(-(b*g) + a*h))^(3/2))/(c*h)","A",1
238,1,487,421,2.4604781,"\int \frac{d+e x+f x^2}{(g+h x)^2 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^2*(a + b*x + c*x^2)^(3/2)),x]","\frac{\frac{c h \left(\frac{\left(4 a c-b^2\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(h \left(2 a h (e h-2 f g)+b h (e g-3 d h)+b f g^2\right)+2 c \left(g h (3 d h-2 e g)+f g^3\right)\right)}{\left(h (a h-b g)+c g^2\right)^{5/2}}-\frac{2 h \sqrt{a+x (b+c x)} \left(4 a^2 f h^2-2 c \left(2 a h (2 d h-3 e g)+4 a f g^2+b g (2 d h+e g)\right)-2 a b h (e h+2 f g)+b^2 \left(h (3 d h-e g)+3 f g^2\right)+4 c^2 d g^2\right)}{(g+h x) \left(h (a h-b g)+c g^2\right)^2}\right)}{b^2-4 a c}+\frac{4 c \left(-a^2 f h^2+a c (2 h (d h-e g+e h x)+f g (g-2 h x))+2 c^2 d g h x\right)-b^2 \left(a f h^2+4 c d h^2+c f g (g-4 h x)\right)-4 b c h (a h (f x-e)+c (-d g+d h x+e g x))+b^3 f g h}{\left(b^2-4 a c\right) (g+h x) \sqrt{a+x (b+c x)} \left(h (b g-a h)-c g^2\right)}-\frac{f}{(g+h x) \sqrt{a+x (b+c x)}}}{2 c h}","-\frac{2 \left(c x \left(2 a^2 f h^2-c \left(2 a \left(d h^2-2 e g h+f g^2\right)+b g (2 d h+e g)\right)-a b h (e h+2 f g)+b^2 \left(d h^2+f g^2\right)+2 c^2 d g^2\right)+b \left(a^2 f h^2+a c \left(-3 d h^2+2 e g h+f g^2\right)+c^2 d g^2\right)-b^2 h (a e h+2 c d g)-2 a c (a h (2 f g-e h)+c g (e g-2 d h))+b^3 d h^2\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a h^2-b g h+c g^2\right)^2}-\frac{h \sqrt{a+b x+c x^2} \left(f g^2-h (e g-d h)\right)}{(g+h x) \left(a h^2-b g h+c g^2\right)^2}+\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(2 c g \left(f g^2-h (2 e g-3 d h)\right)-h \left(2 a h (2 f g-e h)-b \left(-3 d h^2+e g h+f g^2\right)\right)\right)}{2 \left(a h^2-b g h+c g^2\right)^{5/2}}",1,"(-(f/((g + h*x)*Sqrt[a + x*(b + c*x)])) + (b^3*f*g*h - b^2*(4*c*d*h^2 + a*f*h^2 + c*f*g*(g - 4*h*x)) - 4*b*c*h*(a*h*(-e + f*x) + c*(-(d*g) + e*g*x + d*h*x)) + 4*c*(-(a^2*f*h^2) + 2*c^2*d*g*h*x + a*c*(f*g*(g - 2*h*x) + 2*h*(-(e*g) + d*h + e*h*x))))/((b^2 - 4*a*c)*(-(c*g^2) + h*(b*g - a*h))*(g + h*x)*Sqrt[a + x*(b + c*x)]) + (c*h*((-2*h*(4*c^2*d*g^2 + 4*a^2*f*h^2 - 2*a*b*h*(2*f*g + e*h) - 2*c*(4*a*f*g^2 + 2*a*h*(-3*e*g + 2*d*h) + b*g*(e*g + 2*d*h)) + b^2*(3*f*g^2 + h*(-(e*g) + 3*d*h)))*Sqrt[a + x*(b + c*x)])/((c*g^2 + h*(-(b*g) + a*h))^2*(g + h*x)) + ((-b^2 + 4*a*c)*(2*c*(f*g^3 + g*h*(-2*e*g + 3*d*h)) + h*(b*f*g^2 + b*h*(e*g - 3*d*h) + 2*a*h*(-2*f*g + e*h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(c*g^2 + h*(-(b*g) + a*h))^(5/2)))/(b^2 - 4*a*c))/(2*c*h)","A",1
239,1,762,713,5.2905425,"\int \frac{d+e x+f x^2}{(g+h x)^3 \left(a+b x+c x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)^3*(a + b*x + c*x^2)^(3/2)),x]","\frac{\frac{3 c h \left(-\frac{4 h \sqrt{a+x (b+c x)} \left(8 a^2 f h^2-4 c \left(a h (3 d h-5 e g)+3 a f g^2+b g (2 d h+e g)\right)-4 a b h (e h+2 f g)+b^2 \left(h (5 d h-e g)+5 f g^2\right)+8 c^2 d g^2\right)}{(g+h x)^2}+\frac{\left(b^2-4 a c\right) \tanh ^{-1}\left(\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right) \left(h^2 \left(-8 a^2 f h^2+4 a b h (3 e h-2 f g)+b^2 \left(3 h (e g-5 d h)+f g^2\right)\right)-4 c h \left(a h \left(-3 d h^2+9 e g h-11 f g^2\right)+3 b g h (e g-4 d h)+2 b f g^3\right)-8 c^2 g^2 \left(6 d h^2-3 e g h+f g^2\right)\right)}{\left(h (a h-b g)+c g^2\right)^{3/2}}-\frac{2 h \sqrt{a+x (b+c x)} \left(2 c \left(8 a^2 h^2 (5 f g-2 e h)+2 a b h \left(h (13 d h-9 e g)-7 f g^2\right)+b^2 \left(g h (19 d h-5 e g)+7 f g^3\right)\right)+b h \left(-8 a^2 f h^2+4 a b h (3 e h-2 f g)+b^2 \left(3 h (e g-5 d h)+f g^2\right)\right)-8 c^2 g \left(a h (13 d h-11 e g)+5 a f g^2+b g (3 d h+e g)\right)+16 c^3 d g^3\right)}{(g+h x) \left(h (a h-b g)+c g^2\right)}\right)}{8 \left(b^2-4 a c\right) \left(h (a h-b g)+c g^2\right)^2}+\frac{4 c \left(-2 a^2 f h^2+a c (3 h (d h-e g+e h x)+f g (g-3 h x))+3 c^2 d g h x\right)-b^2 \left(a f h^2+6 c d h^2+c f g (g-6 h x)\right)+2 b c h (3 a e h+a f (g-3 h x)-3 c (-d g+d h x+e g x))+b^3 f g h}{\left(b^2-4 a c\right) (g+h x)^2 \sqrt{a+x (b+c x)} \left(h (b g-a h)-c g^2\right)}-\frac{f}{(g+h x)^2 \sqrt{a+x (b+c x)}}}{3 c h}","\frac{\tanh ^{-1}\left(\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right) \left(h^2 \left(8 a^2 f h^2+4 a b h (2 f g-3 e h)-\left(b^2 \left(3 h (e g-5 d h)+f g^2\right)\right)\right)-4 c h \left(a h \left(3 d h^2-9 e g h+11 f g^2\right)-b g \left(3 h (e g-4 d h)+2 f g^2\right)\right)+8 c^2 g^2 \left(6 d h^2-3 e g h+f g^2\right)\right)}{8 \left(a h^2-b g h+c g^2\right)^{7/2}}+\frac{2 \left(b^2 h \left(a^2 f h^2+a c h (3 e g-4 d h)+3 c^2 d g^2\right)-c x \left(c \left(2 a^2 h^2 (3 f g-e h)-3 a b h \left(-d h^2+e g h+f g^2\right)+b^2 \left(3 d g h^2+f g^3\right)\right)-b h^3 \left(a^2 f-a b e+b^2 d\right)-c^2 g \left(2 a \left(3 d h^2-3 e g h+f g^2\right)+b g (3 d h+e g)\right)+2 c^3 d g^3\right)-b c \left(3 a^2 h^2 (f g-e h)+a c g \left(-9 d h^2+3 e g h+f g^2\right)+c^2 d g^3\right)-2 a c \left(a^2 f h^3-a c h \left(d h^2-3 e g h+3 f g^2\right)-c^2 g^2 (e g-3 d h)\right)-b^3 h^2 (a e h+3 c d g)+b^4 d h^3\right)}{\left(b^2-4 a c\right) \sqrt{a+b x+c x^2} \left(a h^2-b g h+c g^2\right)^3}-\frac{h \sqrt{a+b x+c x^2} \left(f g^2-h (e g-d h)\right)}{2 (g+h x)^2 \left(a h^2-b g h+c g^2\right)^2}-\frac{h \sqrt{a+b x+c x^2} \left(2 c g \left(3 f g^2-h (5 e g-7 d h)\right)-h \left(4 a h (2 f g-e h)-b \left(-7 d h^2+3 e g h+f g^2\right)\right)\right)}{4 (g+h x) \left(a h^2-b g h+c g^2\right)^3}",1,"(-(f/((g + h*x)^2*Sqrt[a + x*(b + c*x)])) + (b^3*f*g*h - b^2*(6*c*d*h^2 + a*f*h^2 + c*f*g*(g - 6*h*x)) + 2*b*c*h*(3*a*e*h + a*f*(g - 3*h*x) - 3*c*(-(d*g) + e*g*x + d*h*x)) + 4*c*(-2*a^2*f*h^2 + 3*c^2*d*g*h*x + a*c*(f*g*(g - 3*h*x) + 3*h*(-(e*g) + d*h + e*h*x))))/((b^2 - 4*a*c)*(-(c*g^2) + h*(b*g - a*h))*(g + h*x)^2*Sqrt[a + x*(b + c*x)]) + (3*c*h*((-4*h*(8*c^2*d*g^2 + 8*a^2*f*h^2 - 4*a*b*h*(2*f*g + e*h) - 4*c*(3*a*f*g^2 + b*g*(e*g + 2*d*h) + a*h*(-5*e*g + 3*d*h)) + b^2*(5*f*g^2 + h*(-(e*g) + 5*d*h)))*Sqrt[a + x*(b + c*x)])/(g + h*x)^2 - (2*h*(16*c^3*d*g^3 - 8*c^2*g*(5*a*f*g^2 + b*g*(e*g + 3*d*h) + a*h*(-11*e*g + 13*d*h)) + b*h*(-8*a^2*f*h^2 + 4*a*b*h*(-2*f*g + 3*e*h) + b^2*(f*g^2 + 3*h*(e*g - 5*d*h))) + 2*c*(8*a^2*h^2*(5*f*g - 2*e*h) + 2*a*b*h*(-7*f*g^2 + h*(-9*e*g + 13*d*h)) + b^2*(7*f*g^3 + g*h*(-5*e*g + 19*d*h))))*Sqrt[a + x*(b + c*x)])/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)) + ((b^2 - 4*a*c)*(-8*c^2*g^2*(f*g^2 - 3*e*g*h + 6*d*h^2) - 4*c*h*(2*b*f*g^3 + 3*b*g*h*(e*g - 4*d*h) + a*h*(-11*f*g^2 + 9*e*g*h - 3*d*h^2)) + h^2*(-8*a^2*f*h^2 + 4*a*b*h*(-2*f*g + 3*e*h) + b^2*(f*g^2 + 3*h*(e*g - 5*d*h))))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(c*g^2 + h*(-(b*g) + a*h))^(3/2)))/(8*(b^2 - 4*a*c)*(c*g^2 + h*(-(b*g) + a*h))^2))/(3*c*h)","A",1
240,1,60,120,0.0448867,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\sqrt{2-x+3 x^2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]","\frac{6 \sqrt{3 x^2-x+2} \left(6912 x^4+22032 x^3+26904 x^2+7538 x-22383\right)-46055 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{19440}","\frac{2}{15} \sqrt{3 x^2-x+2} (2 x+1)^4+\frac{19}{60} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{44}{135} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{(6298 x+24897) \sqrt{3 x^2-x+2}}{3240}+\frac{9211 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{1296 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(-22383 + 7538*x + 26904*x^2 + 22032*x^3 + 6912*x^4) - 46055*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/19440","A",1
241,1,55,95,0.0306952,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\sqrt{2-x+3 x^2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]","\frac{6 \sqrt{3 x^2-x+2} \left(432 x^3+1176 x^2+1138 x-243\right)-4147 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{1944}","\frac{1}{6} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{11}{27} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{143}{324} (3-2 x) \sqrt{3 x^2-x+2}+\frac{4147 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{648 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(-243 + 1138*x + 1176*x^2 + 432*x^3) - 4147*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/1944","A",1
242,1,50,70,0.0207663,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\sqrt{2-x+3 x^2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]","\frac{1}{324} \left(6 \sqrt{3 x^2-x+2} \left(48 x^2+110 x+81\right)-251 \sqrt{3} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)\right)","\frac{2}{9} \sqrt{3 x^2-x+2} (2 x+1)^2+\frac{1}{54} (62 x+69) \sqrt{3 x^2-x+2}+\frac{251 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{108 \sqrt{3}}",1,"(6*Sqrt[2 - x + 3*x^2]*(81 + 110*x + 48*x^2) - 251*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/324","A",1
243,1,78,78,0.0459011,"\int \frac{1+3 x+4 x^2}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*Sqrt[2 - x + 3*x^2]),x]","\frac{2}{3} \sqrt{3 x^2-x+2}-\frac{\tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{2 \sqrt{13}}+\frac{5 \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{6 \sqrt{3}}","\frac{2}{3} \sqrt{3 x^2-x+2}-\frac{\tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{2 \sqrt{13}}-\frac{5 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{6 \sqrt{3}}",1,"(2*Sqrt[2 - x + 3*x^2])/3 + (5*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(6*Sqrt[3]) - ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])]/(2*Sqrt[13])","A",1
244,1,82,83,0.053356,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*Sqrt[2 - x + 3*x^2]),x]","-\frac{\sqrt{3 x^2-x+2}}{13 (2 x+1)}+\frac{9 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{26 \sqrt{13}}+\frac{\sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{\sqrt{3}}","-\frac{\sqrt{3 x^2-x+2}}{13 (2 x+1)}+\frac{9 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{26 \sqrt{13}}-\frac{\sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{\sqrt{3}}",1,"-1/13*Sqrt[2 - x + 3*x^2]/(1 + 2*x) + ArcSinh[(-1 + 6*x)/Sqrt[23]]/Sqrt[3] + (9*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(26*Sqrt[13])","A",1
245,1,69,89,0.0392892,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \sqrt{2-x+3 x^2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*Sqrt[2 - x + 3*x^2]),x]","\frac{\frac{26 (28 x+1) \sqrt{3 x^2-x+2}}{(2 x+1)^2}-581 \sqrt{13} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{8788}","\frac{7 \sqrt{3 x^2-x+2}}{169 (2 x+1)}-\frac{\sqrt{3 x^2-x+2}}{26 (2 x+1)^2}-\frac{581 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{676 \sqrt{13}}",1,"((26*(1 + 28*x)*Sqrt[2 - x + 3*x^2])/(1 + 2*x)^2 - 581*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/8788","A",1
246,1,69,103,0.0407965,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]","\frac{6 \left(3312 x^4+13110 x^3+23207 x^2-2974 x+29997\right)-8119 \sqrt{9 x^2-3 x+6} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{5589 \sqrt{3 x^2-x+2}}","\frac{32}{27} \sqrt{3 x^2-x+2} x^2+\frac{412}{81} \sqrt{3 x^2-x+2} x+\frac{746}{81} \sqrt{3 x^2-x+2}+\frac{2 (12839-3871 x)}{1863 \sqrt{3 x^2-x+2}}+\frac{353 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{81 \sqrt{3}}",1,"(6*(29997 - 2974*x + 23207*x^2 + 13110*x^3 + 3312*x^4) - 8119*Sqrt[6 - 3*x + 9*x^2]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(5589*Sqrt[2 - x + 3*x^2])","A",1
247,1,61,82,0.032616,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]","\frac{2 \left(828 x^3+3588 x^2+736 \sqrt{9 x^2-3 x+6} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)-3009 x+3825\right)}{621 \sqrt{3 x^2-x+2}}","\frac{2 (1249-2273 x)}{621 \sqrt{3 x^2-x+2}}+\frac{8}{9} x \sqrt{3 x^2-x+2}+\frac{112}{27} \sqrt{3 x^2-x+2}-\frac{64 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{9 \sqrt{3}}",1,"(2*(3825 - 3009*x + 3588*x^2 + 828*x^3 + 736*Sqrt[6 - 3*x + 9*x^2]*ArcSinh[(-1 + 6*x)/Sqrt[23]]))/(621*Sqrt[2 - x + 3*x^2])","A",1
248,1,50,63,0.1193137,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]","\frac{2 \left(92 x^2-153 x+37\right)}{69 \sqrt{3 x^2-x+2}}+\frac{14 \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)}{3 \sqrt{3}}","-\frac{2 (367 x+73)}{207 \sqrt{3 x^2-x+2}}+\frac{8}{9} \sqrt{3 x^2-x+2}-\frac{14 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{3 \sqrt{3}}",1,"(2*(37 - 153*x + 92*x^2))/(69*Sqrt[2 - x + 3*x^2]) + (14*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(3*Sqrt[3])","A",1
249,1,73,62,0.0226431,"\int \frac{1+3 x+4 x^2}{(1+2 x) \left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 - x + 3*x^2)^(3/2)),x]","-\frac{2 \left(23 \sqrt{13} \sqrt{3 x^2-x+2} \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)-1001 x+1313\right)}{3887 \sqrt{3 x^2-x+2}}","-\frac{2 (101-77 x)}{299 \sqrt{3 x^2-x+2}}-\frac{2 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{13 \sqrt{13}}",1,"(-2*(1313 - 1001*x + 23*Sqrt[13]*Sqrt[2 - x + 3*x^2]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])]))/(3887*Sqrt[2 - x + 3*x^2])","A",1
250,1,74,87,0.0438642,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 - x + 3*x^2)^(3/2)),x]","\frac{2 \left(1536 x^2+489 x-289\right)}{3887 (2 x+1) \sqrt{3 x^2-x+2}}+\frac{2 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{169 \sqrt{13}}","-\frac{2 (197-837 x)}{3887 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{169 (2 x+1)}+\frac{2 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{169 \sqrt{13}}",1,"(2*(-289 + 489*x + 1536*x^2))/(3887*(1 + 2*x)*Sqrt[2 - x + 3*x^2]) + (2*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(169*Sqrt[13])","A",1
251,1,79,112,0.0556704,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left(2-x+3 x^2\right)^{3/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 - x + 3*x^2)^(3/2)),x]","\frac{2 \left(14496 x^3+23281 x^2+13306 x+1673\right)}{50531 (2 x+1)^2 \sqrt{3 x^2-x+2}}-\frac{487 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{2197 \sqrt{13}}","\frac{2 (3693 x+2363)}{50531 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 \sqrt{3 x^2-x+2}}{169 (2 x+1)^2}-\frac{487 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{2197 \sqrt{13}}",1,"(2*(1673 + 13306*x + 23281*x^2 + 14496*x^3))/(50531*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2]) - (487*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2197*Sqrt[13])","A",1
252,1,71,86,0.0608756,"\int \frac{(1+2 x)^3 \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]","\frac{2 \left(228528 x^4-743712 x^3+25890 x^2+78292 \sqrt{3} \left(3 x^2-x+2\right)^{3/2} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)-358377 x-134217\right)}{42849 \left(3 x^2-x+2\right)^{3/2}}","\frac{2 (12839-3871 x)}{5589 \left(3 x^2-x+2\right)^{3/2}}+\frac{32}{27} \sqrt{3 x^2-x+2}-\frac{28 (42240 x+35809)}{128547 \sqrt{3 x^2-x+2}}-\frac{296 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{27 \sqrt{3}}",1,"(2*(-134217 - 358377*x + 25890*x^2 - 743712*x^3 + 228528*x^4 + 78292*Sqrt[3]*(2 - x + 3*x^2)^(3/2)*ArcSinh[(-1 + 6*x)/Sqrt[23]]))/(42849*(2 - x + 3*x^2)^(3/2))","A",1
253,1,66,68,0.1261228,"\int \frac{(1+2 x)^2 \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]","\frac{2 \left(5892 x^3-94992 x^2+4232 \sqrt{3} \left(3 x^2-x+2\right)^{3/2} \sinh ^{-1}\left(\frac{6 x-1}{\sqrt{23}}\right)+17511 x-52443\right)}{14283 \left(3 x^2-x+2\right)^{3/2}}","\frac{2 (1249-2273 x)}{1863 \left(3 x^2-x+2\right)^{3/2}}-\frac{8 (23257-1473 x)}{42849 \sqrt{3 x^2-x+2}}-\frac{16 \sinh ^{-1}\left(\frac{1-6 x}{\sqrt{23}}\right)}{9 \sqrt{3}}",1,"(2*(-52443 + 17511*x - 94992*x^2 + 5892*x^3 + 4232*Sqrt[3]*(2 - x + 3*x^2)^(3/2)*ArcSinh[(-1 + 6*x)/Sqrt[23]]))/(14283*(2 - x + 3*x^2)^(3/2))","A",1
254,1,33,47,0.0585835,"\int \frac{(1+2 x) \left(1+3 x+4 x^2\right)}{\left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]","\frac{2 \left(2860 x^3-3546 x^2+1833 x-1915\right)}{1587 \left(3 x^2-x+2\right)^{3/2}}","-\frac{4 (3889-4290 x)}{14283 \sqrt{3 x^2-x+2}}-\frac{2 (367 x+73)}{621 \left(3 x^2-x+2\right)^{3/2}}",1,"(2*(-1915 + 1833*x - 3546*x^2 + 2860*x^3))/(1587*(2 - x + 3*x^2)^(3/2))","A",1
255,1,72,85,0.050739,"\int \frac{1+3 x+4 x^2}{(1+2 x) \left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 - x + 3*x^2)^(5/2)),x]","\frac{2 \left(82008 x^3-31482 x^2+79077 x-32963\right)}{268203 \left(3 x^2-x+2\right)^{3/2}}-\frac{8 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{169 \sqrt{13}}","-\frac{4 (691-13668 x)}{268203 \sqrt{3 x^2-x+2}}-\frac{2 (101-77 x)}{897 \left(3 x^2-x+2\right)^{3/2}}-\frac{8 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{169 \sqrt{13}}",1,"(2*(-32963 + 79077*x - 31482*x^2 + 82008*x^3))/(268203*(2 - x + 3*x^2)^(3/2)) - (8*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(169*Sqrt[13])","A",1
256,1,111,110,0.0636763,"\int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 - x + 3*x^2)^(5/2)),x]","\frac{26 \left(1318464 x^4+133308 x^3+1021566 x^2+569989 x-170239\right)-88872 \sqrt{13} \sqrt{3 x^2-x+2} \left(6 x^3+x^2+3 x+2\right) \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{45326307 (2 x+1) \left(3 x^2-x+2\right)^{3/2}}","-\frac{24 (841-6633 x)}{1162213 \sqrt{3 x^2-x+2}}-\frac{16 \sqrt{3 x^2-x+2}}{2197 (2 x+1)}-\frac{2 (197-837 x)}{11661 \left(3 x^2-x+2\right)^{3/2}}-\frac{56 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{2197 \sqrt{13}}",1,"(26*(-170239 + 569989*x + 1021566*x^2 + 133308*x^3 + 1318464*x^4) - 88872*Sqrt[13]*Sqrt[2 - x + 3*x^2]*(2 + 3*x + x^2 + 6*x^3)*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(45326307*(1 + 2*x)*(2 - x + 3*x^2)^(3/2))","A",1
257,1,89,135,0.0746214,"\int \frac{1+3 x+4 x^2}{(1+2 x)^3 \left(2-x+3 x^2\right)^{5/2}} \, dx","Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 - x + 3*x^2)^(5/2)),x]","\frac{2 \left(20304864 x^5+20074356 x^4+19381992 x^3+21890266 x^2+10777477 x+847141\right)}{45326307 (2 x+1)^2 \left(3 x^2-x+2\right)^{3/2}}-\frac{2084 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{28561 \sqrt{13}}","\frac{2 (3693 x+2363)}{151593 \left(3 x^2-x+2\right)^{3/2}}-\frac{144 \sqrt{3 x^2-x+2}}{28561 (2 x+1)}-\frac{8 \sqrt{3 x^2-x+2}}{2197 (2 x+1)^2}+\frac{12 (103526 x+25771)}{15108769 \sqrt{3 x^2-x+2}}-\frac{2084 \tanh ^{-1}\left(\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right)}{28561 \sqrt{13}}",1,"(2*(847141 + 10777477*x + 21890266*x^2 + 19381992*x^3 + 20074356*x^4 + 20304864*x^5))/(45326307*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2)) - (2084*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(28561*Sqrt[13])","A",1
258,1,219,208,0.5051264,"\int \frac{d+e x+f x^2}{(g+h x) \left(-c g^2+b g h+b h^2 x+c h^2 x^2\right)^{3/2}} \, dx","Integrate[(d + e*x + f*x^2)/((g + h*x)*(-(c*g^2) + b*g*h + b*h^2*x + c*h^2*x^2)^(3/2)),x]","\frac{2 b^2 h^2 \left(f \left(8 g^2+12 g h x+3 h^2 x^2\right)-h (d h+2 e g+3 e h x)\right)-4 b c h \left(h \left(e \left(g^2+2 g h x+3 h^2 x^2\right)-2 d h (2 g+h x)\right)+2 f g^2 (4 g+5 h x)\right)+8 c^2 \left(h \left(d h \left(-g^2+2 g h x+2 h^2 x^2\right)+e g \left(g^2+g h x+h^2 x^2\right)\right)+f g^2 \left(2 g^2+2 g h x-h^2 x^2\right)\right)}{3 h^3 (g+h x) (b h-2 c g)^3 \sqrt{(g+h x) (b h-c g+c h x)}}","\frac{(b+2 c x) \left(-3 b^2 f h^2+6 b c e h^2+4 c^2 \left(f g^2-h (2 d h+e g)\right)\right)}{3 c h^2 (2 c g-b h)^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}+\frac{2 \left(d h^2-e g h+f g^2\right)}{3 h^3 (g+h x) (2 c g-b h) \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}-\frac{f}{c h^3 \sqrt{-g (c g-b h)+b h^2 x+c h^2 x^2}}",1,"(2*b^2*h^2*(-(h*(2*e*g + d*h + 3*e*h*x)) + f*(8*g^2 + 12*g*h*x + 3*h^2*x^2)) + 8*c^2*(f*g^2*(2*g^2 + 2*g*h*x - h^2*x^2) + h*(e*g*(g^2 + g*h*x + h^2*x^2) + d*h*(-g^2 + 2*g*h*x + 2*h^2*x^2))) - 4*b*c*h*(2*f*g^2*(4*g + 5*h*x) + h*(-2*d*h*(2*g + h*x) + e*(g^2 + 2*g*h*x + 3*h^2*x^2))))/(3*h^3*(-2*c*g + b*h)^3*(g + h*x)*Sqrt[(g + h*x)*(-(c*g) + b*h + c*h*x)])","A",1
259,1,15669,906,14.9901306,"\int \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2),x]","\text{Result too large to show}","\frac{2 C (d+e x)^{3/2} \left(c x^2+b x+a\right)^{3/2}}{9 c e}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left(c x^2+b x+a\right)^{3/2}}{21 c^2 e}+\frac{2 \sqrt{d+e x} \left(d \left(8 C d^2-3 e (4 B d-7 A e)\right) c^3+3 e \left(a e (C d-5 B e)-b \left(C d^2-2 B e d-7 A e^2\right)\right) c^2-3 b e^2 (b C d+4 b B e-a C e) c+3 e \left(-\left(\left(2 C d^2-3 e (B d+7 A e)\right) c^2\right)-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right) x c+8 b^3 C e^3\right) \sqrt{c x^2+b x+a}}{315 c^3 e^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(2 \left(4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right) \left(-\left(\left(2 C d^2-3 e (B d+7 A e)\right) c^2\right)-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right)-5 c e (2 c d-b e) \left(6 C d e b^2+\left(2 a C e^2-c d (C d+9 B e)\right) b+c e (21 A c d-5 a C d-3 a B e)\right)\right) \sqrt{d+e 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c d^2-b e d+a e^2\right) \left(-2 d \left(8 C d^2-3 e (4 B d-7 A e)\right) c^3-3 e^2 (b B d+2 a C d-7 A b e-10 a B e) c^2+3 b e^2 (b C d-4 b B e-9 a C e) c+8 b^3 C e^3\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
260,1,9965,668,14.4102213,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{\sqrt{d+e x}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/Sqrt[d + e*x],x]","\text{Result too large to show}","\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 e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(c e (-10 a C e-7 b B e+8 b C d)+c^2 \left(48 C d^2-14 e (4 B d-5 A e)\right)+4 b^2 C e^2\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left(-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right) (4 b C e-7 B c e+6 c C d)\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left(a+b x+c x^2\right)^{3/2}}{7 c e}",1,"Result too large to show","C",1
261,1,13240,749,14.0394818,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{(d+e x)^{3/2}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(3/2),x]","\text{Result too large to show}","\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 (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(-c e \left(2 a e (9 C d-5 B e)-b \left(32 C d^2-5 e (5 B d-3 A e)\right)\right)+b C e^2 (b d-a e)-2 c^2 d \left(24 C d^2-5 e (4 B d-3 A e)\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(c e (-6 a C e-5 b B e+8 b C d)-\left(c^2 \left(48 C d^2-10 e (4 B d-3 A e)\right)\right)+2 b^2 C e^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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^2 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(3 c e^2 x \left(-a C e-5 A c e+b C d+5 B c d-\frac{6 c C d^2}{e}\right)+c e \left(a e (9 C d-5 B e)-5 b \left(3 A e^2-4 B d e+5 C d^2\right)\right)+b C e^2 (b d-a e)+c^2 d \left(24 C d^2-5 e (4 B d-3 A e)\right)\right)}{15 c e^3 \left(a e^2-b d e+c d^2\right)}-\frac{2 \left(a+b x+c x^2\right)^{3/2} \left(C d^2-e (B d-A e)\right)}{e \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}",1,"Result too large to show","C",1
262,1,8456,712,14.2679846,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{(d+e x)^{5/2}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(5/2),x]","\text{Result too large to show}","-\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 (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(e (-2 a C e-3 b B e+8 b C d)-2 c \left(8 C d^2-e (4 B d-A e)\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(2 \left(4 c d-\frac{b e}{2}\right) \left(-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right)+6 c (e (a B e-a C d+A c d)+b d (C d-B e))\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^3 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \left(a+b x+c x^2\right)^{3/2} \left(C d^2-e (B d-A e)\right)}{3 e (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}-\frac{2 \sqrt{a+b x+c x^2} \left(e^2 x \left(-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right)+e (b d-a e) (7 C d-3 B e)-c d \left(8 C d^2-e (4 B d-A e)\right)\right)}{3 e^3 \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)}",1,"Result too large to show","C",1
263,1,12997,992,14.665695,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{(d+e x)^{7/2}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(C d^2-e (B d-A e)\right) \left(c x^2+b x+a\right)^{3/2}}{5 e \left(c d^2-b e d+a e^2\right) (d+e x)^{5/2}}-\frac{2 \left(c^2 \left(24 C d^2-e (4 B d+A e)\right) d^3-c e \left(b d \left(41 C d^2-6 B e d+A e^2\right)-a e \left(37 C d^2-7 B e d+7 A e^2\right)\right) d+e^2 \left(15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left(22 C d^2+3 B e d+2 A e^2\right)\right)+e \left(5 c^2 \left(6 C d^2-e (B d+A e)\right) d^2+e^2 \left(\left(23 C d^2-3 B e d-2 A e^2\right) b^2-5 a e (8 C d-B e) b+15 a^2 C e^2\right)-c e \left(5 b d \left(11 C d^2-2 B e d-A e^2\right)-a e \left(53 C d^2-13 B e d+3 A e^2\right)\right)\right) x\right) \sqrt{c x^2+b x+a}}{15 e^3 \left(c d^2-b e d+a e^2\right)^2 (d+e x)^{3/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(2 c^2 \left(24 C d^2-e (4 B d+A e)\right) d^2+e^2 \left(\left(38 C d^2-3 B e d-2 A e^2\right) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right)-c e \left(b d \left(88 C d^2-13 B e d-2 A e^2\right)-2 a e \left(43 C d^2-8 B e d+3 A e^2\right)\right)\right) \sqrt{d+e 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^4 \left(c d^2-b e d+a e^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(2 d \left(24 C d^2-e (4 B d+A e)\right) c^2+e \left(10 a e (5 C d-B e)-b \left(64 C d^2-9 B e d-A e^2\right)\right) c+15 b C e^2 (b d-a e)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c e^4 \left(c d^2-b e d+a e^2\right) \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
264,1,19853,1363,16.0151081,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{(d+e x)^{9/2}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(9/2),x]","\text{Result too large to show}","-\frac{2 \left(C d^2-e (B d-A e)\right) \left(c x^2+b x+a\right)^{3/2}}{7 e \left(c d^2-b e d+a e^2\right) (d+e x)^{7/2}}-\frac{2 \left(c^2 \left(24 C d^2+e (4 B d+3 A e)\right) d^3-c e \left(b d \left(43 C d^2+6 B e d+15 A e^2\right)-a e \left(33 C d^2+9 B e d+19 A e^2\right)\right) d-e^2 \left(-d \left(15 C d^2+6 B e d+8 A e^2\right) b^2+a e \left(12 C d^2+23 B e d+12 A e^2\right) b+7 a^2 e^2 (C d-3 B e)\right)+e \left(7 c^2 \left(6 C d^2+e (B d-3 A e)\right) d^2+e^2 \left(\left(45 C d^2-3 B e d-4 A e^2\right) b^2-7 a e (12 C d-B e) b+35 a^2 C e^2\right)+c e \left(a e \left(93 C d^2-9 B e d-5 A e^2\right)-b \left(91 C d^3-21 A d e^2\right)\right)\right) x\right) \sqrt{c x^2+b x+a}}{105 e^3 \left(c d^2-b e d+a e^2\right)^2 (d+e x)^{5/2}}+\frac{2 \left(2 c^3 \left(24 C d^2+e (4 B d+3 A e)\right) d^3+c^2 e \left(2 a e \left(69 C d^2+e (15 B d-29 A e)\right)-b d \left(128 C d^2+e (19 B d+9 A e)\right)\right) d-b e^3 \left(\left(15 C d^2+6 B e d+8 A e^2\right) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right)+c e^2 \left(d \left(103 C d^2+e (9 B d+19 A e)\right) b^2-a e \left(237 C d^2+e (B d-29 A e)\right) b+14 a^2 e^2 (11 C d-3 B e)\right)\right) \sqrt{c x^2+b x+a}}{105 e^3 \left(c d^2-b e d+a e^2\right)^3 \sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(2 c^3 \left(24 C d^2+e (4 B d+3 A e)\right) d^3+c^2 e \left(2 a e \left(69 C d^2+e (15 B d-29 A e)\right)-b d \left(128 C d^2+e (19 B d+9 A e)\right)\right) d-b e^3 \left(\left(15 C d^2+6 B e d+8 A e^2\right) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right)+c e^2 \left(d \left(103 C d^2+e (9 B d+19 A e)\right) b^2-a e \left(237 C d^2+e (B d-29 A e)\right) b+14 a^2 e^2 (11 C d-3 B e)\right)\right) \sqrt{d+e 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 e^4 \left(c d^2-b e d+a e^2\right)^3 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(2 c^2 \left(24 C d^2+e (4 B d+3 A e)\right) d^2+c e \left(2 a e \left(51 C d^2+e (12 B d-5 A e)\right)-b d \left(104 C d^2+3 e (5 B d+2 A e)\right)\right)+e^2 \left(\left(60 C d^2+e (3 B d+4 A e)\right) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\right)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 e^4 \left(c d^2-b e d+a e^2\right)^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
265,1,29140,1904,18.387309,"\int \frac{\sqrt{a+b x+c x^2} \left(A+B x+C x^2\right)}{(d+e x)^{11/2}} \, dx","Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(11/2),x]","\text{Result too large to show}","-\frac{2 \left(C d^2-e (B d-A e)\right) \left(c x^2+b x+a\right)^{3/2}}{9 e \left(c d^2-b e d+a e^2\right) (d+e x)^{9/2}}-\frac{2 \left(c^2 \left(8 C d^2+e (4 B d+5 A e)\right) d^3-c e \left(3 b d \left(5 C d^2+2 B e d+5 A e^2\right)-a e \left(7 C d^2+11 B e d+13 A e^2\right)\right) d-e^2 \left(-d \left(5 C d^2+4 B e d+8 A e^2\right) b^2-a e \left(2 C d^2-17 B e d-10 A e^2\right) b+3 a^2 e^2 (3 C d-5 B e)\right)+e \left(3 c^2 \left(6 C d^2+e (3 B d-5 A e)\right) d^2+c e \left(a e \left(47 C d^2+B e d-7 A e^2\right)-3 b d \left(15 C d^2+2 B e d-5 A e^2\right)\right)+e^2 \left(\left(25 C d^2-e (B d+2 A e)\right) b^2-3 a e (16 C d-B e) b+21 a^2 C e^2\right)\right) x\right) \sqrt{c x^2+b x+a}}{105 e^3 \left(c d^2-b e d+a e^2\right)^2 (d+e x)^{7/2}}+\frac{2 \left(2 c^4 \left(8 C d^2+e (4 B d+5 A e)\right) d^4+c^3 e \left(6 a e \left(11 C d^2+8 B e d-34 A e^2\right)-b d \left(56 C d^2+5 e (5 B d+4 A e)\right)\right) d^2+2 b^2 e^4 \left(\left(5 C d^2+4 B e d+8 A e^2\right) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right)-6 c^2 e^2 \left(-b^2 \left(11 C d^2+3 B e d+11 A e^2\right) d^2+a b e \left(30 C d^2-5 B e d-34 A e^2\right) d-a^2 e^2 \left(30 C d^2-36 B e d+7 A e^2\right)\right)-c e^3 \left(d \left(20 C d^2+25 B e d+56 A e^2\right) b^3-6 a e \left(5 C d^2+7 B e d-12 A e^2\right) b^2-3 a^2 e^2 (12 C d+29 B e) b+126 a^3 C e^3\right)\right) \sqrt{c x^2+b x+a}}{315 e^3 \left(c d^2-b e d+a e^2\right)^4 \sqrt{d+e x}}+\frac{2 \left(2 c^3 \left(8 C d^2+e (4 B d+5 A e)\right) d^3+3 c^2 e \left(2 a e \left(9 C d^2+7 B e d-9 A e^2\right)-b d \left(16 C d^2+7 B e d+5 A e^2\right)\right) d+3 c e^2 \left(d \left(15 C d^2+3 B e d+7 A e^2\right) b^2-a e \left(41 C d^2+5 B e d-9 A e^2\right) b+2 a^2 e^2 (17 C d-5 B e)\right)-b e^3 \left(\left(5 C d^2+4 B e d+8 A e^2\right) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right)\right) \sqrt{c x^2+b x+a}}{315 e^3 \left(c d^2-b e d+a e^2\right)^3 (d+e x)^{3/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(2 c^4 \left(8 C d^2+e (4 B d+5 A e)\right) d^4+c^3 e \left(6 a e \left(11 C d^2+8 B e d-34 A e^2\right)-b d \left(56 C d^2+5 e (5 B d+4 A e)\right)\right) d^2+2 b^2 e^4 \left(\left(5 C d^2+4 B e d+8 A e^2\right) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right)-6 c^2 e^2 \left(-b^2 \left(11 C d^2+3 B e d+11 A e^2\right) d^2+a b e \left(30 C d^2-5 B e d-34 A e^2\right) d-a^2 e^2 \left(30 C d^2-36 B e d+7 A e^2\right)\right)-c e^3 \left(d \left(20 C d^2+25 B e d+56 A e^2\right) b^3-6 a e \left(5 C d^2+7 B e d-12 A e^2\right) b^2-3 a^2 e^2 (12 C d+29 B e) b+126 a^3 C e^3\right)\right) \sqrt{d+e 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 e^4 \left(c d^2-b e d+a e^2\right)^4 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(2 c^3 \left(8 C d^2+e (4 B d+5 A e)\right) d^3+3 c^2 e \left(2 a e \left(9 C d^2+7 B e d-9 A e^2\right)-b d \left(16 C d^2+7 B e d+5 A e^2\right)\right) d+3 c e^2 \left(d \left(15 C d^2+3 B e d+7 A e^2\right) b^2-a e \left(41 C d^2+5 B e d-9 A e^2\right) b+2 a^2 e^2 (17 C d-5 B e)\right)-b e^3 \left(\left(5 C d^2+4 B e d+8 A e^2\right) b^2-6 a e (3 C d+2 B e) b+21 a^2 C e^2\right)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{315 e^4 \left(c d^2-b e d+a e^2\right)^3 \sqrt{d+e x} \sqrt{c x^2+b x+a}}",1,"Result too large to show","C",0
266,1,9972,724,14.5498505,"\int \frac{(d+e x)^{3/2} \left(A+B x+C x^2\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[((d + e*x)^(3/2)*(A + B*x + C*x^2))/Sqrt[a + b*x + c*x^2],x]","\text{Result too large to show}","-\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 e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(-c e (25 a C e+28 b B e+15 b C d)-\left(c^2 \left(6 C d^2-7 e (5 A e+3 B d)\right)\right)+24 b^2 C e^2\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(-c e (25 a C e+28 b B e+15 b C d)-\left(c^2 \left(6 C d^2-7 e (5 A e+3 B d)\right)\right)+24 b^2 C e^2\right)}{105 c^3 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(c^2 e \left(a e (63 B e+82 C d)+b \left(70 A e^2+91 B d e+12 C d^2\right)\right)-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 d \left(6 C d^2-7 e (20 A e+3 B d)\right)+48 b^3 C e^3\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{105 c^4 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}",1,"Result too large to show","C",1
267,1,992,557,11.5881243,"\int \frac{\sqrt{d+e x} \left(A+B x+C x^2\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(Sqrt[d + e*x]*(A + B*x + C*x^2))/Sqrt[a + b*x + c*x^2],x]","\frac{\left(\frac{2 (c C d+5 B c e-4 b C e)}{15 c^2 e}+\frac{2 C x}{5 c}\right) \sqrt{d+e x} \left(c x^2+b x+a\right)}{\sqrt{a+x (b+c x)}}-\frac{2 (d+e x)^{3/2} \sqrt{c x^2+b x+a} \left(\left(\left(2 C d^2-5 e (B d+3 A e)\right) c^2+e (3 b C d+10 b B e+9 a C e) c-8 b^2 C e^2\right) \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{1-\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}+1} \left(\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(\left(5 e (B d+3 A e)-2 C d^2\right) c^2-e (3 b C d+10 b B e+9 a C e) c+8 b^2 C e^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(8 b^3 C e^3-b^2 \left(11 c d C+8 \sqrt{\left(b^2-4 a c\right) e^2} C+10 B c e\right) e^2+b c \left(15 A c e^2-17 a C e^2+5 B \left(3 c d e+2 \sqrt{\left(b^2-4 a c\right) e^2} e\right)+3 C d \sqrt{\left(b^2-4 a c\right) e^2}\right) e+c \left(-15 A c \left(2 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+a \left(14 c d C+9 \sqrt{\left(b^2-4 a c\right) e^2} C+10 B c e\right) e^2+c d \sqrt{\left(b^2-4 a c\right) e^2} (2 C d-5 B e)\right)\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{d+e x}}\right)}{15 c^3 e^3 \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}","\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(-c e (9 a C e+10 b B e+3 b C d)-\left(c^2 \left(2 C d^2-5 e (3 A e+B d)\right)\right)+8 b^2 C e^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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \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}} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} (4 b C e-5 B c e+2 c C d) 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (4 b C e-5 B c e+2 c C d)}{15 c^2 e}+\frac{2 C (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{5 c e}",1,"(((2*(c*C*d + 5*B*c*e - 4*b*C*e))/(15*c^2*e) + (2*C*x)/(5*c))*Sqrt[d + e*x]*(a + b*x + c*x^2))/Sqrt[a + x*(b + c*x)] - (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((-8*b^2*C*e^2 + c*e*(3*b*C*d + 10*b*B*e + 9*a*C*e) + c^2*(2*C*d^2 - 5*e*(B*d + 3*A*e)))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(8*b^2*C*e^2 - c*e*(3*b*C*d + 10*b*B*e + 9*a*C*e) + c^2*(-2*C*d^2 + 5*e*(B*d + 3*A*e)))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (8*b^3*C*e^3 - b^2*e^2*(11*c*C*d + 10*B*c*e + 8*C*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(c*d*Sqrt[(b^2 - 4*a*c)*e^2]*(2*C*d - 5*B*e) - 15*A*c*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(14*c*C*d + 10*B*c*e + 9*C*Sqrt[(b^2 - 4*a*c)*e^2])) + b*c*e*(15*A*c*e^2 - 17*a*C*e^2 + 3*C*d*Sqrt[(b^2 - 4*a*c)*e^2] + 5*B*(3*c*d*e + 2*e*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(15*c^3*e^3*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
268,1,1080,471,12.4997115,"\int \frac{A+B x+C x^2}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]),x]","\frac{\sqrt{c x^2+b x+a} \left(-4 (2 c C d-3 B c e+2 b C e) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)+\frac{i \sqrt{2} (2 c C d-3 B c e+2 b C e) \left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left(-2 b^2 C e^2+b \left(2 \sqrt{\left(b^2-4 a c\right) e^2} C+3 B c e\right) e+c \left(-6 A c e^2+2 a C e^2+\sqrt{\left(b^2-4 a c\right) e^2} (2 C d-3 B e)\right)\right) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left(\frac{2 d}{d+e x}-1\right) e-2 c d \left(\frac{d}{d+e x}-1\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left(\frac{d}{d+e x}-1\right)+b \left(e-\frac{2 d e}{d+e x}\right)+\sqrt{\left(b^2-4 a c\right) e^2}}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)}{\sqrt{d+e x}}\right) (d+e x)^{3/2}}{6 c^2 e^3 \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}+\frac{2 C \left(c x^2+b x+a\right) \sqrt{d+e x}}{3 c e \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}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(C e (b d-a e)+c \left(2 C d^2-3 e (B d-A e)\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} (2 b C e-3 B c e+2 c C d) 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c^2 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{2 C \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c e}",1,"(2*C*Sqrt[d + e*x]*(a + b*x + c*x^2))/(3*c*e*Sqrt[a + x*(b + c*x)]) + ((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*(-4*(2*c*C*d - 3*B*c*e + 2*b*C*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + (I*Sqrt[2]*(2*c*C*d - 3*B*c*e + 2*b*C*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-2*b^2*C*e^2 + b*e*(3*B*c*e + 2*C*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-6*A*c*e^2 + 2*a*C*e^2 + Sqrt[(b^2 - 4*a*c)*e^2]*(2*C*d - 3*B*e)))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x]))/(6*c^2*e^3*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
269,1,772,508,7.2802551,"\int \frac{A+B x+C x^2}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \left(-\frac{i (d+e x)^{3/2} \sqrt{1-\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right)}} \sqrt{\frac{2 \left(e (a e-b d)+c d^2\right)}{(d+e x) \left(\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d\right)}+1} \left(\left(\sqrt{e^2 \left(b^2-4 a c\right)}-b e+2 c d\right) \left(C e (a e-b d)+c e (A e-B d)+2 c C d^2\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)+\left(b \left(C d e \sqrt{e^2 \left(b^2-4 a c\right)}+a C e^3+A c e^3+B c d e^2\right)-A c e^2 \left(\sqrt{e^2 \left(b^2-4 a c\right)}+2 c d\right)+B c d e \sqrt{e^2 \left(b^2-4 a c\right)}-2 c C d^2 \sqrt{e^2 \left(b^2-4 a c\right)}-a C e^2 \sqrt{e^2 \left(b^2-4 a c\right)}-2 a B c e^3+2 a c C d e^2-b^2 C d e^2\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{2 \sqrt{2} c \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left(b^2-4 a c\right)}+b e-2 c d}}}-\left(e^2 (a+x (b+c x)) \left(e (A e-B d)+C d^2\right)\right)+\frac{e^2 (a+x (b+c x)) \left(C e (a e-b d)+c e (A e-B d)+2 c C d^2\right)}{c}\right)}{e^3 \sqrt{d+e x} \sqrt{a+x (b+c x)} \left(e (a e-b d)+c d^2\right)}","-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(C e (b d-a e)-c \left(2 C d^2-e (B d-A e)\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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c e^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right) \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{a+b x+c x^2} \left(C d^2-e (B d-A e)\right)}{e \sqrt{d+e x} \left(a e^2-b d e+c d^2\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}} (2 C d-B e) \sqrt{\frac{c (d+e x)}{2 c d-e \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}",1,"(2*(-(e^2*(C*d^2 + e*(-(B*d) + A*e))*(a + x*(b + c*x))) + (e^2*(2*c*C*d^2 + C*e*(-(b*d) + a*e) + c*e*(-(B*d) + A*e))*(a + x*(b + c*x)))/c - ((I/2)*(d + e*x)^(3/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(2*c*C*d^2 + C*e*(-(b*d) + a*e) + c*e*(-(B*d) + A*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (-(b^2*C*d*e^2) + 2*a*c*C*d*e^2 - 2*a*B*c*e^3 - 2*c*C*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + B*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2] - a*C*e^2*Sqrt[(b^2 - 4*a*c)*e^2] - A*c*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*(B*c*d*e^2 + A*c*e^3 + a*C*e^3 + C*d*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*c*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(e^3*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)])","C",1
270,1,1194,684,12.1556499,"\int \frac{A+B x+C x^2}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2]),x]","\frac{2 \sqrt{c x^2+b x+a} \left(\frac{i \sqrt{1-\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}} \sqrt{\frac{2 \left(c d^2+e (a e-b d)\right)}{\left(-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) (d+e x)}+1} \left(\left(2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}\right) \left(c d \left(2 C d^2+e (B d-4 A e)\right)+e \left(-4 b C d^2+b e (B d+2 A e)-3 a e (B e-2 C d)\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)-\left(-6 a^2 C e^4-8 a B c d e^3-3 a B \sqrt{\left(b^2-4 a c\right) e^2} e^3+2 a c C d^2 e^2+2 A c \left(-3 c d^2-2 \sqrt{\left(b^2-4 a c\right) e^2} d+a e^2\right) e^2-b^2 \left(2 C d^2+e (B d+2 A e)\right) e^2+6 a C d \sqrt{\left(b^2-4 a c\right) e^2} e^2+b \left(2 A \left(3 c d+\sqrt{\left(b^2-4 a c\right) e^2}\right) e^2+B \left(3 c d^2+\sqrt{\left(b^2-4 a c\right) e^2} d+3 a e^2\right) e+2 C d \left(3 a e^2-2 d \sqrt{\left(b^2-4 a c\right) e^2}\right)\right) e+B c d^2 \sqrt{\left(b^2-4 a c\right) e^2} e+2 c C d^3 \sqrt{\left(b^2-4 a c\right) e^2}\right) F\left(i \sinh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}}}{\sqrt{d+e x}}\right)|-\frac{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}{2 c d-b e+\sqrt{\left(b^2-4 a c\right) e^2}}\right)\right)}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left(b^2-4 a c\right) e^2}}} \sqrt{d+e x}}-\left(2 c C d^3+c e (B d-4 A e) d-3 a e^2 (B e-2 C d)+b e \left(e (B d+2 A e)-4 C d^2\right)\right) \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)\right) (d+e x)^{3/2}}{3 e^3 \left(c d^2-b e d+a e^2\right)^2 \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left(c \left(\frac{d}{d+e x}-1\right)^2+\frac{e \left(-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right)}{d+e x}\right)}{e^2}}}+\frac{\left(c x^2+b x+a\right) \left(-\frac{2 \left(C d^2-B e d+A e^2\right)}{3 e \left(c d^2-b e d+a e^2\right) (d+e x)^2}-\frac{2 \left(-2 c C d^3-B c e d^2+4 b C e d^2-b B e^2 d+4 A c e^2 d-6 a C e^2 d-2 A b e^3+3 a B e^3\right)}{3 e \left(c d^2-b e d+a e^2\right)^2 (d+e x)}\right) \sqrt{d+e x}}{\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}} \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}} \left(3 C e (b d-a e)-c \left(e (B d-A e)+2 C d^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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left(a+b x+c x^2\right)}{b^2-4 a c}} \left(e \left(3 a e (2 C d-B e)-b \left(-2 A e^2-B d e+4 C d^2\right)\right)+c d \left(e (B d-4 A e)+2 C d^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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{3 e^2 \sqrt{a+b x+c x^2} \left(a e^2-b d e+c d^2\right)^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{2 \sqrt{a+b x+c x^2} \left(C d^2-e (B d-A e)\right)}{3 e (d+e x)^{3/2} \left(a e^2-b d e+c d^2\right)}+\frac{2 \sqrt{a+b x+c x^2} \left(e \left(3 a e (2 C d-B e)-b \left(-2 A e^2-B d e+4 C d^2\right)\right)+c d \left(e (B d-4 A e)+2 C d^2\right)\right)}{3 e \sqrt{d+e x} \left(a e^2-b d e+c d^2\right)^2}",1,"(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*(C*d^2 - B*d*e + A*e^2))/(3*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (2*(-2*c*C*d^3 - B*c*d^2*e + 4*b*C*d^2*e - b*B*d*e^2 + 4*A*c*d*e^2 - 6*a*C*d*e^2 - 2*A*b*e^3 + 3*a*B*e^3))/(3*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*(-((2*c*C*d^3 + c*d*e*(B*d - 4*A*e) - 3*a*e^2*(-2*C*d + B*e) + b*e*(-4*C*d^2 + e*(B*d + 2*A*e)))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x))) + ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c*d*(2*C*d^2 + e*(B*d - 4*A*e)) + e*(-4*b*C*d^2 + b*e*(B*d + 2*A*e) - 3*a*e*(-2*C*d + B*e)))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (2*a*c*C*d^2*e^2 - 8*a*B*c*d*e^3 - 6*a^2*C*e^4 + 2*c*C*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + B*c*d^2*e*Sqrt[(b^2 - 4*a*c)*e^2] + 6*a*C*d*e^2*Sqrt[(b^2 - 4*a*c)*e^2] - 3*a*B*e^3*Sqrt[(b^2 - 4*a*c)*e^2] + 2*A*c*e^2*(-3*c*d^2 + a*e^2 - 2*d*Sqrt[(b^2 - 4*a*c)*e^2]) - b^2*e^2*(2*C*d^2 + e*(B*d + 2*A*e)) + b*e*(2*A*e^2*(3*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 2*C*d*(3*a*e^2 - 2*d*Sqrt[(b^2 - 4*a*c)*e^2]) + B*e*(3*c*d^2 + 3*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(3*e^3*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2])","C",1
271,1,12295,944,15.0379878,"\int \frac{A+B x+C x^2}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx","Integrate[(A + B*x + C*x^2)/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]","\text{Result too large to show}","-\frac{2 \sqrt{c x^2+b x+a} \left(C d^2-e (B d-A e)\right)}{5 e \left(c d^2-b e d+a e^2\right) (d+e x)^{5/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left(c^2 \left(2 C d^2+e (3 B d-23 A e)\right) d^2-e^2 \left(\left(3 C d^2+2 B e d+8 A e^2\right) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right)-c e \left(b d \left(7 C d^2-7 B e d-23 A e^2\right)-a e \left(19 C d^2-29 B e d+9 A e^2\right)\right)\right) \sqrt{d+e 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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^2 \left(c d^2-b e d+a e^2\right)^3 \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left(c d \left(2 C d^2+e (3 B d-8 A e)\right)+e \left(5 a e (2 C d-B e)-b \left(6 C d^2-B e d-4 A e^2\right)\right)\right) \sqrt{\frac{c (d+e x)}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}} \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} e}{2 c d-\left(b+\sqrt{b^2-4 a c}\right) e}\right)}{15 e^2 \left(c d^2-b e d+a e^2\right)^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}+\frac{2 \left(c^2 \left(2 C d^2+e (3 B d-23 A e)\right) d^2-e^2 \left(\left(3 C d^2+2 B e d+8 A e^2\right) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right)-c e \left(b d \left(7 C d^2-7 B e d-23 A e^2\right)-a e \left(19 C d^2-29 B e d+9 A e^2\right)\right)\right) \sqrt{c x^2+b x+a}}{15 e \left(c d^2-b e d+a e^2\right)^3 \sqrt{d+e x}}+\frac{2 \left(c d \left(2 C d^2+e (3 B d-8 A e)\right)+e \left(5 a e (2 C d-B e)-b \left(6 C d^2-B e d-4 A e^2\right)\right)\right) \sqrt{c x^2+b x+a}}{15 e \left(c d^2-b e d+a e^2\right)^2 (d+e x)^{3/2}}",1,"Result too large to show","C",0
272,0,0,510,2.2768693,"\int (g+h x)^m \left(a+b x+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^m*(a + b*x + c*x^2)^p*(d + e*x + f*x^2),x]","\int (g+h x)^m \left(a+b x+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","\frac{(g+h x)^{m+1} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (g+h x)}{2 c g-h \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (g+h x)}{2 c g-h \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(m+1;-p,-p;m+2;\frac{2 c (g+h x)}{2 c g-\left(b-\sqrt{b^2-4 a c}\right) h},\frac{2 c (g+h x)}{2 c g-\left(b+\sqrt{b^2-4 a c}\right) h}\right) \left(f h (m+1) (b g-a h)+c \left(2 f g^2 (p+1)-h (m+2 p+3) (e g-d h)\right)\right)}{c h^3 (m+1) (m+2 p+3)}-\frac{(g+h x)^{m+2} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (g+h x)}{2 c g-h \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (g+h x)}{2 c g-h \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} (b f h (m+p+2)+c (2 f g (p+1)-e h (m+2 p+3))) F_1\left(m+2;-p,-p;m+3;\frac{2 c (g+h x)}{2 c g-\left(b-\sqrt{b^2-4 a c}\right) h},\frac{2 c (g+h x)}{2 c g-\left(b+\sqrt{b^2-4 a c}\right) h}\right)}{c h^3 (m+2) (m+2 p+3)}+\frac{f (g+h x)^{m+1} \left(a+b x+c x^2\right)^{p+1}}{c h (m+2 p+3)}",1,"Integrate[(g + h*x)^m*(a + b*x + c*x^2)^p*(d + e*x + f*x^2), x]","F",-1
273,0,0,496,1.4623852,"\int (g+h x)^m \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^m*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2),x]","\int (g+h x)^m \sqrt{a+b x+c x^2} \left(d+e x+f x^2\right) \, dx","\frac{\sqrt{a+b x+c x^2} (g+h x)^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (g+h x)}{2 c g-\left(b-\sqrt{b^2-4 a c}\right) h},\frac{2 c (g+h x)}{2 c g-\left(b+\sqrt{b^2-4 a c}\right) h}\right) \left(f h (m+1) (b g-a h)+c \left(3 f g^2-h (m+4) (e g-d h)\right)\right)}{c h^3 (m+1) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left(\sqrt{b^2-4 a c}+b\right)}}}-\frac{\sqrt{a+b x+c x^2} (g+h x)^{m+2} (b f h (2 m+5)+c (6 f g-2 e h (m+4))) F_1\left(m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (g+h x)}{2 c g-\left(b-\sqrt{b^2-4 a c}\right) h},\frac{2 c (g+h x)}{2 c g-\left(b+\sqrt{b^2-4 a c}\right) h}\right)}{2 c h^3 (m+2) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left(b-\sqrt{b^2-4 a c}\right)}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left(\sqrt{b^2-4 a c}+b\right)}}}+\frac{f \left(a+b x+c x^2\right)^{3/2} (g+h x)^{m+1}}{c h (m+4)}",1,"Integrate[(g + h*x)^m*Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2), x]","F",-1
274,0,0,590,3.464011,"\int (g+h x)^{-3-2 p} \left(a+b x+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","Integrate[(g + h*x)^(-3 - 2*p)*(a + b*x + c*x^2)^p*(d + e*x + f*x^2),x]","\int (g+h x)^{-3-2 p} \left(a+b x+c x^2\right)^p \left(d+e x+f x^2\right) \, dx","-\frac{f (g+h x)^{-2 p} \left(a+b x+c x^2\right)^p \left(1-\frac{2 c (g+h x)}{2 c g-h \left(b-\sqrt{b^2-4 a c}\right)}\right)^{-p} \left(1-\frac{2 c (g+h x)}{2 c g-h \left(\sqrt{b^2-4 a c}+b\right)}\right)^{-p} F_1\left(-2 p;-p,-p;1-2 p;\frac{2 c (g+h x)}{2 c g-\left(b-\sqrt{b^2-4 a c}\right) h},\frac{2 c (g+h x)}{2 c g-\left(b+\sqrt{b^2-4 a c}\right) h}\right)}{2 h^3 p}-\frac{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) (g+h x)^{-2 p-1} \left(a+b x+c x^2\right)^p \left(\frac{\left(\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c g-h \left(b-\sqrt{b^2-4 a c}\right)\right)}{\left(-\sqrt{b^2-4 a c}+b+2 c x\right) \left(2 c g-h \left(\sqrt{b^2-4 a c}+b\right)\right)}\right)^{-p} \left(h \left(2 a h (2 f g-e h)-b \left(-d h^2-e g h+3 f g^2\right)\right)+2 c \left(f g^3-d g h^2\right)\right) \, _2F_1\left(-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (g+h x)}{\left(2 c g-\left(b+\sqrt{b^2-4 a c}\right) h\right) \left(b+2 c x-\sqrt{b^2-4 a c}\right)}\right)}{2 h^2 (2 p+1) \left(2 c g-h \left(b-\sqrt{b^2-4 a c}\right)\right) \left(a h^2-b g h+c g^2\right)}-\frac{(g+h x)^{-2 (p+1)} \left(a+b x+c x^2\right)^{p+1} \left(f g^2-h (e g-d h)\right)}{2 h (p+1) \left(a h^2-b g h+c g^2\right)}",1,"Integrate[(g + h*x)^(-3 - 2*p)*(a + b*x + c*x^2)^p*(d + e*x + f*x^2), x]","F",-1
275,1,119,41,0.1057877,"\int \left(d+f x^2\right)^p \left(2 c d f+2 b f^2 (3+2 p) x+2 c f^2 (3+2 p) x^2\right) \, dx","Integrate[(d + f*x^2)^p*(2*c*d*f + 2*b*f^2*(3 + 2*p)*x + 2*c*f^2*(3 + 2*p)*x^2),x]","\frac{f \left(d+f x^2\right)^p \left(\frac{f x^2}{d}+1\right)^{-p} \left((2 p+3) \left(3 b \left(d+f x^2\right) \left(\frac{f x^2}{d}+1\right)^p+2 c f (p+1) x^3 \, _2F_1\left(\frac{3}{2},-p;\frac{5}{2};-\frac{f x^2}{d}\right)\right)+6 c d (p+1) x \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{f x^2}{d}\right)\right)}{3 (p+1)}","\frac{b f (2 p+3) \left(d+f x^2\right)^{p+1}}{p+1}+2 c f x \left(d+f x^2\right)^{p+1}",1,"(f*(d + f*x^2)^p*(6*c*d*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, -((f*x^2)/d)] + (3 + 2*p)*(3*b*(d + f*x^2)*(1 + (f*x^2)/d)^p + 2*c*f*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, -((f*x^2)/d)])))/(3*(1 + p)*(1 + (f*x^2)/d)^p)","C",1
276,1,34,46,0.1299336,"\int \left(d+e x+f x^2\right)^p \left(-2 c e^2+2 c d f-c e^2 p+2 c f^2 (3+2 p) x^2\right) \, dx","Integrate[(d + e*x + f*x^2)^p*(-2*c*e^2 + 2*c*d*f - c*e^2*p + 2*c*f^2*(3 + 2*p)*x^2),x]","\frac{c (2 f (p+1) x-e (p+2)) (d+x (e+f x))^{p+1}}{p+1}","2 c f x \left(d+e x+f x^2\right)^{p+1}-\frac{c e (p+2) \left(d+e x+f x^2\right)^{p+1}}{p+1}",1,"(c*(-(e*(2 + p)) + 2*f*(1 + p)*x)*(d + x*(e + f*x))^(1 + p))/(1 + p)","A",1
277,1,43,57,0.3055813,"\int \left(d+e x+f x^2\right)^p \left(-2 c e^2+2 c d f+3 b e f-c e^2 p+2 b e f p+2 b f^2 (3+2 p) x+2 c f^2 (3+2 p) x^2\right) \, dx","Integrate[(d + e*x + f*x^2)^p*(-2*c*e^2 + 2*c*d*f + 3*b*e*f - c*e^2*p + 2*b*e*f*p + 2*b*f^2*(3 + 2*p)*x + 2*c*f^2*(3 + 2*p)*x^2),x]","\frac{(d+x (e+f x))^{p+1} (b f (2 p+3)-c e (p+2)+2 c f (p+1) x)}{p+1}","2 c f x \left(d+e x+f x^2\right)^{p+1}-\frac{(c e (p+2)-b f (2 p+3)) \left(d+e x+f x^2\right)^{p+1}}{p+1}",1,"((-(c*e*(2 + p)) + b*f*(3 + 2*p) + 2*c*f*(1 + p)*x)*(d + x*(e + f*x))^(1 + p))/(1 + p)","A",1
278,1,167,20,0.4485984,"\int (d+e x)^3 \left(a+b x+c x^2\right)^5 \left(d (6 b d+5 a e)+\left(12 c d^2+17 b d e+5 a e^2\right) x+e (29 c d+11 b e) x^2+17 c e^2 x^3\right) \, dx","Integrate[(d + e*x)^3*(a + b*x + c*x^2)^5*(d*(6*b*d + 5*a*e) + (12*c*d^2 + 17*b*d*e + 5*a*e^2)*x + e*(29*c*d + 11*b*e)*x^2 + 17*c*e^2*x^3),x]","x \left(a^6 e \left(5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right)+6 a^5 (b+c x) (d+e x)^5+15 a^4 x (b+c x)^2 (d+e x)^5+20 a^3 x^2 (b+c x)^3 (d+e x)^5+15 a^2 x^3 (b+c x)^4 (d+e x)^5+6 a x^4 (b+c x)^5 (d+e x)^5+x^5 (b+c x)^6 (d+e x)^5\right)","(d+e x)^5 \left(a+b x+c x^2\right)^6",1,"x*(6*a^5*(b + c*x)*(d + e*x)^5 + 15*a^4*x*(b + c*x)^2*(d + e*x)^5 + 20*a^3*x^2*(b + c*x)^3*(d + e*x)^5 + 15*a^2*x^3*(b + c*x)^4*(d + e*x)^5 + 6*a*x^4*(b + c*x)^5*(d + e*x)^5 + x^5*(b + c*x)^6*(d + e*x)^5 + a^6*e*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))","B",1
279,1,26,26,0.0048826,"\int \frac{x^2+x^3}{-2+x+x^2} \, dx","Integrate[(x^2 + x^3)/(-2 + x + x^2),x]","\frac{x^2}{2}+\frac{2}{3} \log (1-x)+\frac{4}{3} \log (x+2)","\frac{x^2}{2}+\frac{2}{3} \log (1-x)+\frac{4}{3} \log (x+2)",1,"x^2/2 + (2*Log[1 - x])/3 + (4*Log[2 + x])/3","A",1
280,1,282,346,0.7391456,"\int \frac{x^2 \left(d+e x+f x^2+g x^3\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(x^2*(d + e*x + f*x^2 + g*x^3))/Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} \left(16 c^2 \left(64 a^2 g-a c (80 e+x (45 f+32 g x))+2 c^2 x (30 d+x (20 e+3 x (5 f+4 g x)))\right)+4 b^2 c (-735 a g+300 c e+7 c x (25 f+18 g x))-8 b c^2 \left(2 c \left(90 d+x \left(50 e+35 f x+27 g x^2\right)\right)-a (275 f+161 g x)\right)+945 b^4 g-210 b^3 c (5 f+3 g x)\right)}{1920 c^5}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(40 b^3 c (2 c e-7 a g)-48 b^2 c^2 (2 c d-5 a f)+48 a b c^2 (5 a g-4 c e)+32 a c^3 (4 c d-3 a f)+63 b^5 g-70 b^4 c f\right)}{256 c^{11/2}}","\frac{x^2 \sqrt{a+b x+c x^2} \left(-64 a c g+63 b^2 g-70 b c f+80 c^2 e\right)}{240 c^3}-\frac{\sqrt{a+b x+c x^2} \left(-2 c x \left(-40 c^2 (9 a f+10 b e)+14 b c (46 a g+25 b f)-315 b^3 g+480 c^3 d\right)-60 b^2 c (20 c e-49 a g)+40 b c^2 (36 c d-55 a f)+256 a c^2 (5 c e-4 a g)-945 b^4 g+1050 b^3 c f\right)}{1920 c^5}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-40 b^3 c (2 c e-7 a g)+48 b^2 c^2 (2 c d-5 a f)+48 a b c^2 (4 c e-5 a g)-32 a c^3 (4 c d-3 a f)-63 b^5 g+70 b^4 c f\right)}{256 c^{11/2}}+\frac{x^3 \sqrt{a+b x+c x^2} (10 c f-9 b g)}{40 c^2}+\frac{g x^4 \sqrt{a+b x+c x^2}}{5 c}",1,"(Sqrt[a + x*(b + c*x)]*(945*b^4*g - 210*b^3*c*(5*f + 3*g*x) + 4*b^2*c*(300*c*e - 735*a*g + 7*c*x*(25*f + 18*g*x)) - 8*b*c^2*(-(a*(275*f + 161*g*x)) + 2*c*(90*d + x*(50*e + 35*f*x + 27*g*x^2))) + 16*c^2*(64*a^2*g - a*c*(80*e + x*(45*f + 32*g*x)) + 2*c^2*x*(30*d + x*(20*e + 3*x*(5*f + 4*g*x))))))/(1920*c^5) - ((-70*b^4*c*f - 48*b^2*c^2*(2*c*d - 5*a*f) + 32*a*c^3*(4*c*d - 3*a*f) + 63*b^5*g + 40*b^3*c*(2*c*e - 7*a*g) + 48*a*b*c^2*(-4*c*e + 5*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(11/2))","A",1
281,1,199,245,0.4428978,"\int \frac{x \left(d+e x+f x^2+g x^3\right)}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(x*(d + e*x + f*x^2 + g*x^3))/Sqrt[a + b*x + c*x^2],x]","\frac{\sqrt{a+x (b+c x)} \left(-8 c^2 \left(16 a f+9 a g x+18 b e+10 b f x+7 b g x^2\right)+10 b c (22 a g+12 b f+7 b g x)-105 b^3 g+16 c^3 \left(12 d+x \left(6 e+4 f x+3 g x^2\right)\right)\right)}{192 c^4}+\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(24 b^2 c (2 c e-5 a g)+32 b c^2 (3 a f-2 c d)+16 a c^2 (3 a g-4 c e)+35 b^4 g-40 b^3 c f\right)}{128 c^{9/2}}","\frac{\sqrt{a+b x+c x^2} \left(2 c x \left(-36 a c g+35 b^2 g-40 b c f+48 c^2 e\right)-16 c^2 (8 a f+9 b e)+20 b c (11 a g+6 b f)-105 b^3 g+192 c^3 d\right)}{192 c^4}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-24 b^2 c (2 c e-5 a g)+32 b c^2 (2 c d-3 a f)+16 a c^2 (4 c e-3 a g)-35 b^4 g+40 b^3 c f\right)}{128 c^{9/2}}+\frac{x^2 \sqrt{a+b x+c x^2} (8 c f-7 b g)}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}",1,"(Sqrt[a + x*(b + c*x)]*(-105*b^3*g + 10*b*c*(12*b*f + 22*a*g + 7*b*g*x) - 8*c^2*(18*b*e + 16*a*f + 10*b*f*x + 9*a*g*x + 7*b*g*x^2) + 16*c^3*(12*d + x*(6*e + 4*f*x + 3*g*x^2))))/(192*c^4) + ((-40*b^3*c*f + 32*b*c^2*(-2*c*d + 3*a*f) + 35*b^4*g + 24*b^2*c*(2*c*e - 5*a*g) + 16*a*c^2*(-4*c*e + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(128*c^(9/2))","A",1
282,1,141,177,0.2638313,"\int \frac{d+e x+f x^2+g x^3}{\sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/Sqrt[a + b*x + c*x^2],x]","\frac{3 \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right)+2 \sqrt{c} \sqrt{a+x (b+c x)} \left(-2 c (8 a g+9 b f+5 b g x)+15 b^2 g+4 c^2 (6 e+x (3 f+2 g x))\right)}{48 c^{7/2}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right)}{16 c^{7/2}}+\frac{\sqrt{a+b x+c x^2} \left(-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right)}{24 c^3}+\frac{x \sqrt{a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac{g x^2 \sqrt{a+b x+c x^2}}{3 c}",1,"(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^2*g - 2*c*(9*b*f + 8*a*g + 5*b*g*x) + 4*c^2*(6*e + x*(3*f + 2*g*x))) + 3*(16*c^3*d - 8*c^2*(b*e + a*f) - 5*b^3*g + 6*b*c*(b*f + 2*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(48*c^(7/2))","A",1
283,1,134,155,0.3692158,"\int \frac{d+e x+f x^2+g x^3}{x \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x*Sqrt[a + b*x + c*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right) \left(-4 c (a g+b f)+3 b^2 g+8 c^2 e\right)}{8 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} (-3 b g+4 c f+2 c g x)}{4 c^2}-\frac{d \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{\sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right) \left(-4 c (a g+b f)+3 b^2 g+8 c^2 e\right)}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c f-3 b g)}{4 c^2}-\frac{d \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{\sqrt{a}}+\frac{g x \sqrt{a+b x+c x^2}}{2 c}",1,"((4*c*f - 3*b*g + 2*c*g*x)*Sqrt[a + x*(b + c*x)])/(4*c^2) - (d*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/Sqrt[a] + ((8*c^2*e + 3*b^2*g - 4*c*(b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(5/2))","A",1
284,1,127,139,0.3987984,"\int \frac{d+e x+f x^2+g x^3}{x^2 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x^2*Sqrt[a + b*x + c*x^2]),x]","\frac{(b d-2 a e) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right)}{2 a^{3/2}}+\frac{(2 c f-b g) \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{2 c^{3/2}}+\frac{\sqrt{a+x (b+c x)} (a g x-c d)}{a c x}","\frac{(b d-2 a e) \tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right)}{2 a^{3/2}}+\frac{(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 c^{3/2}}-\frac{d \sqrt{a+b x+c x^2}}{a x}+\frac{g \sqrt{a+b x+c x^2}}{c}",1,"((-(c*d) + a*g*x)*Sqrt[a + x*(b + c*x)])/(a*c*x) + ((b*d - 2*a*e)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(2*a^(3/2)) + ((2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(3/2))","A",1
285,1,137,159,0.3585417,"\int \frac{d+e x+f x^2+g x^3}{x^3 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x^3*Sqrt[a + b*x + c*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right) \left(4 a b e+4 a (c d-2 a f)-3 b^2 d\right)}{8 a^{5/2}}+\frac{\sqrt{a+x (b+c x)} (3 b d x-2 a (d+2 e x))}{4 a^2 x^2}+\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right)}{\sqrt{c}}","\frac{\sqrt{a+b x+c x^2} (3 b d-4 a e)}{4 a^2 x}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right) \left(8 a^2 f-4 a b e-4 a c d+3 b^2 d\right)}{8 a^{5/2}}-\frac{d \sqrt{a+b x+c x^2}}{2 a x^2}+\frac{g \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{\sqrt{c}}",1,"(Sqrt[a + x*(b + c*x)]*(3*b*d*x - 2*a*(d + 2*e*x)))/(4*a^2*x^2) + ((-3*b^2*d + 4*a*b*e + 4*a*(c*d - 2*a*f))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(8*a^(5/2)) + (g*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c]","A",1
286,1,150,186,0.3104667,"\int \frac{d+e x+f x^2+g x^3}{x^4 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x^4*Sqrt[a + b*x + c*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right) \left(8 a^2 (c e-2 a g)-6 a b^2 e+4 a b (2 a f-3 c d)+5 b^3 d\right)}{16 a^{7/2}}-\frac{\sqrt{a+x (b+c x)} \left(4 a^2 (2 d+3 x (e+2 f x))-2 a x (5 b d+9 b e x+8 c d x)+15 b^2 d x^2\right)}{24 a^3 x^3}","\frac{\sqrt{a+b x+c x^2} (5 b d-6 a e)}{12 a^2 x^2}+\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right) \left(8 a^2 (c e-2 a g)-6 a b^2 e-4 a b (3 c d-2 a f)+5 b^3 d\right)}{16 a^{7/2}}-\frac{\sqrt{a+b x+c x^2} \left(24 a^2 f-18 a b e-16 a c d+15 b^2 d\right)}{24 a^3 x}-\frac{d \sqrt{a+b x+c x^2}}{3 a x^3}",1,"-1/24*(Sqrt[a + x*(b + c*x)]*(15*b^2*d*x^2 - 2*a*x*(5*b*d + 8*c*d*x + 9*b*e*x) + 4*a^2*(2*d + 3*x*(e + 2*f*x))))/(a^3*x^3) + ((5*b^3*d - 6*a*b^2*e + 4*a*b*(-3*c*d + 2*a*f) + 8*a^2*(c*e - 2*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(16*a^(7/2))","A",1
287,1,212,270,0.5218434,"\int \frac{d+e x+f x^2+g x^3}{x^5 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x^5*Sqrt[a + b*x + c*x^2]),x]","\frac{\sqrt{a+x (b+c x)} \left(-16 a^3 \left(3 d+4 e x+6 x^2 (f+2 g x)\right)+8 a^2 x (7 b d+2 b x (5 e+9 f x)+c x (9 d+16 e x))-10 a b x^2 (7 b d+12 b e x+22 c d x)+105 b^3 d x^3\right)}{192 a^4 x^4}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right) \left(32 a^2 b (3 c e-2 a g)+16 a^2 c (3 c d-4 a f)-40 a b^3 e+24 a b^2 (2 a f-5 c d)+35 b^4 d\right)}{128 a^{9/2}}","\frac{\sqrt{a+b x+c x^2} (7 b d-8 a e)}{24 a^2 x^3}-\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right) \left(32 a^2 b (3 c e-2 a g)+16 a^2 c (3 c d-4 a f)-40 a b^3 e-24 a b^2 (5 c d-2 a f)+35 b^4 d\right)}{128 a^{9/2}}+\frac{\sqrt{a+b x+c x^2} \left(64 a^2 (2 c e-3 a g)-120 a b^2 e-4 a b (55 c d-36 a f)+105 b^3 d\right)}{192 a^4 x}-\frac{\sqrt{a+b x+c x^2} \left(48 a^2 f-40 a b e-36 a c d+35 b^2 d\right)}{96 a^3 x^2}-\frac{d \sqrt{a+b x+c x^2}}{4 a x^4}",1,"(Sqrt[a + x*(b + c*x)]*(105*b^3*d*x^3 - 10*a*b*x^2*(7*b*d + 22*c*d*x + 12*b*e*x) + 8*a^2*x*(7*b*d + c*x*(9*d + 16*e*x) + 2*b*x*(5*e + 9*f*x)) - 16*a^3*(3*d + 4*e*x + 6*x^2*(f + 2*g*x))))/(192*a^4*x^4) - ((35*b^4*d - 40*a*b^3*e + 16*a^2*c*(3*c*d - 4*a*f) + 24*a*b^2*(-5*c*d + 2*a*f) + 32*a^2*b*(3*c*e - 2*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(128*a^(9/2))","A",1
288,1,299,371,0.7312859,"\int \frac{d+e x+f x^2+g x^3}{x^6 \sqrt{a+b x+c x^2}} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3)/(x^6*Sqrt[a + b*x + c*x^2]),x]","\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right) \left(32 a^3 c (4 a g-3 c e)-48 a^2 b^2 (2 a g-5 c e)-48 a^2 b c (4 a f-5 c d)-70 a b^4 e+40 a b^3 (2 a f-7 c d)+63 b^5 d\right)}{256 a^{11/2}}-\frac{\sqrt{a+x (b+c x)} \left(32 a^4 \left(12 d+5 x \left(3 e+4 f x+6 g x^2\right)\right)-16 a^3 x (b (27 d+5 x (7 e+2 x (5 f+9 g x)))+c x (32 d+5 x (9 e+16 f x)))+4 a^2 x^2 \left(b^2 (126 d+25 x (7 e+12 f x))+2 b c x (161 d+275 e x)+256 c^2 d x^2\right)-210 a b^2 x^3 (3 b d+5 b e x+14 c d x)+945 b^4 d x^4\right)}{1920 a^5 x^5}","\frac{\sqrt{a+b x+c x^2} (9 b d-10 a e)}{40 a^2 x^4}-\frac{\sqrt{a+b x+c x^2} \left(40 a^2 b (55 c e-36 a g)+256 a^2 c (4 c d-5 a f)-1050 a b^3 e-60 a b^2 (49 c d-20 a f)+945 b^4 d\right)}{1920 a^5 x}+\frac{\sqrt{a+b x+c x^2} \left(120 a^2 (3 c e-4 a g)-350 a b^2 e-4 a b (161 c d-100 a f)+315 b^3 d\right)}{960 a^4 x^2}-\frac{\sqrt{a+b x+c x^2} \left(80 a^2 f-70 a b e-64 a c d+63 b^2 d\right)}{240 a^3 x^3}+\frac{\tanh ^{-1}\left(\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right) \left(-32 a^3 c (3 c e-4 a g)+48 a^2 b^2 (5 c e-2 a g)+48 a^2 b c (5 c d-4 a f)-70 a b^4 e-40 a b^3 (7 c d-2 a f)+63 b^5 d\right)}{256 a^{11/2}}-\frac{d \sqrt{a+b x+c x^2}}{5 a x^5}",1,"-1/1920*(Sqrt[a + x*(b + c*x)]*(945*b^4*d*x^4 - 210*a*b^2*x^3*(3*b*d + 14*c*d*x + 5*b*e*x) + 32*a^4*(12*d + 5*x*(3*e + 4*f*x + 6*g*x^2)) + 4*a^2*x^2*(256*c^2*d*x^2 + 2*b*c*x*(161*d + 275*e*x) + b^2*(126*d + 25*x*(7*e + 12*f*x))) - 16*a^3*x*(c*x*(32*d + 5*x*(9*e + 16*f*x)) + b*(27*d + 5*x*(7*e + 2*x*(5*f + 9*g*x))))))/(a^5*x^5) + ((63*b^5*d - 70*a*b^4*e + 40*a*b^3*(-7*c*d + 2*a*f) - 48*a^2*b*c*(-5*c*d + 4*a*f) - 48*a^2*b^2*(-5*c*e + 2*a*g) + 32*a^3*c*(-3*c*e + 4*a*g))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(256*a^(11/2))","A",1
289,1,212,258,0.0397869,"\int (d+e x)^3 \left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^3*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","6 d^3 x+\frac{1}{8} e x^8 \left(60 d^2-51 d e+17 e^2\right)+d x^3 \left(7 d^2+7 d e+6 e^2\right)+\frac{1}{2} d^2 x^2 (7 d+18 e)+\frac{1}{7} x^7 \left(20 d^3-51 d^2 e+51 d e^2-4 e^3\right)+\frac{1}{6} x^6 \left(-17 d^3+51 d^2 e-12 d e^2+21 e^3\right)+\frac{1}{5} x^5 \left(17 d^3-12 d^2 e+63 d e^2+7 e^3\right)+\frac{1}{4} x^4 \left(-4 d^3+63 d^2 e+21 d e^2+6 e^3\right)+\frac{1}{9} e^2 x^9 (60 d-17 e)+2 e^3 x^{10}","\frac{\left(300 d^2+85 d e+17 e^2\right) (d+e x)^8}{8 e^7}-\frac{2 \left(200 d^3+85 d^2 e+34 d e^2+2 e^3\right) (d+e x)^7}{7 e^7}+\frac{\left(300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right) (d+e x)^6}{6 e^7}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) (d+e x)^4}{4 e^7}-\frac{\left(120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right) (d+e x)^5}{5 e^7}+\frac{2 (d+e x)^{10}}{e^7}-\frac{(120 d+17 e) (d+e x)^9}{9 e^7}",1,"6*d^3*x + (d^2*(7*d + 18*e)*x^2)/2 + d*(7*d^2 + 7*d*e + 6*e^2)*x^3 + ((-4*d^3 + 63*d^2*e + 21*d*e^2 + 6*e^3)*x^4)/4 + ((17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e^3)*x^5)/5 + ((-17*d^3 + 51*d^2*e - 12*d*e^2 + 21*e^3)*x^6)/6 + ((20*d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7)/7 + (e*(60*d^2 - 51*d*e + 17*e^2)*x^8)/8 + ((60*d - 17*e)*e^2*x^9)/9 + 2*e^3*x^10","A",1
290,1,136,157,0.0321106,"\int (d+e x)^2 \left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^2*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","d^2 \left(\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x\right)+d e \left(5 x^8-\frac{34 x^7}{7}+\frac{17 x^6}{3}-\frac{8 x^5}{5}+\frac{21 x^4}{2}+\frac{14 x^3}{3}+6 x^2\right)+\frac{e^2 \left(5600 x^6-5355 x^5+6120 x^4-1680 x^3+10584 x^2+4410 x+5040\right) x^3}{2520}","\frac{1}{7} x^7 \left(20 d^2-34 d e+17 e^2\right)-\frac{1}{6} x^6 \left(17 d^2-34 d e+4 e^2\right)+\frac{1}{5} x^5 \left(17 d^2-8 d e+21 e^2\right)-\frac{1}{4} x^4 \left(4 d^2-42 d e-7 e^2\right)+\frac{1}{3} x^3 \left(21 d^2+14 d e+6 e^2\right)+6 d^2 x+\frac{1}{8} e x^8 (40 d-17 e)+\frac{1}{2} d x^2 (7 d+12 e)+\frac{20 e^2 x^9}{9}",1,"(e^2*x^3*(5040 + 4410*x + 10584*x^2 - 1680*x^3 + 6120*x^4 - 5355*x^5 + 5600*x^6))/2520 + d^2*(6*x + (7*x^2)/2 + 7*x^3 - x^4 + (17*x^5)/5 - (17*x^6)/6 + (20*x^7)/7) + d*e*(6*x^2 + (14*x^3)/3 + (21*x^4)/2 - (8*x^5)/5 + (17*x^6)/3 - (34*x^7)/7 + 5*x^8)","A",1
291,1,93,93,0.0127742,"\int (d+e x) \left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)+\frac{1}{4} x^4 (21 e-4 d)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2}","\frac{1}{7} x^7 (20 d-17 e)-\frac{17}{6} x^6 (d-e)+\frac{1}{5} x^5 (17 d-4 e)-\frac{1}{4} x^4 (4 d-21 e)+\frac{7}{3} x^3 (3 d+e)+\frac{1}{2} x^2 (7 d+6 e)+6 d x+\frac{5 e x^8}{2}",1,"6*d*x + ((7*d + 6*e)*x^2)/2 + (7*(3*d + e)*x^3)/3 + ((-4*d + 21*e)*x^4)/4 + ((17*d - 4*e)*x^5)/5 - (17*(d - e)*x^6)/6 + ((20*d - 17*e)*x^7)/7 + (5*e*x^8)/2","A",1
292,1,42,42,0.0015232,"\int \left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x","\frac{20 x^7}{7}-\frac{17 x^6}{6}+\frac{17 x^5}{5}-x^4+7 x^3+\frac{7 x^2}{2}+6 x",1,"6*x + (7*x^2)/2 + 7*x^3 - x^4 + (17*x^5)/5 - (17*x^6)/6 + (20*x^7)/7","A",1
293,1,179,228,0.0589618,"\int \frac{\left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{d+e x} \, dx","Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x),x]","\frac{e x \left(-1200 d^5+60 d^4 e (10 x-17)-10 d^3 e^2 \left(40 x^2-51 x+102\right)+10 d^2 e^3 \left(30 x^3-34 x^2+51 x-24\right)-5 d e^4 \left(48 x^4-51 x^3+68 x^2-24 x+252\right)+e^5 \left(200 x^5-204 x^4+255 x^3-80 x^2+630 x+420\right)\right)+60 \left(20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right) \log (d+e x)}{60 e^7}","\frac{x^4 \left(20 d^2+17 d e+17 e^2\right)}{4 e^3}-\frac{x^3 \left(20 d^3+17 d^2 e+17 d e^2+4 e^3\right)}{3 e^4}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{e^7}+\frac{x^2 \left(20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right)}{2 e^5}-\frac{x \left(20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right)}{e^6}-\frac{x^5 (20 d+17 e)}{5 e^2}+\frac{10 x^6}{3 e}",1,"(e*x*(-1200*d^5 + 60*d^4*e*(-17 + 10*x) - 10*d^3*e^2*(102 - 51*x + 40*x^2) + 10*d^2*e^3*(-24 + 51*x - 34*x^2 + 30*x^3) - 5*d*e^4*(252 - 24*x + 68*x^2 - 51*x^3 + 48*x^4) + e^5*(420 + 630*x - 80*x^2 + 255*x^3 - 204*x^4 + 200*x^5)) + 60*(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6)*Log[d + e*x])/(60*e^7)","A",1
294,1,223,228,0.0871884,"\int \frac{\left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{(d+e x)^2} \, dx","Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^2,x]","\frac{4 e^3 x^3 \left(60 d^2+34 d e+17 e^2\right)-6 e^2 x^2 \left(80 d^3+51 d^2 e+34 d e^2+4 e^3\right)+12 e x \left(100 d^4+68 d^3 e+51 d^2 e^2+8 d e^3+21 e^4\right)-12 \left(120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right) \log (d+e x)-\frac{12 \left(20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right)}{d+e x}-3 e^4 x^4 (40 d+17 e)+48 e^5 x^5}{12 e^7}","\frac{x^3 \left(60 d^2+34 d e+17 e^2\right)}{3 e^4}-\frac{x^2 \left(80 d^3+51 d^2 e+34 d e^2+4 e^3\right)}{2 e^5}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{e^7 (d+e x)}+\frac{x \left(100 d^4+68 d^3 e+51 d^2 e^2+8 d e^3+21 e^4\right)}{e^6}-\frac{\left(120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right) \log (d+e x)}{e^7}-\frac{x^4 (40 d+17 e)}{4 e^3}+\frac{4 x^5}{e^2}",1,"(12*e*(100*d^4 + 68*d^3*e + 51*d^2*e^2 + 8*d*e^3 + 21*e^4)*x - 6*e^2*(80*d^3 + 51*d^2*e + 34*d*e^2 + 4*e^3)*x^2 + 4*e^3*(60*d^2 + 34*d*e + 17*e^2)*x^3 - 3*e^4*(40*d + 17*e)*x^4 + 48*e^5*x^5 - (12*(20*d^6 + 17*d^5*e + 17*d^4*e^2 + 4*d^3*e^3 + 21*d^2*e^4 - 7*d*e^5 + 6*e^6))/(d + e*x) - 12*(120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*Log[d + e*x])/(12*e^7)","A",1
295,1,204,231,0.0661742,"\int \frac{\left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{(d+e x)^3} \, dx","Integrate[((3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^3,x]","\frac{660 d^6+d^5 e (459-480 x)-51 d^4 e^2 \left(40 x^2+2 x-7\right)-3 d^3 e^3 \left(200 x^3+357 x^2-34 x-20\right)+d^2 e^4 \left(150 x^4-340 x^3-561 x^2+48 x+189\right)+6 \left(300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right) (d+e x)^2 \log (d+e x)-d e^5 \left(60 x^5-85 x^4+204 x^3+48 x^2-252 x+21\right)+e^6 \left(30 x^6-34 x^5+51 x^4-24 x^3-42 x-18\right)}{6 e^7 (d+e x)^2}","\frac{x^2 \left(120 d^2+51 d e+17 e^2\right)}{2 e^5}-\frac{x \left(200 d^3+102 d^2 e+51 d e^2+4 e^3\right)}{e^6}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{2 e^7 (d+e x)^2}+\frac{\left(300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right) \log (d+e x)}{e^7}+\frac{120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5}{e^7 (d+e x)}-\frac{x^3 (60 d+17 e)}{3 e^4}+\frac{5 x^4}{e^3}",1,"(660*d^6 + d^5*e*(459 - 480*x) - 51*d^4*e^2*(-7 + 2*x + 40*x^2) - 3*d^3*e^3*(-20 - 34*x + 357*x^2 + 200*x^3) + d^2*e^4*(189 + 48*x - 561*x^2 - 340*x^3 + 150*x^4) - d*e^5*(21 - 252*x + 48*x^2 + 204*x^3 - 85*x^4 + 60*x^5) + e^6*(-18 - 42*x - 24*x^3 + 51*x^4 - 34*x^5 + 30*x^6) + 6*(300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^2*Log[d + e*x])/(6*e^7*(d + e*x)^2)","A",1
296,1,277,391,0.0425805,"\int (d+e x)^3 \left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^3*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","18 d^3 x+\frac{3}{10} e x^{10} \left(100 d^2-45 d e+37 e^2\right)+\frac{1}{3} d x^3 \left(107 d^2+99 d e+54 e^2\right)+\frac{3}{2} d^2 x^2 (11 d+18 e)+\frac{1}{9} x^9 \left(100 d^3-135 d^2 e+333 d e^2-37 e^3\right)+\frac{1}{8} x^8 \left(-45 d^3+333 d^2 e-111 d e^2+148 e^3\right)+\frac{1}{7} x^7 \left(111 d^3-111 d^2 e+444 d e^2+65 e^3\right)+\frac{1}{6} x^6 \left(-37 d^3+444 d^2 e+195 d e^2+107 e^3\right)+\frac{1}{5} x^5 \left(148 d^3+195 d^2 e+321 d e^2+33 e^3\right)+\frac{1}{4} x^4 \left(65 d^3+321 d^2 e+99 d e^2+18 e^3\right)+\frac{15}{11} e^2 x^{11} (20 d-3 e)+\frac{25 e^3 x^{12}}{3}","\frac{\left(2800 d^2+315 d e+111 e^2\right) (d+e x)^{10}}{10 e^9}-\frac{\left(5600 d^3+945 d^2 e+666 d e^2+37 e^3\right) (d+e x)^9}{9 e^9}+\frac{\left(7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right) (d+e x)^8}{8 e^9}+\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) (d+e x)^4}{4 e^9}-\frac{\left(5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right) (d+e x)^7}{7 e^9}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right) (d+e x)^5}{5 e^9}+\frac{\left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right) (d+e x)^6}{6 e^9}+\frac{25 (d+e x)^{12}}{3 e^9}-\frac{5 (160 d+9 e) (d+e x)^{11}}{11 e^9}",1,"18*d^3*x + (3*d^2*(11*d + 18*e)*x^2)/2 + (d*(107*d^2 + 99*d*e + 54*e^2)*x^3)/3 + ((65*d^3 + 321*d^2*e + 99*d*e^2 + 18*e^3)*x^4)/4 + ((148*d^3 + 195*d^2*e + 321*d*e^2 + 33*e^3)*x^5)/5 + ((-37*d^3 + 444*d^2*e + 195*d*e^2 + 107*e^3)*x^6)/6 + ((111*d^3 - 111*d^2*e + 444*d*e^2 + 65*e^3)*x^7)/7 + ((-45*d^3 + 333*d^2*e - 111*d*e^2 + 148*e^3)*x^8)/8 + ((100*d^3 - 135*d^2*e + 333*d*e^2 - 37*e^3)*x^9)/9 + (3*e*(100*d^2 - 45*d*e + 37*e^2)*x^10)/10 + (15*(20*d - 3*e)*e^2*x^11)/11 + (25*e^3*x^12)/3","A",1
297,1,201,201,0.0261056,"\int (d+e x)^2 \left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^2*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{1}{9} x^9 \left(100 d^2-90 d e+111 e^2\right)+\frac{1}{8} x^8 \left(-45 d^2+222 d e-37 e^2\right)+\frac{37}{7} x^7 \left(3 d^2-2 d e+4 e^2\right)+\frac{1}{6} x^6 \left(-37 d^2+296 d e+65 e^2\right)+\frac{1}{5} x^5 \left(148 d^2+130 d e+107 e^2\right)+\frac{1}{4} x^4 \left(65 d^2+214 d e+33 e^2\right)+\frac{1}{3} x^3 \left(107 d^2+66 d e+18 e^2\right)+18 d^2 x+\frac{1}{2} e x^{10} (40 d-9 e)+\frac{3}{2} d x^2 (11 d+12 e)+\frac{100 e^2 x^{11}}{11}","\frac{1}{9} x^9 \left(100 d^2-90 d e+111 e^2\right)-\frac{1}{8} x^8 \left(45 d^2-222 d e+37 e^2\right)+\frac{37}{7} x^7 \left(3 d^2-2 d e+4 e^2\right)-\frac{1}{6} x^6 \left(37 d^2-296 d e-65 e^2\right)+\frac{1}{5} x^5 \left(148 d^2+130 d e+107 e^2\right)+\frac{1}{4} x^4 \left(65 d^2+214 d e+33 e^2\right)+\frac{1}{3} x^3 \left(107 d^2+66 d e+18 e^2\right)+18 d^2 x+\frac{1}{2} e x^{10} (40 d-9 e)+\frac{3}{2} d x^2 (11 d+12 e)+\frac{100 e^2 x^{11}}{11}",1,"18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^5)/5 + ((-37*d^2 + 296*d*e + 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2)*x^7)/7 + ((-45*d^2 + 222*d*e - 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e^2)*x^9)/9 + ((40*d - 9*e)*e*x^10)/2 + (100*e^2*x^11)/11","A",1
298,1,121,121,0.0162701,"\int (d+e x) \left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{5}{9} x^9 (20 d-9 e)-\frac{3}{8} x^8 (15 d-37 e)+\frac{37}{7} x^7 (3 d-e)-\frac{37}{6} x^6 (d-4 e)+\frac{1}{5} x^5 (148 d+65 e)+\frac{1}{4} x^4 (65 d+107 e)+\frac{1}{3} x^3 (107 d+33 e)+\frac{3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}","\frac{5}{9} x^9 (20 d-9 e)-\frac{3}{8} x^8 (15 d-37 e)+\frac{37}{7} x^7 (3 d-e)-\frac{37}{6} x^6 (d-4 e)+\frac{1}{5} x^5 (148 d+65 e)+\frac{1}{4} x^4 (65 d+107 e)+\frac{1}{3} x^3 (107 d+33 e)+\frac{3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}",1,"18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10","A",1
299,1,60,60,0.001146,"\int \left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{100 x^9}{9}-\frac{45 x^8}{8}+\frac{111 x^7}{7}-\frac{37 x^6}{6}+\frac{148 x^5}{5}+\frac{65 x^4}{4}+\frac{107 x^3}{3}+\frac{33 x^2}{2}+18 x","\frac{100 x^9}{9}-\frac{45 x^8}{8}+\frac{111 x^7}{7}-\frac{37 x^6}{6}+\frac{148 x^5}{5}+\frac{65 x^4}{4}+\frac{107 x^3}{3}+\frac{33 x^2}{2}+18 x",1,"18*x + (33*x^2)/2 + (107*x^3)/3 + (65*x^4)/4 + (148*x^5)/5 - (37*x^6)/6 + (111*x^7)/7 - (45*x^8)/8 + (100*x^9)/9","A",1
300,1,262,352,0.1218742,"\int \frac{\left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{d+e x} \, dx","Integrate[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x),x]","\frac{\left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \left(5 d^2-2 d e+3 e^2\right)^2 \log (d+e x)}{e^9}+\frac{x \left(-42000 d^7+2100 d^6 e (10 x-9)-70 d^5 e^2 \left(200 x^2-135 x+666\right)+210 d^4 e^3 \left(50 x^3-30 x^2+111 x-74\right)-105 d^3 e^4 \left(80 x^4-45 x^3+148 x^2-74 x+592\right)+35 d^2 e^5 \left(200 x^5-108 x^4+333 x^3-148 x^2+888 x+780\right)-d e^6 \left(6000 x^6-3150 x^5+9324 x^4-3885 x^3+20720 x^2+13650 x+44940\right)+2 e^7 \left(2625 x^7-1350 x^6+3885 x^5-1554 x^4+7770 x^3+4550 x^2+11235 x+6930\right)\right)}{420 e^8}","\frac{x^6 \left(100 d^2+45 d e+111 e^2\right)}{6 e^3}-\frac{x^5 \left(100 d^3+45 d^2 e+111 d e^2+37 e^3\right)}{5 e^4}+\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{e^9}+\frac{x^4 \left(100 d^4+45 d^3 e+111 d^2 e^2+37 d e^3+148 e^4\right)}{4 e^5}-\frac{x^3 \left(100 d^5+45 d^4 e+111 d^3 e^2+37 d^2 e^3+148 d e^4-65 e^5\right)}{3 e^6}+\frac{x^2 \left(100 d^6+45 d^5 e+111 d^4 e^2+37 d^3 e^3+148 d^2 e^4-65 d e^5+107 e^6\right)}{2 e^7}-\frac{x \left(100 d^7+45 d^6 e+111 d^5 e^2+37 d^4 e^3+148 d^3 e^4-65 d^2 e^5+107 d e^6-33 e^7\right)}{e^8}-\frac{5 x^7 (20 d+9 e)}{7 e^2}+\frac{25 x^8}{2 e}",1,"(x*(-42000*d^7 + 2100*d^6*e*(-9 + 10*x) - 70*d^5*e^2*(666 - 135*x + 200*x^2) + 210*d^4*e^3*(-74 + 111*x - 30*x^2 + 50*x^3) - 105*d^3*e^4*(592 - 74*x + 148*x^2 - 45*x^3 + 80*x^4) + 35*d^2*e^5*(780 + 888*x - 148*x^2 + 333*x^3 - 108*x^4 + 200*x^5) - d*e^6*(44940 + 13650*x + 20720*x^2 - 3885*x^3 + 9324*x^4 - 3150*x^5 + 6000*x^6) + 2*e^7*(6930 + 11235*x + 4550*x^2 + 7770*x^3 - 1554*x^4 + 3885*x^5 - 1350*x^6 + 2625*x^7)))/(420*e^8) + ((5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/e^9","A",1
301,1,342,353,0.1385569,"\int \frac{\left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{(d+e x)^2} \, dx","Integrate[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^2,x]","\frac{252 e^5 x^5 \left(100 d^2+30 d e+37 e^2\right)-105 e^4 x^4 \left(400 d^3+135 d^2 e+222 d e^2+37 e^3\right)+140 e^3 x^3 \left(500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right)-\frac{420 \left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{d+e x}-210 e^2 x^2 \left(600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right)+420 e x \left(700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right)-420 \left(800 d^7+315 d^6 e+666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+214 d e^6-33 e^7\right) \log (d+e x)-350 e^6 x^6 (40 d+9 e)+6000 e^7 x^7}{420 e^9}","\frac{3 x^5 \left(100 d^2+30 d e+37 e^2\right)}{5 e^4}-\frac{x^4 \left(400 d^3+135 d^2 e+222 d e^2+37 e^3\right)}{4 e^5}-\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{e^9 (d+e x)}+\frac{x^3 \left(500 d^4+180 d^3 e+333 d^2 e^2+74 d e^3+148 e^4\right)}{3 e^6}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right) \log (d+e x)}{e^9}-\frac{x^2 \left(600 d^5+225 d^4 e+444 d^3 e^2+111 d^2 e^3+296 d e^4-65 e^5\right)}{2 e^7}+\frac{x \left(700 d^6+270 d^5 e+555 d^4 e^2+148 d^3 e^3+444 d^2 e^4-130 d e^5+107 e^6\right)}{e^8}-\frac{5 x^6 (40 d+9 e)}{6 e^3}+\frac{100 x^7}{7 e^2}",1,"(420*e*(700*d^6 + 270*d^5*e + 555*d^4*e^2 + 148*d^3*e^3 + 444*d^2*e^4 - 130*d*e^5 + 107*e^6)*x - 210*e^2*(600*d^5 + 225*d^4*e + 444*d^3*e^2 + 111*d^2*e^3 + 296*d*e^4 - 65*e^5)*x^2 + 140*e^3*(500*d^4 + 180*d^3*e + 333*d^2*e^2 + 74*d*e^3 + 148*e^4)*x^3 - 105*e^4*(400*d^3 + 135*d^2*e + 222*d*e^2 + 37*e^3)*x^4 + 252*e^5*(100*d^2 + 30*d*e + 37*e^2)*x^5 - 350*e^6*(40*d + 9*e)*x^6 + 6000*e^7*x^7 - (420*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(d + e*x) - 420*(800*d^7 + 315*d^6*e + 666*d^5*e^2 + 185*d^4*e^3 + 592*d^3*e^4 - 195*d^2*e^5 + 214*d*e^6 - 33*e^7)*Log[d + e*x])/(420*e^9)","A",1
302,1,311,354,0.101971,"\int \frac{\left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{(d+e x)^3} \, dx","Integrate[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^3,x]","\frac{9000 d^8-390 d^7 e (40 x-9)-18 d^6 e^2 \left(2300 x^2+240 x-407\right)-2 d^5 e^3 \left(5600 x^3+6750 x^2+2664 x-999\right)+4 d^4 e^4 \left(700 x^4-945 x^3-5661 x^2-111 x+1554\right)-d^3 e^5 \left(1120 x^5-945 x^4+6660 x^3+4662 x^2-1776 x+1950\right)+d^2 e^6 \left(560 x^6-378 x^5+1665 x^4-1480 x^3-9768 x^2-1560 x+1926\right)+12 \left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right) (d+e x)^2 \log (d+e x)+d e^7 \left(-320 x^7+189 x^6-666 x^5+370 x^4-3552 x^3+1560 x^2+2568 x-198\right)+e^8 \left(200 x^8-108 x^7+333 x^6-148 x^5+888 x^4+780 x^3-396 x-108\right)}{12 e^9 (d+e x)^2}","\frac{3 x^4 \left(200 d^2+45 d e+37 e^2\right)}{4 e^5}-\frac{x^3 \left(1000 d^3+270 d^2 e+333 d e^2+37 e^3\right)}{3 e^6}-\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{2 e^9 (d+e x)^2}+\frac{x^2 \left(1500 d^4+450 d^3 e+666 d^2 e^2+111 d e^3+148 e^4\right)}{2 e^7}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right)}{e^9 (d+e x)}-\frac{x \left(2100 d^5+675 d^4 e+1110 d^3 e^2+222 d^2 e^3+444 d e^4-65 e^5\right)}{e^8}+\frac{\left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right) \log (d+e x)}{e^9}-\frac{3 x^5 (20 d+3 e)}{e^4}+\frac{50 x^6}{3 e^3}",1,"(9000*d^8 - 390*d^7*e*(-9 + 40*x) - 18*d^6*e^2*(-407 + 240*x + 2300*x^2) - 2*d^5*e^3*(-999 + 2664*x + 6750*x^2 + 5600*x^3) + 4*d^4*e^4*(1554 - 111*x - 5661*x^2 - 945*x^3 + 700*x^4) - d^3*e^5*(1950 - 1776*x + 4662*x^2 + 6660*x^3 - 945*x^4 + 1120*x^5) + d^2*e^6*(1926 - 1560*x - 9768*x^2 - 1480*x^3 + 1665*x^4 - 378*x^5 + 560*x^6) + d*e^7*(-198 + 2568*x + 1560*x^2 - 3552*x^3 + 370*x^4 - 666*x^5 + 189*x^6 - 320*x^7) + e^8*(-108 - 396*x + 780*x^3 + 888*x^4 - 148*x^5 + 333*x^6 - 108*x^7 + 200*x^8) + 12*(2800*d^6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)*(d + e*x)^2*Log[d + e*x])/(12*e^9*(d + e*x)^2)","A",1
303,1,344,360,0.1228786,"\int \frac{\left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{(d+e x)^4} \, dx","Integrate[((3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(d + e*x)^4,x]","\frac{4 e^3 x^3 \left(1000 d^2+180 d e+111 e^2\right)-6 e^2 x^2 \left(2000 d^3+450 d^2 e+444 d e^2+37 e^3\right)+24 e x \left(1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right)-\frac{4 \left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{(d+e x)^3}-12 \left(5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right) \log (d+e x)-\frac{12 \left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right)}{d+e x}+\frac{6 \left(800 d^7+315 d^6 e+666 d^5 e^2+185 d^4 e^3+592 d^3 e^4-195 d^2 e^5+214 d e^6-33 e^7\right)}{(d+e x)^2}-15 e^4 x^4 (80 d+9 e)+240 e^5 x^5}{12 e^9}","\frac{x^3 \left(1000 d^2+180 d e+111 e^2\right)}{3 e^6}-\frac{x^2 \left(2000 d^3+450 d^2 e+444 d e^2+37 e^3\right)}{2 e^7}-\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{3 e^9 (d+e x)^3}+\frac{2 x \left(1750 d^4+450 d^3 e+555 d^2 e^2+74 d e^3+74 e^4\right)}{e^8}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right)}{2 e^9 (d+e x)^2}-\frac{\left(5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right) \log (d+e x)}{e^9}-\frac{2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6}{e^9 (d+e x)}-\frac{5 x^4 (80 d+9 e)}{4 e^5}+\frac{20 x^5}{e^4}",1,"(24*e*(1750*d^4 + 450*d^3*e + 555*d^2*e^2 + 74*d*e^3 + 74*e^4)*x - 6*e^2*(2000*d^3 + 450*d^2*e + 444*d*e^2 + 37*e^3)*x^2 + 4*e^3*(1000*d^2 + 180*d*e + 111*e^2)*x^3 - 15*e^4*(80*d + 9*e)*x^4 + 240*e^5*x^5 - (4*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(d + e*x)^3 + (6*(800*d^7 + 315*d^6*e + 666*d^5*e^2 + 185*d^4*e^3 + 592*d^3*e^4 - 195*d^2*e^5 + 214*d*e^6 - 33*e^7))/(d + e*x)^2 - (12*(2800*d^6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6))/(d + e*x) - 12*(5600*d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*Log[d + e*x])/(12*e^9)","A",1
304,1,178,221,0.120702,"\int \frac{(d+e x)^3 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{3+2 x+5 x^2} \, dx","Integrate[((d + e*x)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]","\frac{42 \left(57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right) \log \left(5 x^2+2 x+3\right)+35 x \left(250 d^3 \left(200 x^2-495 x+486\right)+450 d^2 e \left(250 x^3-550 x^2+405 x+916\right)+45 d e^2 \left(2000 x^4-4125 x^3+2700 x^2+4580 x-3524\right)+e^3 \left(25000 x^5-49500 x^4+30375 x^3+45800 x^2-26430 x-61296\right)\right)-6 \sqrt{14} \left(52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{6562500}","\frac{3}{500} e x^4 \left(100 d^2-165 d e+27 e^2\right)+\frac{x^3 \left(500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right)}{1875}-\frac{x^2 \left(4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right)}{6250}+\frac{\left(57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right) \log \left(5 x^2+2 x+3\right)}{156250}+\frac{x \left(10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right)}{15625}-\frac{\left(52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{78125 \sqrt{14}}+\frac{3}{125} e^2 x^5 (20 d-11 e)+\frac{2 e^3 x^6}{15}",1,"(35*x*(250*d^3*(486 - 495*x + 200*x^2) + 450*d^2*e*(916 + 405*x - 550*x^2 + 250*x^3) + 45*d*e^2*(-3524 + 4580*x + 2700*x^2 - 4125*x^3 + 2000*x^4) + e^3*(-61296 - 26430*x + 45800*x^2 + 30375*x^3 - 49500*x^4 + 25000*x^5)) - 6*Sqrt[14]*(52875*d^3 + 449175*d^2*e - 274845*d*e^2 - 53189*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 42*(57250*d^3 - 66075*d^2*e - 76620*d*e^2 + 23431*e^3)*Log[3 + 2*x + 5*x^2])/6562500","A",1
305,1,130,156,0.0846651,"\int \frac{(d+e x)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{3+2 x+5 x^2} \, dx","Integrate[((d + e*x)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]","\frac{84 \left(5725 d^2-4405 d e-2554 e^2\right) \log \left(5 x^2+2 x+3\right)+35 x \left(50 d^2 \left(200 x^2-495 x+486\right)+60 d e \left(250 x^3-550 x^2+405 x+916\right)+3 e^2 \left(2000 x^4-4125 x^3+2700 x^2+4580 x-3524\right)\right)-6 \sqrt{14} \left(10575 d^2+59890 d e-18323 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1312500}","\frac{1}{375} x^3 \left(100 d^2-330 d e+81 e^2\right)-\frac{x^2 \left(825 d^2-810 d e-458 e^2\right)}{1250}+\frac{\left(5725 d^2-4405 d e-2554 e^2\right) \log \left(5 x^2+2 x+3\right)}{15625}+\frac{x \left(2025 d^2+4580 d e-881 e^2\right)}{3125}-\frac{\left(10575 d^2+59890 d e-18323 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{15625 \sqrt{14}}+\frac{1}{100} e x^4 (40 d-33 e)+\frac{4 e^2 x^5}{25}",1,"(35*x*(50*d^2*(486 - 495*x + 200*x^2) + 60*d*e*(916 + 405*x - 550*x^2 + 250*x^3) + 3*e^2*(-3524 + 4580*x + 2700*x^2 - 4125*x^3 + 2000*x^4)) - 6*Sqrt[14]*(10575*d^2 + 59890*d*e - 18323*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 84*(5725*d^2 - 4405*d*e - 2554*e^2)*Log[3 + 2*x + 5*x^2])/1312500","A",1
306,1,86,99,0.0536677,"\int \frac{(d+e x) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{3+2 x+5 x^2} \, dx","Integrate[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]","\frac{21 (2290 d-881 e) \log \left(5 x^2+2 x+3\right)+35 x \left(5 d \left(200 x^2-495 x+486\right)+3 e \left(250 x^3-550 x^2+405 x+916\right)\right)-3 \sqrt{14} (2115 d+5989 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{131250}","\frac{1}{75} x^3 (20 d-33 e)-\frac{3}{250} x^2 (55 d-27 e)+\frac{(2290 d-881 e) \log \left(5 x^2+2 x+3\right)}{6250}+\frac{1}{625} x (405 d+458 e)-\frac{(2115 d+5989 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{3125 \sqrt{14}}+\frac{e x^4}{5}",1,"(35*x*(5*d*(486 - 495*x + 200*x^2) + 3*e*(916 + 405*x - 550*x^2 + 250*x^3)) - 3*Sqrt[14]*(2115*d + 5989*e)*ArcTan[(1 + 5*x)/Sqrt[14]] + 21*(2290*d - 881*e)*Log[3 + 2*x + 5*x^2])/131250","A",1
307,1,50,56,0.0181245,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{3+2 x+5 x^2} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2),x]","\frac{35 x \left(200 x^2-495 x+486\right)+9618 \log \left(5 x^2+2 x+3\right)-1269 \sqrt{14} \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{26250}","\frac{4 x^3}{15}-\frac{33 x^2}{50}+\frac{229}{625} \log \left(5 x^2+2 x+3\right)+\frac{81 x}{125}-\frac{423 \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{625 \sqrt{14}}",1,"(35*x*(486 - 495*x + 200*x^2) - 1269*Sqrt[14]*ArcTan[(1 + 5*x)/Sqrt[14]] + 9618*Log[3 + 2*x + 5*x^2])/26250","A",1
308,1,146,168,0.1096277,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) \left(3+2 x+5 x^2\right)} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)*(3 + 2*x + 5*x^2)),x]","\frac{70 e x \left(5 d^2-2 d e+3 e^2\right) (e (10 x-33)-20 d)+1750 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)+7 e^3 (458 d-7 e) \log \left(5 x^2+2 x+3\right)-\sqrt{14} e^3 (423 d-1367 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1750 e^3 \left(5 d^2-2 d e+3 e^2\right)}","\frac{(458 d-7 e) \log \left(5 x^2+2 x+3\right)}{250 \left(5 d^2-2 d e+3 e^2\right)}-\frac{(423 d-1367 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{125 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)}+\frac{\left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{e^3 \left(5 d^2-2 d e+3 e^2\right)}-\frac{x (20 d+33 e)}{25 e^2}+\frac{2 x^2}{5 e}",1,"(70*e*(5*d^2 - 2*d*e + 3*e^2)*x*(-20*d + e*(-33 + 10*x)) - Sqrt[14]*(423*d - 1367*e)*e^3*ArcTan[(1 + 5*x)/Sqrt[14]] + 1750*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x] + 7*(458*d - 7*e)*e^3*Log[3 + 2*x + 5*x^2])/(1750*e^3*(5*d^2 - 2*d*e + 3*e^2))","A",1
309,1,233,233,0.1603056,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left(3+2 x+5 x^2\right)} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^2*(3 + 2*x + 5*x^2)),x]","\frac{\left(229 d^2-7 d e-136 e^2\right) \log \left(5 x^2+2 x+3\right)}{25 \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{\left(-423 d^2+2734 d e-293 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{25 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{-4 d^4-5 d^3 e-3 d^2 e^2+d e^3-2 e^4}{e^3 \left(5 d^2-2 d e+3 e^2\right) (d+e x)}+\frac{\left(-40 d^5-d^4 e-28 d^3 e^2-44 d^2 e^3+2 d e^4-e^5\right) \log (d+e x)}{e^3 \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{4 x}{5 e^2}","\frac{\left(229 d^2-7 d e-136 e^2\right) \log \left(5 x^2+2 x+3\right)}{25 \left(5 d^2-2 d e+3 e^2\right)^2}-\frac{\left(423 d^2-2734 d e+293 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{25 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^2}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^3 \left(5 d^2-2 d e+3 e^2\right) (d+e x)}-\frac{\left(40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right) \log (d+e x)}{e^3 \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{4 x}{5 e^2}",1,"(4*x)/(5*e^2) + (-4*d^4 - 5*d^3*e - 3*d^2*e^2 + d*e^3 - 2*e^4)/(e^3*(5*d^2 - 2*d*e + 3*e^2)*(d + e*x)) + ((-423*d^2 + 2734*d*e - 293*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]])/(25*Sqrt[14]*(5*d^2 - 2*d*e + 3*e^2)^2) + ((-40*d^5 - d^4*e - 28*d^3*e^2 - 44*d^2*e^3 + 2*d*e^4 - e^5)*Log[d + e*x])/(e^3*(5*d^2 - 2*d*e + 3*e^2)^2) + ((229*d^2 - 7*d*e - 136*e^2)*Log[3 + 2*x + 5*x^2])/(25*(5*d^2 - 2*d*e + 3*e^2)^2)","A",1
310,1,278,317,0.4344892,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left(3+2 x+5 x^2\right)} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^3*(3 + 2*x + 5*x^2)),x]","-\frac{-7 \left(458 d^3-21 d^2 e-816 d e^2+113 e^3\right) \log \left(5 x^2+2 x+3\right)+\sqrt{14} \left(423 d^3-4101 d^2 e+879 d e^2+703 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+\frac{35 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \left(5 d^2-2 d e+3 e^2\right)^2}{e^3 (d+e x)^2}-\frac{70 \left(40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5\right) \left(5 d^2-2 d e+3 e^2\right)}{e^3 (d+e x)}+\frac{70 \left(-100 d^6+120 d^5 e-228 d^4 e^2+242 d^3 e^3-141 d^2 e^4-120 d e^5+e^6\right) \log (d+e x)}{e^3}}{70 \left(5 d^2-2 d e+3 e^2\right)^3}","\frac{\left(458 d^3-21 d^2 e-816 d e^2+113 e^3\right) \log \left(5 x^2+2 x+3\right)}{10 \left(5 d^2-2 d e+3 e^2\right)^3}-\frac{\left(423 d^3-4101 d^2 e+879 d e^2+703 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{5 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^3}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left(5 d^2-2 d e+3 e^2\right) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left(5 d^2-2 d e+3 e^2\right)^2 (d+e x)}+\frac{\left(100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right) \log (d+e x)}{e^3 \left(5 d^2-2 d e+3 e^2\right)^3}",1,"-1/70*((35*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e^3*(d + e*x)^2) - (70*(5*d^2 - 2*d*e + 3*e^2)*(40*d^5 + d^4*e + 28*d^3*e^2 + 44*d^2*e^3 - 2*d*e^4 + e^5))/(e^3*(d + e*x)) + Sqrt[14]*(423*d^3 - 4101*d^2*e + 879*d*e^2 + 703*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + (70*(-100*d^6 + 120*d^5*e - 228*d^4*e^2 + 242*d^3*e^3 - 141*d^2*e^4 - 120*d*e^5 + e^6)*Log[d + e*x])/e^3 - 7*(458*d^3 - 21*d^2*e - 816*d*e^2 + 113*e^3)*Log[3 + 2*x + 5*x^2])/(5*d^2 - 2*d*e + 3*e^2)^3","A",1
311,1,209,189,0.1581219,"\int \frac{(d+e x)^3 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^2} \, dx","Integrate[((d + e*x)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]","\frac{14700 e x^2 \left(300 d^2-615 d e+103 e^2\right)-\frac{42 \left(125 d^3 (423 x+1367)+75 d^2 e (5989 x-1269)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right)}{5 x^2+2 x+3}+2940 \left(-1025 d^3+1545 d^2 e+2601 d e^2-832 e^3\right) \log \left(5 x^2+2 x+3\right)+5880 x \left(500 d^3-3075 d^2 e+1545 d e^2+867 e^3\right)+15 \sqrt{14} \left(32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+49000 e^2 x^3 (60 d-41 e)+735000 e^3 x^4}{18375000}","\frac{e x^2 \left(840 d^2-1722 d e+373 e^2\right)}{3500}-\frac{\left(1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right) \log \left(5 x^2+2 x+3\right)}{6250}+\frac{x \left(2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right)}{17500}+\frac{\left(32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{87500 \sqrt{14}}+\frac{1}{375} e^2 x^3 (60 d-41 e)-\frac{(423 x+1367) (d+e x)^3}{3500 \left(5 x^2+2 x+3\right)}+\frac{e^3 x^4}{25}",1,"(5880*(500*d^3 - 3075*d^2*e + 1545*d*e^2 + 867*e^3)*x + 14700*e*(300*d^2 - 615*d*e + 103*e^2)*x^2 + 49000*(60*d - 41*e)*e^2*x^3 + 735000*e^3*x^4 - (42*(e^3*(54969 - 53189*x) + 125*d^3*(1367 + 423*x) + 75*d^2*e*(-1269 + 5989*x) - 15*d*e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2) + 15*Sqrt[14]*(32825*d^3 + 317565*d^2*e - 221643*d*e^2 - 67499*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 2940*(-1025*d^3 + 1545*d^2*e + 2601*d*e^2 - 832*e^3)*Log[3 + 2*x + 5*x^2])/18375000","A",1
312,1,150,140,0.1127205,"\int \frac{(d+e x)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^2} \, dx","Integrate[((d + e*x)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]","\frac{-\frac{42 \left(25 d^2 (423 x+1367)+10 d e (5989 x-1269)-e^2 (18323 x+17967)\right)}{5 x^2+2 x+3}+588 \left(-1025 d^2+1030 d e+867 e^2\right) \log \left(5 x^2+2 x+3\right)+5880 x \left(100 d^2-410 d e+103 e^2\right)+3 \sqrt{14} \left(32825 d^2+211710 d e-73881 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+14700 e x^2 (40 d-41 e)+196000 e^2 x^3}{3675000}","-\frac{\left(1025 d^2-1030 d e-867 e^2\right) \log \left(5 x^2+2 x+3\right)}{6250}+\frac{x \left(2800 d^2-11480 d e+3307 e^2\right)}{17500}+\frac{\left(32825 d^2+211710 d e-73881 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{87500 \sqrt{14}}+\frac{1}{250} e x^2 (40 d-41 e)-\frac{(423 x+1367) (d+e x)^2}{3500 \left(5 x^2+2 x+3\right)}+\frac{4 e^2 x^3}{75}",1,"(5880*(100*d^2 - 410*d*e + 103*e^2)*x + 14700*(40*d - 41*e)*e*x^2 + 196000*e^2*x^3 - (42*(25*d^2*(1367 + 423*x) + 10*d*e*(-1269 + 5989*x) - e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2) + 3*Sqrt[14]*(32825*d^2 + 211710*d*e - 73881*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 588*(-1025*d^2 + 1030*d*e + 867*e^2)*Log[3 + 2*x + 5*x^2])/3675000","A",1
313,1,96,97,0.0659414,"\int \frac{(d+e x) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^2} \, dx","Integrate[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]","\frac{-\frac{14 (5 d (423 x+1367)+e (5989 x-1269))}{5 x^2+2 x+3}+196 (103 e-205 d) \log \left(5 x^2+2 x+3\right)+1960 x (20 d-41 e)+\sqrt{14} (6565 d+21171 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+19600 e x^2}{245000}","-\frac{(423 x+1367) (d+e x)}{3500 \left(5 x^2+2 x+3\right)}-\frac{(205 d-103 e) \log \left(5 x^2+2 x+3\right)}{1250}+\frac{1}{125} x (20 d-41 e)+\frac{(6565 d+21171 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{17500 \sqrt{14}}+\frac{2 e x^2}{25}",1,"(1960*(20*d - 41*e)*x + 19600*e*x^2 - (14*(5*d*(1367 + 423*x) + e*(-1269 + 5989*x)))/(3 + 2*x + 5*x^2) + Sqrt[14]*(6565*d + 21171*e)*ArcTan[(1 + 5*x)/Sqrt[14]] + 196*(-205*d + 103*e)*Log[3 + 2*x + 5*x^2])/245000","A",1
314,1,59,63,0.0371828,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{\left(3+2 x+5 x^2\right)^2} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2)^2,x]","\frac{-\frac{14 (423 x+1367)}{5 x^2+2 x+3}-8036 \log \left(5 x^2+2 x+3\right)+7840 x+1313 \sqrt{14} \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{49000}","-\frac{423 x+1367}{3500 \left(5 x^2+2 x+3\right)}-\frac{41}{250} \log \left(5 x^2+2 x+3\right)+\frac{4 x}{25}+\frac{1313 \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{3500 \sqrt{14}}",1,"(7840*x - (14*(1367 + 423*x))/(3 + 2*x + 5*x^2) + 1313*Sqrt[14]*ArcTan[(1 + 5*x)/Sqrt[14]] - 8036*Log[3 + 2*x + 5*x^2])/49000","A",1
315,1,186,224,0.1639397,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) \left(3+2 x+5 x^2\right)^2} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)*(3 + 2*x + 5*x^2)^2),x]","\frac{\frac{14 \left(5 d^2-2 d e+3 e^2\right) (e (1367 x+293)-d (423 x+1367))}{5 x^2+2 x+3}-196 \left(205 d^3-61 d^2 e+23 d e^2+14 e^3\right) \log \left(5 x^2+2 x+3\right)+\sqrt{14} \left(6565 d^3-26423 d^2 e+11089 d e^2-6623 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+\frac{9800 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{e}}{9800 \left(5 d^2-2 d e+3 e^2\right)^2}","-\frac{x (423 d-1367 e)+1367 d-293 e}{700 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)}-\frac{\left(205 d^3-61 d^2 e+23 d e^2+14 e^3\right) \log \left(5 x^2+2 x+3\right)}{50 \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{\left(6565 d^3-26423 d^2 e+11089 d e^2-6623 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{700 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{\left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{e \left(5 d^2-2 d e+3 e^2\right)^2}",1,"((14*(5*d^2 - 2*d*e + 3*e^2)*(-(d*(1367 + 423*x)) + e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + Sqrt[14]*(6565*d^3 - 26423*d^2*e + 11089*d*e^2 - 6623*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + (9800*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x])/e - 196*(205*d^3 - 61*d^2*e + 23*d*e^2 + 14*e^3)*Log[3 + 2*x + 5*x^2])/(9800*(5*d^2 - 2*d*e + 3*e^2)^2)","A",1
316,1,270,313,0.2622387,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left(3+2 x+5 x^2\right)^2} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^2*(3 + 2*x + 5*x^2)^2),x]","\frac{-\frac{14 \left(5 d^2-2 d e+3 e^2\right) \left(d^2 (423 x+1367)-2 d e (1367 x+293)+e^2 (293 x-703)\right)}{5 x^2+2 x+3}+980 \left(-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right) \log \left(5 x^2+2 x+3\right)-\frac{1960 \left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{e (d+e x)}+1960 \left(41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right) \log (d+e x)+5 \sqrt{14} \left(1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1960 \left(5 d^2-2 d e+3 e^2\right)^3}","-\frac{x \left(423 d^2-2734 d e+293 e^2\right)+1367 d^2-586 d e-703 e^2}{140 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)^2}-\frac{\left(41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right) \log \left(5 x^2+2 x+3\right)}{2 \left(5 d^2-2 d e+3 e^2\right)^3}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e \left(5 d^2-2 d e+3 e^2\right)^2 (d+e x)}+\frac{\left(41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4\right) \log (d+e x)}{\left(5 d^2-2 d e+3 e^2\right)^3}+\frac{\left(1313 d^4-10044 d^3 e+4290 d^2 e^2+156 d e^3-271 e^4\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{28 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^3}",1,"((-1960*(5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e*(d + e*x)) - (14*(5*d^2 - 2*d*e + 3*e^2)*(e^2*(-703 + 293*x) + d^2*(1367 + 423*x) - 2*d*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + 5*Sqrt[14]*(1313*d^4 - 10044*d^3*e + 4290*d^2*e^2 + 156*d*e^3 - 271*e^4)*ArcTan[(1 + 5*x)/Sqrt[14]] + 1960*(41*d^4 - 8*d^3*e - 60*d^2*e^2 + 24*d*e^3 - 5*e^4)*Log[d + e*x] + 980*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4)*Log[3 + 2*x + 5*x^2])/(1960*(5*d^2 - 2*d*e + 3*e^2)^3)","A",1
317,1,363,412,0.3961155,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left(3+2 x+5 x^2\right)^2} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^3*(3 + 2*x + 5*x^2)^2),x]","\frac{-\frac{14 \left(5 d^2-2 d e+3 e^2\right) \left(d^3 (423 x+1367)-3 d^2 e (1367 x+293)+3 d e^2 (293 x-703)+e^3 (703 x+457)\right)}{5 x^2+2 x+3}-\frac{196 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \left(5 d^2-2 d e+3 e^2\right)^2}{e (d+e x)^2}+\frac{392 \left(-41 d^4+8 d^3 e+60 d^2 e^2-24 d e^3+5 e^4\right) \left(5 d^2-2 d e+3 e^2\right)}{d+e x}+196 \left(-205 d^5+19 d^4 e+846 d^3 e^2-396 d^2 e^3-57 d e^4+21 e^5\right) \log \left(5 x^2+2 x+3\right)+392 \left(205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right) \log (d+e x)+\sqrt{14} \left(6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{392 \left(5 d^2-2 d e+3 e^2\right)^4}","-\frac{1367 d^3-879 d^2 e+x \left(423 d^3-4101 d^2 e+879 d e^2+703 e^3\right)-2109 d e^2+457 e^3}{28 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)^3}-\frac{41 d^4-8 d^3 e-60 d^2 e^2+24 d e^3-5 e^4}{\left(5 d^2-2 d e+3 e^2\right)^3 (d+e x)}-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e \left(5 d^2-2 d e+3 e^2\right)^2 (d+e x)^2}-\frac{\left(205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right) \log \left(5 x^2+2 x+3\right)}{2 \left(5 d^2-2 d e+3 e^2\right)^4}+\frac{\left(205 d^5-19 d^4 e-846 d^3 e^2+396 d^2 e^3+57 d e^4-21 e^5\right) \log (d+e x)}{\left(5 d^2-2 d e+3 e^2\right)^4}+\frac{\left(6565 d^5-74017 d^4 e+35022 d^3 e^2+42858 d^2 e^3-17247 d e^4+579 e^5\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{28 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^4}",1,"((-196*(5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(e*(d + e*x)^2) + (392*(5*d^2 - 2*d*e + 3*e^2)*(-41*d^4 + 8*d^3*e + 60*d^2*e^2 - 24*d*e^3 + 5*e^4))/(d + e*x) - (14*(5*d^2 - 2*d*e + 3*e^2)*(3*d*e^2*(-703 + 293*x) + d^3*(1367 + 423*x) + e^3*(457 + 703*x) - 3*d^2*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2) + Sqrt[14]*(6565*d^5 - 74017*d^4*e + 35022*d^3*e^2 + 42858*d^2*e^3 - 17247*d*e^4 + 579*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]] + 392*(205*d^5 - 19*d^4*e - 846*d^3*e^2 + 396*d^2*e^3 + 57*d*e^4 - 21*e^5)*Log[d + e*x] + 196*(-205*d^5 + 19*d^4*e + 846*d^3*e^2 - 396*d^2*e^3 - 57*d*e^4 + 21*e^5)*Log[3 + 2*x + 5*x^2])/(392*(5*d^2 - 2*d*e + 3*e^2)^4)","A",1
318,1,209,171,0.2006191,"\int \frac{(d+e x)^3 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^3} \, dx","Integrate[((d + e*x)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^3,x]","\frac{164640 e \left(100 d^2-245 d e+47 e^2\right) \log \left(5 x^2+2 x+3\right)-\frac{392 \left(125 d^3 (423 x+1367)+75 d^2 e (5989 x-1269)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right)}{\left(5 x^2+2 x+3\right)^2}+\frac{14 \left(125 d^3 (11015 x+34347)+75 d^2 e (181765 x-44399)-15 d e^2 (647195 x+809167)+e^3 (2639639-3109005 x)\right)}{5 x^2+2 x+3}+15 \sqrt{14} \left(353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)+548800 e^2 x (60 d-49 e)+5488000 e^3 x^2}{343000000}","\frac{3 e \left(100 d^2-245 d e+47 e^2\right) \log \left(5 x^2+2 x+3\right)}{6250}+\frac{3 \left(353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{4900000 \sqrt{14}}+\frac{e^2 x (83065 d-126009 e)}{980000}+\frac{(d+e x)^2 (x (11015 d+49177 e)+3 (11449 d-2105 e))}{196000 \left(5 x^2+2 x+3\right)}-\frac{(423 x+1367) (d+e x)^3}{7000 \left(5 x^2+2 x+3\right)^2}+\frac{2 e^3 x^2}{125}",1,"(548800*(60*d - 49*e)*e^2*x + 5488000*e^3*x^2 - (392*(e^3*(54969 - 53189*x) + 125*d^3*(1367 + 423*x) + 75*d^2*e*(-1269 + 5989*x) - 15*d*e^2*(17967 + 18323*x)))/(3 + 2*x + 5*x^2)^2 + (14*(e^3*(2639639 - 3109005*x) + 125*d^3*(34347 + 11015*x) + 75*d^2*e*(-44399 + 181765*x) - 15*d*e^2*(809167 + 647195*x)))/(3 + 2*x + 5*x^2) + 15*Sqrt[14]*(353125*d^3 - 855175*d^2*e + 74085*d*e^2 + 556349*e^3)*ArcTan[(1 + 5*x)/Sqrt[14]] + 164640*e*(100*d^2 - 245*d*e + 47*e^2)*Log[3 + 2*x + 5*x^2])/343000000","A",1
319,1,146,134,0.1816833,"\int \frac{(d+e x)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^3} \, dx","Integrate[((d + e*x)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^3,x]","\frac{70 \left(\frac{5 \left(5 d^2 \left(11015 x^3+38753 x^2+17979 x+12953\right)+2 d e \left(181765 x^3+28307 x^2+57761 x-19533\right)+e^2 \left(156800 x^5+125440 x^4+83809 x^3-138345 x^2-65427 x-76977\right)\right)}{\left(5 x^2+2 x+3\right)^2}+784 e (40 d-49 e) \log \left(5 x^2+2 x+3\right)\right)+5 \sqrt{14} \left(211875 d^2-342070 d e+14817 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{68600000}","\frac{\left(211875 d^2-342070 d e+14817 e^2\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{980000 \sqrt{14}}+\frac{(d+e x) (5 x (2203 d+8553 e)+34347 d-6413 e)}{196000 \left(5 x^2+2 x+3\right)}-\frac{(423 x+1367) (d+e x)^2}{7000 \left(5 x^2+2 x+3\right)^2}+\frac{e (40 d-49 e) \log \left(5 x^2+2 x+3\right)}{1250}+\frac{4 e^2 x}{125}",1,"(5*Sqrt[14]*(211875*d^2 - 342070*d*e + 14817*e^2)*ArcTan[(1 + 5*x)/Sqrt[14]] + 70*((5*(5*d^2*(12953 + 17979*x + 38753*x^2 + 11015*x^3) + 2*d*e*(-19533 + 57761*x + 28307*x^2 + 181765*x^3) + e^2*(-76977 - 65427*x - 138345*x^2 + 83809*x^3 + 125440*x^4 + 156800*x^5)))/(3 + 2*x + 5*x^2)^2 + 784*(40*d - 49*e)*e*Log[3 + 2*x + 5*x^2]))/68600000","A",1
320,1,107,103,0.0851961,"\int \frac{(d+e x) \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^3} \, dx","Integrate[((d + e*x)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^3,x]","\frac{-2115 d x-6835 d-5989 e x+1269 e}{35000 \left(5 x^2+2 x+3\right)^2}+\frac{55075 d x+171735 d+181765 e x-44399 e}{980000 \left(5 x^2+2 x+3\right)}+\frac{(42375 d-34207 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{196000 \sqrt{14}}+\frac{2}{125} e \log \left(5 x^2+2 x+3\right)","-\frac{(423 x+1367) (d+e x)}{7000 \left(5 x^2+2 x+3\right)^2}+\frac{x (11015 d+36353 e)+34347 d-6511 e}{196000 \left(5 x^2+2 x+3\right)}+\frac{(42375 d-34207 e) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{196000 \sqrt{14}}+\frac{2}{125} e \log \left(5 x^2+2 x+3\right)",1,"(-6835*d + 1269*e - 2115*d*x - 5989*e*x)/(35000*(3 + 2*x + 5*x^2)^2) + (171735*d - 44399*e + 55075*d*x + 181765*e*x)/(980000*(3 + 2*x + 5*x^2)) + ((42375*d - 34207*e)*ArcTan[(1 + 5*x)/Sqrt[14]])/(196000*Sqrt[14]) + (2*e*Log[3 + 2*x + 5*x^2])/125","A",1
321,1,53,64,0.039181,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{\left(3+2 x+5 x^2\right)^3} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2)^3,x]","\frac{\frac{14 \left(11015 x^3+38753 x^2+17979 x+12953\right)}{\left(5 x^2+2 x+3\right)^2}+8475 \sqrt{14} \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{548800}","-\frac{423 x+1367}{7000 \left(5 x^2+2 x+3\right)^2}+\frac{11015 x+34347}{196000 \left(5 x^2+2 x+3\right)}+\frac{339 \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1568 \sqrt{14}}",1,"((14*(12953 + 17979*x + 38753*x^2 + 11015*x^3))/(3 + 2*x + 5*x^2)^2 + 8475*Sqrt[14]*ArcTan[(1 + 5*x)/Sqrt[14]])/548800","A",1
322,1,282,329,0.3053607,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x) \left(3+2 x+5 x^2\right)^3} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)*(3 + 2*x + 5*x^2)^3),x]","\frac{\frac{392 \left(5 d^2-2 d e+3 e^2\right)^2 (e (1367 x+293)-d (423 x+1367))}{\left(5 x^2+2 x+3\right)^2}+\frac{14 \left(5 d^2-2 d e+3 e^2\right) \left(5 d^3 (11015 x+34347)-d^2 e (225825 x+92989)+d e^2 (90875 x+36207)+e^3 (1831-45725 x)\right)}{5 x^2+2 x+3}-274400 e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log \left(5 x^2+2 x+3\right)+548800 e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)+25 \sqrt{14} \left(42375 d^5-16643 d^4 e+58530 d^3 e^2-56058 d^2 e^3+31811 d e^4-8623 e^5\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{548800 \left(5 d^2-2 d e+3 e^2\right)^3}","-\frac{x (423 d-1367 e)+1367 d-293 e}{1400 \left(5 x^2+2 x+3\right)^2 \left(5 d^2-2 d e+3 e^2\right)}+\frac{171735 d^3-92989 d^2 e+25 x \left(2203 d^3-9033 d^2 e+3635 d e^2-1829 e^3\right)+36207 d e^2+1831 e^3}{39200 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)^2}-\frac{e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log \left(5 x^2+2 x+3\right)}{2 \left(5 d^2-2 d e+3 e^2\right)^3}+\frac{e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \log (d+e x)}{\left(5 d^2-2 d e+3 e^2\right)^3}+\frac{\left(42375 d^5-16643 d^4 e+58530 d^3 e^2-56058 d^2 e^3+31811 d e^4-8623 e^5\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1568 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^3}",1,"((392*(5*d^2 - 2*d*e + 3*e^2)^2*(-(d*(1367 + 423*x)) + e*(293 + 1367*x)))/(3 + 2*x + 5*x^2)^2 + (14*(5*d^2 - 2*d*e + 3*e^2)*(e^3*(1831 - 45725*x) + 5*d^3*(34347 + 11015*x) + d*e^2*(36207 + 90875*x) - d^2*e*(92989 + 225825*x)))/(3 + 2*x + 5*x^2) + 25*Sqrt[14]*(42375*d^5 - 16643*d^4*e + 58530*d^3*e^2 - 56058*d^2*e^3 + 31811*d*e^4 - 8623*e^5)*ArcTan[(1 + 5*x)/Sqrt[14]] + 548800*e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[d + e*x] - 274400*e*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*Log[3 + 2*x + 5*x^2])/(548800*(5*d^2 - 2*d*e + 3*e^2)^3)","A",1
323,1,389,443,0.5255237,"\int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^2 \left(3+2 x+5 x^2\right)^3} \, dx","Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/((d + e*x)^2*(3 + 2*x + 5*x^2)^3),x]","\frac{-\frac{392 \left(5 d^2-2 d e+3 e^2\right)^2 \left(d^2 (423 x+1367)-2 d e (1367 x+293)+e^2 (293 x-703)\right)}{\left(5 x^2+2 x+3\right)^2}+\frac{14 \left(5 d^2-2 d e+3 e^2\right) \left(5 d^4 (11015 x+34347)-4 d^3 e (107405 x+29321)+6 d^2 e^2 (28915 x-33417)+4 d e^3 (12935 x+26107)-e^4 (17945 x+23189)\right)}{5 x^2+2 x+3}-\frac{109760 e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) \left(5 d^2-2 d e+3 e^2\right)}{d+e x}-54880 e \left(40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right) \log \left(5 x^2+2 x+3\right)+109760 e \left(40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right) \log (d+e x)+5 \sqrt{14} \left(211875 d^6+3070 d^5 e+209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4-49586 d e^5-43695 e^6\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{109760 \left(5 d^2-2 d e+3 e^2\right)^4}","-\frac{x \left(423 d^2-2734 d e+293 e^2\right)+1367 d^2-586 d e-703 e^2}{280 \left(5 x^2+2 x+3\right)^2 \left(5 d^2-2 d e+3 e^2\right)^2}+\frac{171735 d^4-117284 d^3 e-200502 d^2 e^2+5 x \left(11015 d^4-85924 d^3 e+34698 d^2 e^2+10348 d e^3-3589 e^4\right)+104428 d e^3-23189 e^4}{7840 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)^3}-\frac{e \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{\left(5 d^2-2 d e+3 e^2\right)^3 (d+e x)}-\frac{e \left(40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right) \log \left(5 x^2+2 x+3\right)}{2 \left(5 d^2-2 d e+3 e^2\right)^4}+\frac{e \left(40 d^5+83 d^4 e+12 d^3 e^2-76 d^2 e^3+46 d e^4-9 e^5\right) \log (d+e x)}{\left(5 d^2-2 d e+3 e^2\right)^4}+\frac{\left(211875 d^6+3070 d^5 e+209039 d^4 e^2-921444 d^3 e^3+380621 d^2 e^4-49586 d e^5-43695 e^6\right) \tan ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1568 \sqrt{14} \left(5 d^2-2 d e+3 e^2\right)^4}",1,"((-109760*e*(5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(d + e*x) - (392*(5*d^2 - 2*d*e + 3*e^2)^2*(e^2*(-703 + 293*x) + d^2*(1367 + 423*x) - 2*d*e*(293 + 1367*x)))/(3 + 2*x + 5*x^2)^2 + (14*(5*d^2 - 2*d*e + 3*e^2)*(5*d^4*(34347 + 11015*x) + 4*d*e^3*(26107 + 12935*x) - e^4*(23189 + 17945*x) + 6*d^2*e^2*(-33417 + 28915*x) - 4*d^3*e*(29321 + 107405*x)))/(3 + 2*x + 5*x^2) + 5*Sqrt[14]*(211875*d^6 + 3070*d^5*e + 209039*d^4*e^2 - 921444*d^3*e^3 + 380621*d^2*e^4 - 49586*d*e^5 - 43695*e^6)*ArcTan[(1 + 5*x)/Sqrt[14]] + 109760*e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[d + e*x] - 54880*e*(40*d^5 + 83*d^4*e + 12*d^3*e^2 - 76*d^2*e^3 + 46*d*e^4 - 9*e^5)*Log[3 + 2*x + 5*x^2])/(109760*(5*d^2 - 2*d*e + 3*e^2)^4)","A",1
324,1,70,143,0.1538656,"\int (5+2 x) \sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right) \, dx","Integrate[(5 + 2*x)*Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4),x]","\frac{4 \sqrt{2 x^2-x+3} \left(4915200 x^6+12984320 x^5+1390592 x^4+20304768 x^3+11357024 x^2+14742332 x+6231117\right)-124215525 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{13762560}","\frac{5}{112} \left(2 x^2-x+3\right)^{3/2} (2 x+5)^4-\frac{823 \left(2 x^2-x+3\right)^{3/2} (2 x+5)^3}{1344}+\frac{11433 \left(2 x^2-x+3\right)^{3/2} (2 x+5)^2}{4480}-\frac{(295276 x+1005757) \left(2 x^2-x+3\right)^{3/2}}{71680}-\frac{51435 (1-4 x) \sqrt{2 x^2-x+3}}{32768}-\frac{1183005 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{65536 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(6231117 + 14742332*x + 11357024*x^2 + 20304768*x^3 + 1390592*x^4 + 12984320*x^5 + 4915200*x^6) - 124215525*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/13762560","A",1
325,1,65,124,0.0947108,"\int \sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right) \, dx","Integrate[Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4),x]","\frac{4 \sqrt{2 x^2-x+3} \left(204800 x^5-59392 x^4+258432 x^3+105696 x^2+221868 x-27807\right)-1590105 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{983040}","\frac{7}{80} \left(2 x^2-x+3\right)^{3/2} x^2-\frac{71 \left(2 x^2-x+3\right)^{3/2} x}{1280}+\frac{287 \left(2 x^2-x+3\right)^{3/2}}{5120}-\frac{4609 (1-4 x) \sqrt{2 x^2-x+3}}{16384}+\frac{5}{12} \left(2 x^2-x+3\right)^{3/2} x^3-\frac{106007 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32768 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(-27807 + 221868*x + 105696*x^2 + 258432*x^3 - 59392*x^4 + 204800*x^5) - 1590105*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/983040","A",1
326,1,91,149,0.1472858,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{5+2 x} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x),x]","\frac{-16897536 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+4 \sqrt{2 x^2-x+3} \left(6144 x^4-21120 x^3+79840 x^2-300404 x+1561161\right)+16883967 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{49152}","\frac{1}{16} \left(2 x^2-x+3\right)^{3/2} (2 x+5)^2-\frac{127}{128} \left(2 x^2-x+3\right)^{3/2} (2 x+5)+\frac{4535}{768} \left(2 x^2-x+3\right)^{3/2}+\frac{(489587-80844 x) \sqrt{2 x^2-x+3}}{4096}-\frac{11001 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{16 \sqrt{2}}+\frac{5627989 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{8192 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(1561161 - 300404*x + 79840*x^2 - 21120*x^3 + 6144*x^4) + 16883967*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 16897536*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/49152","A",1
327,1,98,149,0.1618119,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^2} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^2,x]","\frac{7654432 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{4 \sqrt{2 x^2-x+3} \left(3840 x^4-17344 x^3+94936 x^2-728410 x-3539439\right)}{2 x+5}-7655541 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{24576}","\frac{5}{64} (2 x+5) \left(2 x^2-x+3\right)^{3/2}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{576 (2 x+5)}-\frac{541}{384} \left(2 x^2-x+3\right)^{3/2}-\frac{(1996953-333380 x) \sqrt{2 x^2-x+3}}{18432}+\frac{239201 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{384 \sqrt{2}}-\frac{2551847 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{4096 \sqrt{2}}",1,"((4*Sqrt[3 - x + 2*x^2]*(-3539439 - 728410*x + 94936*x^2 - 17344*x^3 + 3840*x^4))/(5 + 2*x) - 7655541*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 7654432*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/24576","A",1
328,1,98,151,0.1479521,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^3} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^3,x]","\frac{-12670805 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(3840 x^4-25632 x^3+272520 x^2+2959330 x+4880551\right)}{(2 x+5)^2}+12670020 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{110592}","\frac{357391 \left(2 x^2-x+3\right)^{3/2}}{82944 (2 x+5)}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{1152 (2 x+5)^2}+\frac{5}{48} \left(2 x^2-x+3\right)^{3/2}+\frac{5 (661065-110099 x) \sqrt{2 x^2-x+3}}{82944}-\frac{12670805 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{55296 \sqrt{2}}+\frac{117315 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{512 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(4880551 + 2959330*x + 272520*x^2 - 25632*x^3 + 3840*x^4))/(5 + 2*x)^2 + 12670020*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 12670805*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/110592","A",1
329,1,98,158,0.1585492,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^4} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^4,x]","\frac{170114729 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(414720 x^4-5453568 x^3-97682900 x^2-329667508 x-327735797\right)}{(2 x+5)^3}-170123328 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{7962624}","-\frac{6467659 \left(2 x^2-x+3\right)^{3/2}}{5971968 (2 x+5)}+\frac{158527 \left(2 x^2-x+3\right)^{3/2}}{82944 (2 x+5)^2}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{1728 (2 x+5)^3}-\frac{(44378877-7400779 x) \sqrt{2 x^2-x+3}}{5971968}+\frac{170114729 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{3981312 \sqrt{2}}-\frac{10939 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{256 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(-327735797 - 329667508*x - 97682900*x^2 - 5453568*x^3 + 414720*x^4))/(5 + 2*x)^3 - 170123328*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 170114729*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/7962624","A",1
330,1,98,165,0.1763635,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^5} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^5,x]","\frac{-4640586097 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(238878720 x^4+6105343976 x^3+31323229164 x^2+62847867486 x+44676885233\right)}{(2 x+5)^4}+4640219136 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{2293235712}","-\frac{9363383 \left(2 x^2-x+3\right)^{3/2}}{23887872 (2 x+5)^2}+\frac{593771 \left(2 x^2-x+3\right)^{3/2}}{497664 (2 x+5)^3}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{2304 (2 x+5)^4}+\frac{7 (9616196 x+52836655) \sqrt{2 x^2-x+3}}{95551488 (2 x+5)}-\frac{4640586097 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{1146617856 \sqrt{2}}+\frac{259 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{64 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(44676885233 + 62847867486*x + 31323229164*x^2 + 6105343976*x^3 + 238878720*x^4))/(5 + 2*x)^4 + 4640219136*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 4640586097*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/2293235712","A",1
331,1,98,165,0.1997361,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^6} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]","\frac{64477987315 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)-\frac{24 \sqrt{2 x^2-x+3} \left(186470433136 x^4+1285267446304 x^3+3919478861832 x^2+5608297138216 x+3110673952831\right)}{(2 x+5)^5}-64497254400 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{825564856320}","-\frac{38732321 \left(2 x^2-x+3\right)^{3/2}}{179159040 (2 x+5)^3}+\frac{711961 \left(2 x^2-x+3\right)^{3/2}}{829440 (2 x+5)^4}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{2880 (2 x+5)^5}-\frac{(3174439702 x+4583087983) \sqrt{2 x^2-x+3}}{6879707136 (2 x+5)^2}+\frac{12895597463 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{82556485632 \sqrt{2}}-\frac{5 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32 \sqrt{2}}",1,"((-24*Sqrt[3 - x + 2*x^2]*(3110673952831 + 5608297138216*x + 3919478861832*x^2 + 1285267446304*x^3 + 186470433136*x^4))/(5 + 2*x)^5 - 64497254400*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 64477987315*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/825564856320","A",1
332,1,91,169,0.169177,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^7} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^7,x]","\frac{24 \sqrt{2 x^2-x+3} \left(271409942624 x^5+12256250416 x^4+397498825328 x^3+158340720344 x^2+27245373694 x-219337079305\right)-134863375 \sqrt{2} (2 x+5)^6 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)}{39627113103360 (2 x+5)^6}","\frac{87677717 \left(2 x^2-x+3\right)^{3/2}}{8599633920 (2 x+5)^3}-\frac{5703277 \left(2 x^2-x+3\right)^{3/2}}{39813120 (2 x+5)^4}+\frac{92239 \left(2 x^2-x+3\right)^{3/2}}{138240 (2 x+5)^5}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{3456 (2 x+5)^6}-\frac{1172725 (17-22 x) \sqrt{2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac{26972675 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{3962711310336 \sqrt{2}}",1,"(24*Sqrt[3 - x + 2*x^2]*(-219337079305 + 27245373694*x + 158340720344*x^2 + 397498825328*x^3 + 12256250416*x^4 + 271409942624*x^5) - 134863375*Sqrt[2]*(5 + 2*x)^6*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(39627113103360*(5 + 2*x)^6)","A",1
333,1,96,194,0.1979805,"\int \frac{\sqrt{3-x+2 x^2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^8} \, dx","Integrate[(Sqrt[3 - x + 2*x^2]*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^8,x]","\frac{24 \sqrt{2 x^2-x+3} \left(1574342277056 x^6+27976951397184 x^5+4982916071952 x^4+41058010262368 x^3+14716683780036 x^2+590492177460 x-20465234808721\right)-2023498715 \sqrt{2} (2 x+5)^7 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)}{3994413000818688 (2 x+5)^7}","\frac{246159769 \left(2 x^2-x+3\right)^{3/2}}{866843099136 (2 x+5)^3}+\frac{19414831 \left(2 x^2-x+3\right)^{3/2}}{4013162496 (2 x+5)^4}-\frac{1464037 \left(2 x^2-x+3\right)^{3/2}}{13934592 (2 x+5)^5}+\frac{948341 \left(2 x^2-x+3\right)^{3/2}}{1741824 (2 x+5)^6}-\frac{3667 \left(2 x^2-x+3\right)^{3/2}}{4032 (2 x+5)^7}-\frac{12568315 (17-22 x) \sqrt{2 x^2-x+3}}{23776267862016 (2 x+5)^2}-\frac{289071245 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{285315214344192 \sqrt{2}}",1,"(24*Sqrt[3 - x + 2*x^2]*(-20465234808721 + 590492177460*x + 14716683780036*x^2 + 41058010262368*x^3 + 4982916071952*x^4 + 27976951397184*x^5 + 1574342277056*x^6) - 2023498715*Sqrt[2]*(5 + 2*x)^7*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(3994413000818688*(5 + 2*x)^7)","A",1
334,1,80,166,0.1842019,"\int (5+2 x) \left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right) \, dx","Integrate[(5 + 2*x)*(3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4),x]","\frac{4 \sqrt{2 x^2-x+3} \left(1468006400 x^8+2926837760 x^7+1033175040 x^6+12117893120 x^5+379086848 x^4+12669290112 x^3+4870637856 x^2+12357760788 x+1592737263\right)-46354690935 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{2642411520}","\frac{5}{144} \left(2 x^2-x+3\right)^{5/2} (2 x+5)^4-\frac{1121 \left(2 x^2-x+3\right)^{5/2} (2 x+5)^3}{2304}+\frac{69415 \left(2 x^2-x+3\right)^{5/2} (2 x+5)^2}{32256}-\frac{3 (215900 x+661397) \left(2 x^2-x+3\right)^{5/2}}{143360}-\frac{92727 (1-4 x) \left(2 x^2-x+3\right)^{3/2}}{131072}-\frac{6398163 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{147157749 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{4194304 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(1592737263 + 12357760788*x + 4870637856*x^2 + 12669290112*x^3 + 379086848*x^4 + 12117893120*x^5 + 1033175040*x^6 + 2926837760*x^7 + 1468006400*x^8) - 46354690935*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/2642411520","A",1
335,1,75,147,0.1241361,"\int \left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right) \, dx","Integrate[(3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4),x]","\frac{4 \sqrt{2 x^2-x+3} \left(9175040 x^7-7667712 x^6+29335552 x^5-7497728 x^4+27023744 x^3+3845856 x^2+27845612 x-1663407\right)-95504073 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{29360128}","\frac{23}{448} \left(2 x^2-x+3\right)^{5/2} x^2+\frac{125 \left(2 x^2-x+3\right)^{5/2} x}{3584}+\frac{1167 \left(2 x^2-x+3\right)^{5/2}}{14336}-\frac{8597 (1-4 x) \left(2 x^2-x+3\right)^{3/2}}{65536}-\frac{593193 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}+\frac{5}{16} \left(2 x^2-x+3\right)^{5/2} x^3-\frac{13643439 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{2097152 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(-1663407 + 27845612*x + 3845856*x^2 + 27023744*x^3 - 7497728*x^4 + 29335552*x^5 - 7667712*x^6 + 9175040*x^7) - 95504073*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/29360128","A",1
336,1,101,172,0.1828926,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{5+2 x} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x),x]","\frac{-34065432576 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+4 \sqrt{2 x^2-x+3} \left(983040 x^6-3710976 x^5+14493696 x^4-46476672 x^3+159973408 x^2-609499532 x+3149403255\right)+34062209601 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{5505024}","\frac{5}{112} (2 x+5)^2 \left(2 x^2-x+3\right)^{5/2}-\frac{311}{448} (2 x+5) \left(2 x^2-x+3\right)^{5/2}+\frac{3505}{896} \left(2 x^2-x+3\right)^{5/2}+\frac{(500141-123060 x) \left(2 x^2-x+3\right)^{3/2}}{12288}+\frac{(141051019-23482924 x) \sqrt{2 x^2-x+3}}{65536}-\frac{99009 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{8 \sqrt{2}}+\frac{1622009981 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{131072 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(3149403255 - 609499532*x + 159973408*x^2 - 46476672*x^3 + 14493696*x^4 - 3710976*x^5 + 983040*x^6) + 34062209601*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 34065432576*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/5505024","A",1
337,1,108,172,0.2097956,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^2} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^2,x]","\frac{14739840000 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{4 \sqrt{2 x^2-x+3} \left(409600 x^6-1798144 x^5+8283904 x^4-35369408 x^3+182033816 x^2-1404323114 x-6814208295\right)}{2 x+5}-14740041885 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{1966080}","\frac{5}{96} (2 x+5) \left(2 x^2-x+3\right)^{5/2}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{576 (2 x+5)}-\frac{839}{960} \left(2 x^2-x+3\right)^{5/2}-\frac{(909513-226052 x) \left(2 x^2-x+3\right)^{3/2}}{18432}-\frac{(85448933-14243732 x) \sqrt{2 x^2-x+3}}{32768}+\frac{959625 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{64 \sqrt{2}}-\frac{982669459 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{65536 \sqrt{2}}",1,"((4*Sqrt[3 - x + 2*x^2]*(-6814208295 - 1404323114*x + 182033816*x^2 - 35369408*x^3 + 8283904*x^4 - 1798144*x^5 + 409600*x^6))/(5 + 2*x) - 14740041885*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 14739840000*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/1966080","A",1
338,1,108,174,0.2160771,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^3} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^3,x]","\frac{-129342640 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{4 \sqrt{2 x^2-x+3} \left(8192 x^6-43520 x^5+253312 x^4-1620944 x^3+16667188 x^2+181223072 x+298966737\right)}{(2 x+5)^2}+129342063 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32768}","\frac{438065 \left(2 x^2-x+3\right)^{5/2}}{82944 (2 x+5)}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{1152 (2 x+5)^2}+\frac{1}{16} \left(2 x^2-x+3\right)^{5/2}+\frac{(2154633-534617 x) \left(2 x^2-x+3\right)^{3/2}}{82944}+\frac{(33741483-5623292 x) \sqrt{2 x^2-x+3}}{24576}-\frac{8083915 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{1024 \sqrt{2}}+\frac{129342063 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{16384 \sqrt{2}}",1,"((4*Sqrt[3 - x + 2*x^2]*(298966737 + 181223072*x + 16667188*x^2 - 1620944*x^3 + 253312*x^4 - 43520*x^5 + 8192*x^6))/(5 + 2*x)^2 + 129342063*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 129342640*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/32768","A",1
339,1,108,181,0.21878,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^4} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^4,x]","\frac{517762327 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{12 \sqrt{2 x^2-x+3} \left(46080 x^6-315648 x^5+2626848 x^4-33595416 x^3-594798908 x^2-2006873194 x-1994650739\right)}{(2 x+5)^3}-517763637 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{442368}","-\frac{32865365 \left(2 x^2-x+3\right)^{5/2}}{17915904 (2 x+5)}+\frac{556255 \left(2 x^2-x+3\right)^{5/2}}{248832 (2 x+5)^2}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{1728 (2 x+5)^3}-\frac{(138006843-34265045 x) \left(2 x^2-x+3\right)^{3/2}}{17915904}-\frac{(135068604-22512089 x) \sqrt{2 x^2-x+3}}{331776}+\frac{517762327 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{221184 \sqrt{2}}-\frac{19176431 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{8192 \sqrt{2}}",1,"((12*Sqrt[3 - x + 2*x^2]*(-1994650739 - 2006873194*x - 594798908*x^2 - 33595416*x^3 + 2626848*x^4 - 315648*x^5 + 46080*x^6))/(5 + 2*x)^3 - 517763637*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 517762327*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/442368","A",1
340,1,108,188,0.2279879,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^5} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^5,x]","\frac{-8969688643 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(2949120 x^6-29270016 x^5+468043776 x^4+11761910072 x^3+60528581892 x^2+121473790266 x+86386856771\right)}{(2 x+5)^4}+8969667840 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{42467328}","-\frac{14477995 \left(2 x^2-x+3\right)^{5/2}}{23887872 (2 x+5)^2}+\frac{224815 \left(2 x^2-x+3\right)^{5/2}}{165888 (2 x+5)^3}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{2304 (2 x+5)^4}+\frac{(67865260 x+762984903) \left(2 x^2-x+3\right)^{3/2}}{95551488 (2 x+5)}+\frac{(2339916063-389975609 x) \sqrt{2 x^2-x+3}}{31850496}-\frac{8969688643 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{21233664 \sqrt{2}}+\frac{432565 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{1024 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(86386856771 + 121473790266*x + 60528581892*x^2 + 11761910072*x^3 + 468043776*x^4 - 29270016*x^5 + 2949120*x^6))/(5 + 2*x)^4 + 8969667840*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 8969688643*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/42467328","A",1
341,1,108,195,0.236613,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^6} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^6,x]","\frac{354957625835 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(1592524800 x^6-30496849920 x^5-1023534029552 x^4-7117092892448 x^3-21590439797064 x^2-30872393829992 x-17093312738327\right)}{(2 x+5)^5}-354958848000 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{15288238080}","-\frac{3730507 \left(2 x^2-x+3\right)^{5/2}}{11943936 (2 x+5)^3}+\frac{158527 \left(2 x^2-x+3\right)^{5/2}}{165888 (2 x+5)^4}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{2880 (2 x+5)^5}+\frac{(44773976 x+246012435) \left(2 x^2-x+3\right)^{3/2}}{95551488 (2 x+5)^2}-\frac{(1028823716 x+5658774871) \sqrt{2 x^2-x+3}}{127401984 (2 x+5)}+\frac{70991525167 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{1528823808 \sqrt{2}}-\frac{23775 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{512 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(-17093312738327 - 30872393829992*x - 21590439797064*x^2 - 7117092892448*x^3 - 1023534029552*x^4 - 30496849920*x^5 + 1592524800*x^6))/(5 + 2*x)^5 - 354958848000*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 354957625835*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/15288238080","A",1
342,1,108,195,0.2525034,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^7} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^7,x]","\frac{-1903976002333 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \sqrt{2 x^2-x+3} \left(275188285440 x^6+11854023276320 x^5+103803827945872 x^4+422554114856528 x^3+910256842473992 x^2+1011372787716826 x+458411625354581\right)}{(2 x+5)^6}+1903958949888 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{1320903770112}","-\frac{14087245 \left(2 x^2-x+3\right)^{5/2}}{71663616 (2 x+5)^4}+\frac{182165 \left(2 x^2-x+3\right)^{5/2}}{248832 (2 x+5)^5}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{3456 (2 x+5)^6}-\frac{(6793718806 x+9802984711) \left(2 x^2-x+3\right)^{3/2}}{13759414272 (2 x+5)^3}+\frac{(27596573612 x+151764102421) \sqrt{2 x^2-x+3}}{55037657088 (2 x+5)}-\frac{1903976002333 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{660451885056 \sqrt{2}}+\frac{369 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{128 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(458411625354581 + 1011372787716826*x + 910256842473992*x^2 + 422554114856528*x^3 + 103803827945872*x^4 + 11854023276320*x^5 + 275188285440*x^6))/(5 + 2*x)^6 + 1903958949888*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 1903976002333*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/1320903770112","A",1
343,1,108,195,0.2594248,"\int \frac{\left(3-x+2 x^2\right)^{3/2} \left(2+x+3 x^2-x^3+5 x^4\right)}{(5+2 x)^8} \, dx","Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^8,x]","\frac{2889323929457 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)-\frac{24 \sqrt{2 x^2-x+3} \left(38463671680832 x^6+402255822731712 x^5+2069947287085104 x^4+5966329646300704 x^3+9976065367498188 x^2+9065154700300572 x+3479517268702637\right)}{(2 x+5)^7}-2889476997120 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{73970611126272}","-\frac{1930441 \left(2 x^2-x+3\right)^{5/2}}{13934592 (2 x+5)^5}+\frac{114335 \left(2 x^2-x+3\right)^{5/2}}{193536 (2 x+5)^6}-\frac{3667 \left(2 x^2-x+3\right)^{5/2}}{4032 (2 x+5)^7}-\frac{(411822458 x+463558457) \left(2 x^2-x+3\right)^{3/2}}{2293235712 (2 x+5)^4}-\frac{(101679102454 x+146583836191) \sqrt{2 x^2-x+3}}{440301256704 (2 x+5)^2}+\frac{412760561351 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{5283615080448 \sqrt{2}}-\frac{5 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{64 \sqrt{2}}",1,"((-24*Sqrt[3 - x + 2*x^2]*(3479517268702637 + 9065154700300572*x + 9976065367498188*x^2 + 5966329646300704*x^3 + 2069947287085104*x^4 + 402255822731712*x^5 + 38463671680832*x^6))/(5 + 2*x)^7 - 2889476997120*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 2889323929457*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/73970611126272","A",1
344,1,60,120,0.1169407,"\int \frac{(5+2 x) \left(2+x+3 x^2-x^3+5 x^4\right)}{\sqrt{3-x+2 x^2}} \, dx","Integrate[((5 + 2*x)*(2 + x + 3*x^2 - x^3 + 5*x^4))/Sqrt[3 - x + 2*x^2],x]","\frac{4 \sqrt{2 x^2-x+3} \left(2048 x^4+7040 x^3+352 x^2-6916 x+2973\right)-85429 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{8192}","\frac{1}{16} \sqrt{2 x^2-x+3} (2 x+5)^4-\frac{105}{128} \sqrt{2 x^2-x+3} (2 x+5)^3+\frac{761}{256} \sqrt{2 x^2-x+3} (2 x+5)^2-\frac{(4676 x+19227) \sqrt{2 x^2-x+3}}{2048}-\frac{85429 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{4096 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(2973 - 6916*x + 352*x^2 + 7040*x^3 + 2048*x^4) - 85429*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/8192","A",1
345,1,55,101,0.0733325,"\int \frac{2+x+3 x^2-x^3+5 x^4}{\sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/Sqrt[3 - x + 2*x^2],x]","\frac{4 \sqrt{2 x^2-x+3} \left(1920 x^3+608 x^2-1636 x-1515\right)-20589 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{12288}","\frac{19}{96} \sqrt{2 x^2-x+3} x^2-\frac{409}{768} \sqrt{2 x^2-x+3} x-\frac{505 \sqrt{2 x^2-x+3}}{1024}+\frac{5}{8} \sqrt{2 x^2-x+3} x^3-\frac{6863 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{2048 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(-1515 - 1636*x + 608*x^2 + 1920*x^3) - 20589*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/12288","A",1
346,1,81,126,0.1020733,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)*Sqrt[3 - x + 2*x^2]),x]","\frac{4 \sqrt{2 x^2-x+3} \left(160 x^2-548 x+2637\right)-29336 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+28971 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{1536}","\frac{5}{48} \sqrt{2 x^2-x+3} (2 x+5)^2-\frac{337}{192} \sqrt{2 x^2-x+3} (2 x+5)+\frac{1669}{128} \sqrt{2 x^2-x+3}-\frac{3667 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{96 \sqrt{2}}+\frac{9657 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{256 \sqrt{2}}",1,"(4*Sqrt[3 - x + 2*x^2]*(2637 - 548*x + 160*x^2) + 28971*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 29336*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/1536","A",1
347,1,88,126,0.1150058,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^2*Sqrt[3 - x + 2*x^2]),x]","\frac{\frac{48 \sqrt{2 x^2-x+3} \left(180 x^2-1287 x-6176\right)}{2 x+5}+158527 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)-158922 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{13824}","\frac{5}{32} \sqrt{2 x^2-x+3} (2 x+5)-\frac{243}{64} \sqrt{2 x^2-x+3}-\frac{3667 \sqrt{2 x^2-x+3}}{576 (2 x+5)}+\frac{158527 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{6912 \sqrt{2}}-\frac{2943 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{128 \sqrt{2}}",1,"((48*Sqrt[3 - x + 2*x^2]*(-6176 - 1287*x + 180*x^2))/(5 + 2*x) - 158922*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 158527*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/13824","A",1
348,1,88,128,0.1290038,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^3*Sqrt[3 - x + 2*x^2]),x]","\frac{\frac{24 \sqrt{2 x^2-x+3} \left(34560 x^2+357278 x+589187\right)}{(2 x+5)^2}-1546507 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+1544832 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{663552}","\frac{92239 \sqrt{2 x^2-x+3}}{27648 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{1152 (2 x+5)^2}+\frac{5}{16} \sqrt{2 x^2-x+3}-\frac{1546507 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{331776 \sqrt{2}}+\frac{149 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32 \sqrt{2}}",1,"((24*Sqrt[3 - x + 2*x^2]*(589187 + 357278*x + 34560*x^2))/(5 + 2*x)^2 + 1544832*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - 1546507*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/663552","A",1
349,1,88,135,0.1487719,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^4*Sqrt[3 - x + 2*x^2]),x]","\frac{-\frac{24 \sqrt{2 x^2-x+3} \left(12653660 x^2+44312764 x+44369687\right)}{(2 x+5)^3}+22389491 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)-22394880 \sqrt{2} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{143327232}","-\frac{3163415 \sqrt{2 x^2-x+3}}{5971968 (2 x+5)}+\frac{394907 \sqrt{2 x^2-x+3}}{248832 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{1728 (2 x+5)^3}+\frac{22389491 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{71663616 \sqrt{2}}-\frac{5 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{16 \sqrt{2}}",1,"((-24*Sqrt[3 - x + 2*x^2]*(44369687 + 44312764*x + 12653660*x^2))/(5 + 2*x)^3 - 22394880*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 22389491*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/143327232","A",1
350,1,81,139,0.134847,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt{3-x+2 x^2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^5*Sqrt[3 - x + 2*x^2]),x]","\frac{2053207 \sqrt{2} (2 x+5)^4 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+24 \sqrt{2 x^2-x+3} \left(214400680 x^3+43592076 x^2-255525906 x-298655447\right)}{41278242816 (2 x+5)^4}","\frac{26800085 \sqrt{2 x^2-x+3}}{1719926784 (2 x+5)}-\frac{16295969 \sqrt{2 x^2-x+3}}{71663616 (2 x+5)^2}+\frac{513097 \sqrt{2 x^2-x+3}}{497664 (2 x+5)^3}-\frac{3667 \sqrt{2 x^2-x+3}}{2304 (2 x+5)^4}+\frac{2053207 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{20639121408 \sqrt{2}}",1,"(24*Sqrt[3 - x + 2*x^2]*(-298655447 - 255525906*x + 43592076*x^2 + 214400680*x^3) + 2053207*Sqrt[2]*(5 + 2*x)^4*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(41278242816*(5 + 2*x)^4)","A",1
351,1,74,124,0.470333,"\int \frac{(5+2 x)^2 \left(2+x+3 x^2-x^3+5 x^4\right)}{\left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[((5 + 2*x)^2*(2 + x + 3*x^2 - x^3 + 5*x^4))/(3 - x + 2*x^2)^(3/2),x]","\frac{3316991 \sqrt{4 x^2-2 x+6} \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)+4 \left(29440 x^5+210496 x^4+418232 x^3-510554 x^2+2124123 x-1616165\right)}{47104 \sqrt{2 x^2-x+3}}","\frac{153}{16} \sqrt{2 x^2-x+3} x^2+\frac{2645}{128} \sqrt{2 x^2-x+3} x-\frac{13153}{512} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{23 \sqrt{2 x^2-x+3}}+\frac{5}{4} \sqrt{2 x^2-x+3} x^3+\frac{144217 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{1024 \sqrt{2}}",1,"(4*(-1616165 + 2124123*x - 510554*x^2 + 418232*x^3 + 210496*x^4 + 29440*x^5) + 3316991*Sqrt[6 - 2*x + 4*x^2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/(47104*Sqrt[3 - x + 2*x^2])","A",1
352,1,60,103,0.1851986,"\int \frac{(5+2 x) \left(2+x+3 x^2-x^3+5 x^4\right)}{\left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[((5 + 2*x)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(3 - x + 2*x^2)^(3/2),x]","\frac{7360 x^4+31832 x^3-2162 x^2+122607 x-3345}{4416 \sqrt{2 x^2-x+3}}-\frac{3111 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{128 \sqrt{2}}","\frac{5}{6} \sqrt{2 x^2-x+3} x^2+\frac{193}{48} \sqrt{2 x^2-x+3} x+\frac{33}{64} \sqrt{2 x^2-x+3}+\frac{373 x-53}{23 \sqrt{2 x^2-x+3}}+\frac{3111 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{128 \sqrt{2}}",1,"(-3345 + 122607*x - 2162*x^2 + 31832*x^3 + 7360*x^4)/(4416*Sqrt[3 - x + 2*x^2]) - (3111*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(128*Sqrt[2])","A",1
353,1,55,82,0.1308931,"\int \frac{2+x+3 x^2-x^3+5 x^4}{\left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(3/2),x]","\frac{920 x^3+782 x^2+2511 x+2575}{736 \sqrt{2 x^2-x+3}}-\frac{213 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{64 \sqrt{2}}","\frac{5}{8} \sqrt{2 x^2-x+3} x+\frac{27}{32} \sqrt{2 x^2-x+3}+\frac{219 x+89}{92 \sqrt{2 x^2-x+3}}+\frac{213 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{64 \sqrt{2}}",1,"(2575 + 2511*x + 782*x^2 + 920*x^3)/(736*Sqrt[3 - x + 2*x^2]) - (213*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(64*Sqrt[2])","A",1
354,1,86,101,0.353671,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)*(3 - x + 2*x^2)^(3/2)),x]","\frac{\frac{12 \left(4140 x^2-1153 x+7401\right)}{23 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}-3667 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)+3667 \log (2 x+5)-4212 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{1728 \sqrt{2}}","\frac{917 x+1191}{3312 \sqrt{2 x^2-x+3}}+\frac{5}{8} \sqrt{2 x^2-x+3}-\frac{3667 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{1728 \sqrt{2}}+\frac{39 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{16 \sqrt{2}}",1,"((12*(7401 - 1153*x + 4140*x^2))/(23*Sqrt[3/2 - x/2 + x^2]) - 4212*ArcSinh[(-1 + 4*x)/Sqrt[23]] + 3667*Log[5 + 2*x] - 3667*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(1728*Sqrt[2])","A",1
355,1,104,108,0.3234014,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^2*(3 - x + 2*x^2)^(3/2)),x]","\frac{\frac{8 (2203 x+9897)}{23 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}-\frac{14668 \sqrt{4 x^2-2 x+6}}{2 x+5}+25951 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)-25951 \log (2 x+5)+25920 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{41472 \sqrt{2}}","\frac{2203 x+9897}{119232 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{10368 (2 x+5)}+\frac{25951 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{41472 \sqrt{2}}-\frac{5 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{8 \sqrt{2}}",1,"((8*(9897 + 2203*x))/(23*Sqrt[3/2 - x/2 + x^2]) - (14668*Sqrt[6 - 2*x + 4*x^2])/(5 + 2*x) + 25920*ArcSinh[(-1 + 4*x)/Sqrt[23]] - 25951*Log[5 + 2*x] + 25951*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(41472*Sqrt[2])","A",1
356,1,84,112,0.2979795,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^3*(3 - x + 2*x^2)^(3/2)),x]","\frac{-52631 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)+\frac{12 \left(3444340 x^3+3263288 x^2+5842933 x+11594283\right)}{23 (2 x+5)^2 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}+52631 \log (2 x+5)}{5971968 \sqrt{2}}","\frac{65991-8779 x}{4292352 \sqrt{2 x^2-x+3}}+\frac{115369 \sqrt{2 x^2-x+3}}{1492992 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{20736 (2 x+5)^2}-\frac{52631 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{5971968 \sqrt{2}}",1,"((12*(11594283 + 5842933*x + 3263288*x^2 + 3444340*x^3))/(23*(5 + 2*x)^2*Sqrt[3/2 - x/2 + x^2]) + 52631*Log[5 + 2*x] - 52631*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(5971968*Sqrt[2])","A",1
357,1,95,137,0.1581168,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left(3-x+2 x^2\right)^{3/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^4*(3 - x + 2*x^2)^(3/2)),x]","\frac{24 \left(56754760 x^4+572739684 x^3+441046842 x^2+1257975811 x+1873786587\right)-80633837 (2 x+5)^3 \sqrt{4 x^2-2 x+6} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)}{59337474048 (2 x+5)^3 \sqrt{2 x^2-x+3}}","\frac{369609-175877 x}{154524672 \sqrt{2 x^2-x+3}}+\frac{430799 \sqrt{2 x^2-x+3}}{107495424 (2 x+5)}+\frac{152885 \sqrt{2 x^2-x+3}}{4478976 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{31104 (2 x+5)^3}-\frac{3505819 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{1289945088 \sqrt{2}}",1,"(24*(1873786587 + 1257975811*x + 441046842*x^2 + 572739684*x^3 + 56754760*x^4) - 80633837*(5 + 2*x)^3*Sqrt[6 - 2*x + 4*x^2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(59337474048*(5 + 2*x)^3*Sqrt[3 - x + 2*x^2])","A",1
358,1,65,105,0.6867788,"\int \frac{(5+2 x)^2 \left(2+x+3 x^2-x^3+5 x^4\right)}{\left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[((5 + 2*x)^2*(2 + x + 3*x^2 - x^3 + 5*x^4))/(3 - x + 2*x^2)^(5/2),x]","\frac{126960 x^5+1440996 x^4-3764360 x^3+8639625 x^2-6410082 x+6663133}{25392 \left(2 x^2-x+3\right)^{3/2}}-\frac{1471 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32 \sqrt{2}}","\frac{4 (18982-20383 x)}{1587 \sqrt{2 x^2-x+3}}+\frac{5}{4} x \sqrt{2 x^2-x+3}+\frac{247}{16} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{69 \left(2 x^2-x+3\right)^{3/2}}-\frac{1471 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{32 \sqrt{2}}",1,"(6663133 - 6410082*x + 8639625*x^2 - 3764360*x^3 + 1440996*x^4 + 126960*x^5)/(25392*(3 - x + 2*x^2)^(3/2)) - (1471*ArcSinh[(1 - 4*x)/Sqrt[23]])/(32*Sqrt[2])","A",1
359,1,60,86,0.2534448,"\int \frac{(5+2 x) \left(2+x+3 x^2-x^3+5 x^4\right)}{\left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[((5 + 2*x)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(3 - x + 2*x^2)^(5/2),x]","\frac{31740 x^4-147664 x^3+185337 x^2-199290 x+102869}{6348 \left(2 x^2-x+3\right)^{3/2}}+\frac{71 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{8 \sqrt{2}}","\frac{6055-28981 x}{3174 \sqrt{2 x^2-x+3}}+\frac{5}{4} \sqrt{2 x^2-x+3}+\frac{373 x-53}{69 \left(2 x^2-x+3\right)^{3/2}}-\frac{71 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{8 \sqrt{2}}",1,"(102869 - 199290*x + 185337*x^2 - 147664*x^3 + 31740*x^4)/(6348*(3 - x + 2*x^2)^(3/2)) + (71*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(8*Sqrt[2])","A",1
360,1,55,68,0.2293853,"\int \frac{2+x+3 x^2-x^3+5 x^4}{\left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(5/2),x]","\frac{5 \sinh ^{-1}\left(\frac{4 x-1}{\sqrt{23}}\right)}{4 \sqrt{2}}-\frac{7812 x^3+489 x^2+7002 x+5569}{3174 \left(2 x^2-x+3\right)^{3/2}}","\frac{219 x+89}{276 \left(2 x^2-x+3\right)^{3/2}}-\frac{2604 x+1465}{2116 \sqrt{2 x^2-x+3}}-\frac{5 \sinh ^{-1}\left(\frac{1-4 x}{\sqrt{23}}\right)}{4 \sqrt{2}}",1,"-1/3174*(5569 + 7002*x + 489*x^2 + 7812*x^3)/(3 - x + 2*x^2)^(3/2) + (5*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2])","A",1
361,1,80,85,0.4698201,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)*(3 - x + 2*x^2)^(5/2)),x]","\frac{-3667 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)-\frac{12 \sqrt{2} \left(293458 x^3+523945 x^2-21696 x+841653\right)}{529 \left(2 x^2-x+3\right)^{3/2}}+3667 \log (2 x+5)}{31104 \sqrt{2}}","\frac{917 x+1191}{9936 \left(2 x^2-x+3\right)^{3/2}}-\frac{146729 x+335337}{1371168 \sqrt{2 x^2-x+3}}-\frac{3667 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{31104 \sqrt{2}}",1,"((-12*Sqrt[2]*(841653 - 21696*x + 523945*x^2 + 293458*x^3))/(529*(3 - x + 2*x^2)^(3/2)) + 3667*Log[5 + 2*x] - 3667*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(31104*Sqrt[2])","A",1
362,1,92,110,0.4079379,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^2*(3 - x + 2*x^2)^(5/2)),x]","\frac{-2821 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)-\frac{12 \sqrt{2} \left(6767036 x^4+10350004 x^3+63941915 x^2-18840090 x+79153407\right)}{529 (2 x+5) \left(2 x^2-x+3\right)^{3/2}}+2821 \log (2 x+5)}{2239488 \sqrt{2}}","-\frac{1255878-62021 x}{24681024 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{186624 (2 x+5)}+\frac{2203 x+9897}{357696 \left(2 x^2-x+3\right)^{3/2}}-\frac{2821 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{2239488 \sqrt{2}}",1,"((-12*Sqrt[2]*(79153407 - 18840090*x + 63941915*x^2 + 10350004*x^3 + 6767036*x^4))/(529*(5 + 2*x)*(3 - x + 2*x^2)^(3/2)) + 2821*Log[5 + 2*x] - 2821*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(2239488*Sqrt[2])","A",1
363,1,97,135,0.3097319,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 \left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^3*(3 - x + 2*x^2)^(5/2)),x]","\frac{774079 \log \left(12 \sqrt{4 x^2-2 x+6}-22 x+17\right)+\frac{12 \sqrt{2} \left(217883368 x^5+107028732 x^4-1503926130 x^3-5919924791 x^2+2280511668 x-8953831359\right)}{529 (2 x+5)^2 \left(2 x^2-x+3\right)^{3/2}}-774079 \log (2 x+5)}{322486272 \sqrt{2}}","-\frac{4679797-2148263 x}{592344576 \sqrt{2 x^2-x+3}}-\frac{45979 \sqrt{2 x^2-x+3}}{26873856 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{373248 (2 x+5)^2}+\frac{65991-8779 x}{12877056 \left(2 x^2-x+3\right)^{3/2}}+\frac{774079 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{322486272 \sqrt{2}}",1,"((12*Sqrt[2]*(-8953831359 + 2280511668*x - 5919924791*x^2 - 1503926130*x^3 + 107028732*x^4 + 217883368*x^5))/(529*(5 + 2*x)^2*(3 - x + 2*x^2)^(3/2)) - 774079*Log[5 + 2*x] + 774079*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(322486272*Sqrt[2])","A",1
364,1,89,160,0.2358038,"\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left(3-x+2 x^2\right)^{5/2}} \, dx","Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/((5 + 2*x)^4*(3 - x + 2*x^2)^(5/2)),x]","\frac{2527979381 \sqrt{2} \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right)+\frac{24 \left(6664404208 x^6+34872810880 x^5+46210466520 x^4+27484986184 x^3-6702882569 x^2+73621973154 x-95241881529\right)}{(2 x+5)^3 \left(2 x^2-x+3\right)^{3/2}}}{8188571418624}","-\frac{27754539-31190998 x}{31986607104 \sqrt{2 x^2-x+3}}+\frac{475357 \sqrt{2 x^2-x+3}}{1934917632 (2 x+5)}-\frac{89137 \sqrt{2 x^2-x+3}}{80621568 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{559872 (2 x+5)^3}+\frac{369609-175877 x}{463574016 \left(2 x^2-x+3\right)^{3/2}}+\frac{4778789 \tanh ^{-1}\left(\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right)}{7739670528 \sqrt{2}}",1,"((24*(-95241881529 + 73621973154*x - 6702882569*x^2 + 27484986184*x^3 + 46210466520*x^4 + 34872810880*x^5 + 6664404208*x^6))/((5 + 2*x)^3*(3 - x + 2*x^2)^(3/2)) + 2527979381*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/8188571418624","A",1
365,1,316,354,1.1447546,"\int \frac{f+g x+h x^2+i x^3+j x^4}{\left(a+b x+c x^2\right)^{5/2}} \, dx","Integrate[(f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x + c*x^2)^(5/2),x]","\frac{-\frac{2 \left(b c \left(-3 a^2 j+a c (h+3 i x)+c^2 (f-g x)\right)+2 c^2 \left(a^2 (i+j x)-a c (g+h x)+c^2 f x\right)+b^3 (a j-c i x)+b^2 c (c h x-a (i+4 j x))+b^4 j x\right)}{\left(b^2-4 a c\right) (a+x (b+c x))^{3/2}}+\frac{2 \left(4 b c^2 \left(8 a^2 j+a c (h-3 i x)+2 c^2 (f-g x)\right)+8 c^3 \left(-\left(a^2 (3 i+4 j x)\right)+a c h x+2 c^2 f x\right)+b^3 c (c (h+i x)-10 a j)+2 b^2 c^2 (3 a i+14 a j x-2 c g+c h x)+b^5 j-b^4 c (i+4 j x)\right)}{\left(b^2-4 a c\right)^2 \sqrt{a+x (b+c x)}}+3 \sqrt{c} j \log \left(2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right)}{3 c^3}","\frac{2 \left(-x \left(c^2 \left(2 a^2 j+3 a b i+b^2 h\right)-b^2 c (4 a j+b i)-c^3 (2 a h+b g)+b^4 j+2 c^4 f\right)-b c \left(-3 a^2 j+a c h+c^2 f\right)-a b^3 j+a b^2 c i+2 a c^2 (c g-a i)\right)}{3 c^3 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^{3/2}}-\frac{2 \left(-c x \left(2 c^2 \left(-16 a^2 j-6 a b i+b^2 h\right)+b^2 c (28 a j+b i)-c^3 (8 b g-8 a h)-4 b^4 j+16 c^4 f\right)-4 b c^2 \left(8 a^2 j+a c h+2 c^2 f\right)+24 a^2 c^3 i-b^3 c (c h-10 a j)+2 b^2 c^2 (2 c g-3 a i)+b^5 (-j)+b^4 c i\right)}{3 c^3 \left(b^2-4 a c\right)^2 \sqrt{a+b x+c x^2}}+\frac{j \tanh ^{-1}\left(\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right)}{c^{5/2}}",1,"((-2*(b^4*j*x + b^3*(a*j - c*i*x) + b*c*(-3*a^2*j + c^2*(f - g*x) + a*c*(h + 3*i*x)) + 2*c^2*(c^2*f*x - a*c*(g + h*x) + a^2*(i + j*x)) + b^2*c*(c*h*x - a*(i + 4*j*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2)) + (2*(b^5*j - b^4*c*(i + 4*j*x) + 2*b^2*c^2*(-2*c*g + 3*a*i + c*h*x + 14*a*j*x) + 4*b*c^2*(8*a^2*j + 2*c^2*(f - g*x) + a*c*(h - 3*i*x)) + b^3*c*(-10*a*j + c*(h + i*x)) + 8*c^3*(2*c^2*f*x + a*c*h*x - a^2*(3*i + 4*j*x))))/((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]) + 3*Sqrt[c]*j*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3*c^3)","A",1
366,1,319,353,1.1109231,"\int \frac{f+g x+h x^2+i x^3+j x^4}{\left(a+b x-c x^2\right)^{5/2}} \, dx","Integrate[(f + g*x + h*x^2 + i*x^3 + j*x^4)/(a + b*x - c*x^2)^(5/2),x]","-\frac{2 \left(b^3 \left(3 a^2 j+18 a c j x^2+c^2 \left(f+3 g x-\left(x^2 (3 h+i x)\right)\right)\right)+2 b^2 c \left(21 a^2 j x+a c \left(g+x \left(-6 h+3 i x-14 j x^2\right)\right)+c^2 x (3 f+x (h x-6 g))\right)+4 b c \left(5 a^3 j-2 a^2 c (h-3 i x)+3 a c^2 \left(f-x \left(g-h x+i x^2\right)\right)+2 c^3 x^2 (g x-3 f)\right)+8 c^2 \left(a^3 (2 i+3 j x)-a^2 c \left(g+x^2 (3 i+4 j x)\right)-a c^2 x \left(3 f+h x^2\right)+2 c^3 f x^3\right)+b^4 \left(6 a j x-4 c j x^3\right)+3 b^5 j x^2\right)}{3 c^2 \left(4 a c+b^2\right)^2 (a+x (b-c x))^{3/2}}+\frac{i j \log \left(2 \sqrt{a+x (b-c x)}+\frac{i (b-2 c x)}{\sqrt{c}}\right)}{c^{5/2}}","\frac{2 \left(x \left(c^2 \left(2 a^2 j+3 a b i+b^2 h\right)+b^2 c (4 a j+b i)+c^3 (2 a h+b g)+b^4 j+2 c^4 f\right)-b c \left(-3 a^2 j-a c h+c^2 f\right)+a b^3 j+a b^2 c i+2 a c^2 (a i+c g)\right)}{3 c^3 \left(4 a c+b^2\right) \left(a+b x-c x^2\right)^{3/2}}-\frac{2 \left(-c x \left(2 c^2 \left(-16 a^2 j-6 a b i+b^2 h\right)-b^2 c (28 a j+b i)+8 c^3 (b g-a h)-4 b^4 j+16 c^4 f\right)+4 b c^2 \left(8 a^2 j-a c h+2 c^2 f\right)+24 a^2 c^3 i+b^3 c (10 a j+c h)+2 b^2 c^2 (3 a i+2 c g)+b^5 j+b^4 c i\right)}{3 c^3 \left(4 a c+b^2\right)^2 \sqrt{a+b x-c x^2}}-\frac{j \tan ^{-1}\left(\frac{b-2 c x}{2 \sqrt{c} \sqrt{a+b x-c x^2}}\right)}{c^{5/2}}",1,"(-2*(3*b^5*j*x^2 + b^4*(6*a*j*x - 4*c*j*x^3) + b^3*(3*a^2*j + 18*a*c*j*x^2 + c^2*(f + 3*g*x - x^2*(3*h + i*x))) + 8*c^2*(2*c^3*f*x^3 + a^3*(2*i + 3*j*x) - a*c^2*x*(3*f + h*x^2) - a^2*c*(g + x^2*(3*i + 4*j*x))) + 4*b*c*(5*a^3*j + 2*c^3*x^2*(-3*f + g*x) - 2*a^2*c*(h - 3*i*x) + 3*a*c^2*(f - x*(g - h*x + i*x^2))) + 2*b^2*c*(21*a^2*j*x + c^2*x*(3*f + x*(-6*g + h*x)) + a*c*(g + x*(-6*h + 3*i*x - 14*j*x^2)))))/(3*c^2*(b^2 + 4*a*c)^2*(a + x*(b - c*x))^(3/2)) + (I*j*Log[(I*(b - 2*c*x))/Sqrt[c] + 2*Sqrt[a + x*(b - c*x)]])/c^(5/2)","C",1
367,1,537,588,0.3823425,"\int (d+e x)^m \left(3+2 x+5 x^2\right)^3 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^m*(3 + 2*x + 5*x^2)^3*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{(d+e x)^{m+1} \left(\frac{45 \left(500 d^2+5 d e+17 e^2\right) (d+e x)^8}{m+9}-\frac{2 \left(30000 d^3+450 d^2 e+3060 d e^2+49 e^3\right) (d+e x)^7}{m+8}+\frac{\left(105000 d^4+2100 d^3 e+21420 d^2 e^2+686 d e^3+999 e^4\right) (d+e x)^6}{m+7}+\frac{\left(5 d^2-2 d e+3 e^2\right)^3 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{m+1}-\frac{6 \left(21000 d^5+525 d^4 e+7140 d^3 e^2+343 d^2 e^3+999 d e^4-85 e^5\right) (d+e x)^5}{m+6}-\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(200 d^5+169 d^4 e+108 d^3 e^2-20 d^2 e^3+86 d e^4-15 e^5\right) (d+e x)}{m+2}+\frac{\left(105000 d^6+3150 d^5 e+53550 d^4 e^2+3430 d^3 e^3+14985 d^2 e^4-2550 d e^5+1109 e^6\right) (d+e x)^4}{m+5}+\frac{3 \left(5 d^2-2 d e+3 e^2\right) \left(1500 d^6+660 d^5 e+792 d^4 e^2+58 d^3 e^3+547 d^2 e^4-156 d e^5+53 e^6\right) (d+e x)^2}{m+3}-\frac{2 \left(30000 d^7+1050 d^6 e+21420 d^5 e^2+1715 d^4 e^3+9990 d^3 e^4-2550 d^2 e^5+2218 d e^6-287 e^7\right) (d+e x)^3}{m+4}+\frac{500 (d+e x)^{10}}{m+11}-\frac{25 (200 d+e) (d+e x)^9}{m+10}\right)}{e^{11}}","\frac{45 \left(500 d^2+5 d e+17 e^2\right) (d+e x)^{m+9}}{e^{11} (m+9)}-\frac{2 \left(30000 d^3+450 d^2 e+3060 d e^2+49 e^3\right) (d+e x)^{m+8}}{e^{11} (m+8)}+\frac{\left(5 d^2-2 d e+3 e^2\right)^3 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) (d+e x)^{m+1}}{e^{11} (m+1)}+\frac{\left(105000 d^4+2100 d^3 e+21420 d^2 e^2+686 d e^3+999 e^4\right) (d+e x)^{m+7}}{e^{11} (m+7)}-\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(200 d^5+169 d^4 e+108 d^3 e^2-20 d^2 e^3+86 d e^4-15 e^5\right) (d+e x)^{m+2}}{e^{11} (m+2)}-\frac{6 \left(21000 d^5+525 d^4 e+7140 d^3 e^2+343 d^2 e^3+999 d e^4-85 e^5\right) (d+e x)^{m+6}}{e^{11} (m+6)}+\frac{3 \left(5 d^2-2 d e+3 e^2\right) \left(1500 d^6+660 d^5 e+792 d^4 e^2+58 d^3 e^3+547 d^2 e^4-156 d e^5+53 e^6\right) (d+e x)^{m+3}}{e^{11} (m+3)}+\frac{\left(105000 d^6+3150 d^5 e+53550 d^4 e^2+3430 d^3 e^3+14985 d^2 e^4-2550 d e^5+1109 e^6\right) (d+e x)^{m+5}}{e^{11} (m+5)}-\frac{2 \left(30000 d^7+1050 d^6 e+21420 d^5 e^2+1715 d^4 e^3+9990 d^3 e^4-2550 d^2 e^5+2218 d e^6-287 e^7\right) (d+e x)^{m+4}}{e^{11} (m+4)}-\frac{25 (200 d+e) (d+e x)^{m+10}}{e^{11} (m+10)}+\frac{500 (d+e x)^{m+11}}{e^{11} (m+11)}",1,"((d + e*x)^(1 + m)*(((5*d^2 - 2*d*e + 3*e^2)^3*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(1 + m) - ((5*d^2 - 2*d*e + 3*e^2)^2*(200*d^5 + 169*d^4*e + 108*d^3*e^2 - 20*d^2*e^3 + 86*d*e^4 - 15*e^5)*(d + e*x))/(2 + m) + (3*(5*d^2 - 2*d*e + 3*e^2)*(1500*d^6 + 660*d^5*e + 792*d^4*e^2 + 58*d^3*e^3 + 547*d^2*e^4 - 156*d*e^5 + 53*e^6)*(d + e*x)^2)/(3 + m) - (2*(30000*d^7 + 1050*d^6*e + 21420*d^5*e^2 + 1715*d^4*e^3 + 9990*d^3*e^4 - 2550*d^2*e^5 + 2218*d*e^6 - 287*e^7)*(d + e*x)^3)/(4 + m) + ((105000*d^6 + 3150*d^5*e + 53550*d^4*e^2 + 3430*d^3*e^3 + 14985*d^2*e^4 - 2550*d*e^5 + 1109*e^6)*(d + e*x)^4)/(5 + m) - (6*(21000*d^5 + 525*d^4*e + 7140*d^3*e^2 + 343*d^2*e^3 + 999*d*e^4 - 85*e^5)*(d + e*x)^5)/(6 + m) + ((105000*d^4 + 2100*d^3*e + 21420*d^2*e^2 + 686*d*e^3 + 999*e^4)*(d + e*x)^6)/(7 + m) - (2*(30000*d^3 + 450*d^2*e + 3060*d*e^2 + 49*e^3)*(d + e*x)^7)/(8 + m) + (45*(500*d^2 + 5*d*e + 17*e^2)*(d + e*x)^8)/(9 + m) - (25*(200*d + e)*(d + e*x)^9)/(10 + m) + (500*(d + e*x)^10)/(11 + m)))/e^11","A",1
368,1,391,432,0.2423622,"\int (d+e x)^m \left(3+2 x+5 x^2\right)^2 \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^m*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{(d+e x)^{m+1} \left(\frac{\left(2800 d^2+315 d e+111 e^2\right) (d+e x)^6}{m+7}-\frac{\left(5600 d^3+945 d^2 e+666 d e^2+37 e^3\right) (d+e x)^5}{m+6}+\frac{\left(7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right) (d+e x)^4}{m+5}+\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{m+1}-\frac{\left(5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right) (d+e x)^3}{m+4}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right) (d+e x)}{m+2}+\frac{\left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right) (d+e x)^2}{m+3}+\frac{100 (d+e x)^8}{m+9}-\frac{5 (160 d+9 e) (d+e x)^7}{m+8}\right)}{e^9}","\frac{\left(2800 d^2+315 d e+111 e^2\right) (d+e x)^{m+7}}{e^9 (m+7)}-\frac{\left(5600 d^3+945 d^2 e+666 d e^2+37 e^3\right) (d+e x)^{m+6}}{e^9 (m+6)}+\frac{\left(5 d^2-2 d e+3 e^2\right)^2 \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) (d+e x)^{m+1}}{e^9 (m+1)}+\frac{\left(7000 d^4+1575 d^3 e+1665 d^2 e^2+185 d e^3+148 e^4\right) (d+e x)^{m+5}}{e^9 (m+5)}-\frac{\left(5 d^2-2 d e+3 e^2\right) \left(160 d^5+127 d^4 e+88 d^3 e^2-4 d^2 e^3+64 d e^4-11 e^5\right) (d+e x)^{m+2}}{e^9 (m+2)}-\frac{\left(5600 d^5+1575 d^4 e+2220 d^3 e^2+370 d^2 e^3+592 d e^4-65 e^5\right) (d+e x)^{m+4}}{e^9 (m+4)}+\frac{\left(2800 d^6+945 d^5 e+1665 d^4 e^2+370 d^3 e^3+888 d^2 e^4-195 d e^5+107 e^6\right) (d+e x)^{m+3}}{e^9 (m+3)}-\frac{5 (160 d+9 e) (d+e x)^{m+8}}{e^9 (m+8)}+\frac{100 (d+e x)^{m+9}}{e^9 (m+9)}",1,"((d + e*x)^(1 + m)*(((5*d^2 - 2*d*e + 3*e^2)^2*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(1 + m) - ((5*d^2 - 2*d*e + 3*e^2)*(160*d^5 + 127*d^4*e + 88*d^3*e^2 - 4*d^2*e^3 + 64*d*e^4 - 11*e^5)*(d + e*x))/(2 + m) + ((2800*d^6 + 945*d^5*e + 1665*d^4*e^2 + 370*d^3*e^3 + 888*d^2*e^4 - 195*d*e^5 + 107*e^6)*(d + e*x)^2)/(3 + m) - ((5600*d^5 + 1575*d^4*e + 2220*d^3*e^2 + 370*d^2*e^3 + 592*d*e^4 - 65*e^5)*(d + e*x)^3)/(4 + m) + ((7000*d^4 + 1575*d^3*e + 1665*d^2*e^2 + 185*d*e^3 + 148*e^4)*(d + e*x)^4)/(5 + m) - ((5600*d^3 + 945*d^2*e + 666*d*e^2 + 37*e^3)*(d + e*x)^5)/(6 + m) + ((2800*d^2 + 315*d*e + 111*e^2)*(d + e*x)^6)/(7 + m) - (5*(160*d + 9*e)*(d + e*x)^7)/(8 + m) + (100*(d + e*x)^8)/(9 + m)))/e^9","A",1
369,1,261,292,0.1685029,"\int (d+e x)^m \left(3+2 x+5 x^2\right) \left(2+x+3 x^2-5 x^3+4 x^4\right) \, dx","Integrate[(d + e*x)^m*(3 + 2*x + 5*x^2)*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]","\frac{(d+e x)^{m+1} \left(\frac{\left(300 d^2+85 d e+17 e^2\right) (d+e x)^4}{m+5}-\frac{2 \left(200 d^3+85 d^2 e+34 d e^2+2 e^3\right) (d+e x)^3}{m+4}+\frac{\left(300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right) (d+e x)^2}{m+3}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right)}{m+1}-\frac{\left(120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right) (d+e x)}{m+2}+\frac{20 (d+e x)^6}{m+7}-\frac{(120 d+17 e) (d+e x)^5}{m+6}\right)}{e^7}","\frac{\left(300 d^2+85 d e+17 e^2\right) (d+e x)^{m+5}}{e^7 (m+5)}-\frac{2 \left(200 d^3+85 d^2 e+34 d e^2+2 e^3\right) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{\left(5 d^2-2 d e+3 e^2\right) \left(4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right) (d+e x)^{m+1}}{e^7 (m+1)}+\frac{\left(300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{\left(120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right) (d+e x)^{m+2}}{e^7 (m+2)}-\frac{(120 d+17 e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{20 (d+e x)^{m+7}}{e^7 (m+7)}",1,"((d + e*x)^(1 + m)*(((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4))/(1 + m) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d*e^4 - 7*e^5)*(d + e*x))/(2 + m) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^2)/(3 + m) - (2*(200*d^3 + 85*d^2*e + 34*d*e^2 + 2*e^3)*(d + e*x)^3)/(4 + m) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^4)/(5 + m) - ((120*d + 17*e)*(d + e*x)^5)/(6 + m) + (20*(d + e*x)^6)/(7 + m)))/e^7","A",1
370,1,221,255,0.7313236,"\int \frac{(d+e x)^m \left(2+x+3 x^2-5 x^3+4 x^4\right)}{3+2 x+5 x^2} \, dx","Integrate[((d + e*x)^m*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2),x]","\frac{(d+e x)^{m+1} \left(\frac{28 \left(100 d^2+165 d e+81 e^2\right)}{e^3 (m+1)}+\frac{2800 (d+e x)^2}{e^3 (m+3)}-\frac{140 (40 d+33 e) (d+e x)}{e^3 (m+2)}-\frac{\left(423 \sqrt{14}+6412 i\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+\left(-1-i \sqrt{14}\right) e}\right)}{(m+1) \left(5 i d+\left(\sqrt{14}-i\right) e\right)}-\frac{\left(423 \sqrt{14}-6412 i\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \left(i+\sqrt{14}\right) e}\right)}{(m+1) \left(\left(\sqrt{14}+i\right) e-5 i d\right)}\right)}{3500}","\frac{\left(100 d^2+165 d e+81 e^2\right) (d+e x)^{m+1}}{125 e^3 (m+1)}-\frac{(40 d+33 e) (d+e x)^{m+2}}{25 e^3 (m+2)}+\frac{4 (d+e x)^{m+3}}{5 e^3 (m+3)}-\frac{\left(-423 \sqrt{14}+6412 i\right) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \sqrt{14} e-e}\right)}{3500 (m+1) \left(5 i d-\left(\sqrt{14}+i\right) e\right)}-\frac{\left(423 \sqrt{14}+6412 i\right) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d-\left(1+i \sqrt{14}\right) e}\right)}{3500 (m+1) \left(5 i d-\left(-\sqrt{14}+i\right) e\right)}",1,"((d + e*x)^(1 + m)*((28*(100*d^2 + 165*d*e + 81*e^2))/(e^3*(1 + m)) - (140*(40*d + 33*e)*(d + e*x))/(e^3*(2 + m)) + (2800*(d + e*x)^2)/(e^3*(3 + m)) - ((6412*I + 423*Sqrt[14])*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + (-1 - I*Sqrt[14])*e)])/(((5*I)*d + (-I + Sqrt[14])*e)*(1 + m)) - ((-6412*I + 423*Sqrt[14])*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + I*(I + Sqrt[14])*e)])/(((-5*I)*d + (I + Sqrt[14])*e)*(1 + m))))/3500","A",1
371,1,441,377,1.8028257,"\int \frac{(d+e x)^m \left(2+x+3 x^2-5 x^3+4 x^4\right)}{\left(3+2 x+5 x^2\right)^2} \, dx","Integrate[((d + e*x)^m*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]","\frac{(d+e x)^{m+1} \left(-\frac{\sqrt{14} \left(\frac{\left(2115 d^2+d e \left(-846+\left(-6412+423 i \sqrt{14}\right) m\right)+e^2 \left(1269+\left(98-1367 i \sqrt{14}\right) m\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+\left(-1-i \sqrt{14}\right) e}\right)}{5 i d+\left(\sqrt{14}-i\right) e}-\frac{\left(2115 d^2-d e \left(846+\left(6412+423 i \sqrt{14}\right) m\right)+e^2 \left(1269+\left(98+1367 i \sqrt{14}\right) m\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \left(i+\sqrt{14}\right) e}\right)}{5 i d-\left(\sqrt{14}+i\right) e}\right)}{(m+1) \left(5 d^2-2 d e+3 e^2\right)}-\frac{28 (d (423 x+1367)-e (1367 x+293))}{\left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)}+\frac{56 \left(31 \sqrt{14}+287 i\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+\left(-1-i \sqrt{14}\right) e}\right)}{(m+1) \left(5 i d+\left(\sqrt{14}-i\right) e\right)}+\frac{56 \left(31 \sqrt{14}-287 i\right) \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \left(i+\sqrt{14}\right) e}\right)}{(m+1) \left(\left(\sqrt{14}+i\right) e-5 i d\right)}+\frac{3136}{e m+e}\right)}{19600}","\frac{\left(i \sqrt{14} \left(6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right)+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d+i \sqrt{14} e-e}\right)}{19600 (m+1) \left(5 d+i \left(\sqrt{14}+i\right) e\right) \left(5 d^2-2 d e+3 e^2\right)}+\frac{\left(-i \sqrt{14} \left(6565 d^2-2 d e (1313-3206 m)+e^2 (3939-98 m)\right)+80360 d^2-5922 d e m-32144 d e+19138 e^2 m+48216 e^2\right) (d+e x)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{5 (d+e x)}{5 d-\left(1+i \sqrt{14}\right) e}\right)}{19600 (m+1) \left(5 d-\left(1+i \sqrt{14}\right) e\right) \left(5 d^2-2 d e+3 e^2\right)}-\frac{(x (423 d-1367 e)+1367 d-293 e) (d+e x)^{m+1}}{700 \left(5 x^2+2 x+3\right) \left(5 d^2-2 d e+3 e^2\right)}+\frac{4 (d+e x)^{m+1}}{25 e (m+1)}",1,"((d + e*x)^(1 + m)*(3136/(e + e*m) - (28*(d*(1367 + 423*x) - e*(293 + 1367*x)))/((5*d^2 - 2*d*e + 3*e^2)*(3 + 2*x + 5*x^2)) + (56*(287*I + 31*Sqrt[14])*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + (-1 - I*Sqrt[14])*e)])/(((5*I)*d + (-I + Sqrt[14])*e)*(1 + m)) + (56*(-287*I + 31*Sqrt[14])*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + I*(I + Sqrt[14])*e)])/(((-5*I)*d + (I + Sqrt[14])*e)*(1 + m)) - (Sqrt[14]*(((2115*d^2 + d*e*(-846 + (-6412 + (423*I)*Sqrt[14])*m) + e^2*(1269 + (98 - (1367*I)*Sqrt[14])*m))*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + (-1 - I*Sqrt[14])*e)])/((5*I)*d + (-I + Sqrt[14])*e) - ((2115*d^2 - d*e*(846 + (6412 + (423*I)*Sqrt[14])*m) + e^2*(1269 + (98 + (1367*I)*Sqrt[14])*m))*Hypergeometric2F1[1, 1 + m, 2 + m, (5*(d + e*x))/(5*d + I*(I + Sqrt[14])*e)])/((5*I)*d - (I + Sqrt[14])*e)))/((5*d^2 - 2*d*e + 3*e^2)*(1 + m))))/19600","A",1
372,1,488,528,1.0789896,"\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{\left(a+b x+c x^2\right)^3} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x]","\frac{\frac{2 c \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(2 c^3 \left(6 a^2 h-3 a b g+b^2 f\right)-30 a^2 b c^2 i+10 a b^3 c i+c^4 (4 a f-6 b e)+b^5 (-i)+12 c^5 d\right)}{\left(4 a c-b^2\right)^{5/2}}+\frac{b^2 c \left(-4 a^2 i+a c (g+4 h x)-c^2 f x\right)+b c^2 \left(a^2 (3 h+5 i x)-a c (f+3 g x)+c^2 (e x-d)\right)+2 c^2 \left(a^3 i-a^2 c (g+h x)+a c^2 (e+f x)-c^3 d x\right)+b^4 (a i-c h x)+b^3 c (c g x-a (h+5 i x))+b^5 i x}{\left(b^2-4 a c\right) (a+x (b+c x))^2}+\frac{b^2 c^2 \left(-39 a^2 i+a c (5 g+16 h x)+c^2 (2 f x-3 e)\right)+2 b c^3 \left(a^2 (11 h+25 i x)+a c (f-3 g x)+3 c^2 (d-e x)\right)+4 c^3 \left(8 a^3 i-a^2 c (4 g+5 h x)+a c^2 f x+3 c^3 d x\right)-b^4 c (c (g+2 h x)-11 a i)+b^3 c^2 (-8 a h-30 a i x+c f)+b^6 (-i)+b^5 c (h+4 i x)}{\left(b^2-4 a c\right)^2 (a+x (b+c x))}+c i \log (a+x (b+c x))}{2 c^4}","-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(2 c^3 \left(6 a^2 h-3 a b g+b^2 f\right)-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right)}{c^3 \left(b^2-4 a c\right)^{5/2}}-\frac{x \left(c^3 \left(2 a^2 h+3 a b g+b^2 f\right)-b c^2 \left(5 a^2 i+4 a b h+b^2 g\right)+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right)+b c^2 \left(-3 a^2 h+a c f+c^2 d\right)-2 a c^2 \left(a^2 i-a c g+c^2 e\right)-a b^4 i+a b^3 c h-a b^2 c (c g-4 a i)}{2 c^4 \left(b^2-4 a c\right) \left(a+b x+c x^2\right)^2}+\frac{-b^2 c^2 \left(39 a^2 i-5 a c g+3 c^2 e\right)+2 c x \left(c^3 \left(-10 a^2 h-3 a b g+b^2 f\right)-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right)+2 b c^3 \left(11 a^2 h+a c f+3 c^2 d\right)-16 a^2 c^3 (c g-2 a i)-b^4 c (c g-11 a i)+b^3 c^2 (c f-8 a h)+b^6 (-i)+b^5 c h}{2 c^4 \left(b^2-4 a c\right)^2 \left(a+b x+c x^2\right)}+\frac{i \log \left(a+b x+c x^2\right)}{2 c^3}",1,"((b^5*i*x + b^4*(a*i - c*h*x) + 2*c^2*(a^3*i - c^3*d*x + a*c^2*(e + f*x) - a^2*c*(g + h*x)) + b^2*c*(-4*a^2*i - c^2*f*x + a*c*(g + 4*h*x)) + b^3*c*(c*g*x - a*(h + 5*i*x)) + b*c^2*(c^2*(-d + e*x) - a*c*(f + 3*g*x) + a^2*(3*h + 5*i*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*i) + b^5*c*(h + 4*i*x) + b^3*c^2*(c*f - 8*a*h - 30*a*i*x) - b^4*c*(-11*a*i + c*(g + 2*h*x)) + 4*c^3*(8*a^3*i + 3*c^3*d*x + a*c^2*f*x - a^2*c*(4*g + 5*h*x)) + b^2*c^2*(-39*a^2*i + c^2*(-3*e + 2*f*x) + a*c*(5*g + 16*h*x)) + 2*b*c^3*(3*c^2*(d - e*x) + a*c*(f - 3*g*x) + a^2*(11*h + 25*i*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(12*c^5*d + c^4*(-6*b*e + 4*a*f) + 2*c^3*(b^2*f - 3*a*b*g + 6*a^2*h) - b^5*i + 10*a*b^3*c*i - 30*a^2*b*c^2*i)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*i*Log[a + x*(b + c*x)])/(2*c^4)","A",1
373,1,754,765,0.7307049,"\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x+c x^2} \, dx","Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x + c*x^2),x]","\frac{140 c^3 x^3 \left(c^2 \left(a^2 m+2 a b l+b^2 k\right)-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right)+210 c^2 x^2 \left(c^3 \left(a^2 l+2 a b k+b^2 j\right)-b c^2 \left(3 a^2 m+3 a b l+b^2 k\right)+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right)+210 \log (a+x (b+c x)) \left(c^5 \left(a^2 j+2 a b h+b^2 g\right)-b^3 c^2 \left(10 a^2 m+5 a b l+b^2 k\right)-c^4 \left(a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right)+b c^3 \left(4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right)+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right)+420 c x \left(c^4 \left(a^2 k+2 a b j+b^2 h\right)+b^2 c^2 \left(6 a^2 m+4 a b l+b^2 k\right)-c^3 \left(a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right)-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right)+\frac{420 \tan ^{-1}\left(\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right) \left(c^6 \left(2 a^2 h+3 a b g+b^2 f\right)+b^4 c^2 \left(20 a^2 m+7 a b l+b^2 k\right)-c^5 \left(2 a^3 k+5 a^2 b j+4 a b^2 h+b^3 g\right)-b^2 c^3 \left(16 a^3 m+14 a^2 b l+6 a b^2 k+b^3 j\right)+c^4 \left(2 a^4 m+7 a^3 b l+9 a^2 b^2 k+5 a b^3 j+b^4 h\right)-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right)}{\sqrt{4 a c-b^2}}+105 c^4 x^4 \left(-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right)+84 c^5 x^5 \left(-c (a m+b l)+b^2 m+c^2 k\right)+70 c^6 x^6 (c l-b m)+60 c^7 m x^7}{420 c^8}","\frac{x^3 \left(c^2 \left(a^2 m+2 a b l+b^2 k\right)-b^2 c (3 a m+b l)-c^3 (a k+b j)+b^4 m+c^4 h\right)}{3 c^5}+\frac{x^2 \left(c^3 \left(a^2 l+2 a b k+b^2 j\right)-b c^2 \left(3 a^2 m+3 a b l+b^2 k\right)+b^3 c (4 a m+b l)-c^4 (a j+b h)+b^5 (-m)+c^5 g\right)}{2 c^6}+\frac{\log \left(a+b x+c x^2\right) \left(c^5 \left(a^2 j+2 a b h+b^2 g\right)-b^3 c^2 \left(10 a^2 m+5 a b l+b^2 k\right)-c^4 \left(a^3 l+3 a^2 b k+3 a b^2 j+b^3 h\right)+b c^3 \left(4 a^3 m+6 a^2 b l+4 a b^2 k+b^3 j\right)+b^5 c (6 a m+b l)-c^6 (a g+b f)+b^7 (-m)+c^7 e\right)}{2 c^8}+\frac{x \left(c^4 \left(a^2 k+2 a b j+b^2 h\right)+b^2 c^2 \left(6 a^2 m+4 a b l+b^2 k\right)-c^3 \left(a^3 m+3 a^2 b l+3 a b^2 k+b^3 j\right)-b^4 c (5 a m+b l)-c^5 (a h+b g)+b^6 m+c^6 f\right)}{c^7}-\frac{\tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right) \left(c^6 \left(2 a^2 h+3 a b g+b^2 f\right)+b^4 c^2 \left(20 a^2 m+7 a b l+b^2 k\right)-c^5 \left(2 a^3 k+5 a^2 b j+4 a b^2 h+b^3 g\right)-b^2 c^3 \left(16 a^3 m+14 a^2 b l+6 a b^2 k+b^3 j\right)+c^4 \left(2 a^4 m+7 a^3 b l+9 a^2 b^2 k+5 a b^3 j+b^4 h\right)-b^6 c (8 a m+b l)-c^7 (2 a f+b e)+b^8 m+2 c^8 d\right)}{c^8 \sqrt{b^2-4 a c}}+\frac{x^4 \left(-c^2 (a l+b k)+b c (2 a m+b l)+b^3 (-m)+c^3 j\right)}{4 c^4}+\frac{x^5 \left(-c (a m+b l)+b^2 m+c^2 k\right)}{5 c^3}+\frac{x^6 (c l-b m)}{6 c^2}+\frac{m x^7}{7 c}",1,"(420*c*(c^6*f - c^5*(b*g + a*h) + c^4*(b^2*h + 2*a*b*j + a^2*k) + b^6*m - b^4*c*(b*l + 5*a*m) + b^2*c^2*(b^2*k + 4*a*b*l + 6*a^2*m) - c^3*(b^3*j + 3*a*b^2*k + 3*a^2*b*l + a^3*m))*x + 210*c^2*(c^5*g - c^4*(b*h + a*j) + c^3*(b^2*j + 2*a*b*k + a^2*l) - b^5*m + b^3*c*(b*l + 4*a*m) - b*c^2*(b^2*k + 3*a*b*l + 3*a^2*m))*x^2 + 140*c^3*(c^4*h - c^3*(b*j + a*k) + b^4*m - b^2*c*(b*l + 3*a*m) + c^2*(b^2*k + 2*a*b*l + a^2*m))*x^3 + 105*c^4*(c^3*j - c^2*(b*k + a*l) - b^3*m + b*c*(b*l + 2*a*m))*x^4 + 84*c^5*(c^2*k + b^2*m - c*(b*l + a*m))*x^5 + 70*c^6*(c*l - b*m)*x^6 + 60*c^7*m*x^7 + (420*(2*c^8*d - c^7*(b*e + 2*a*f) + c^6*(b^2*f + 3*a*b*g + 2*a^2*h) - c^5*(b^3*g + 4*a*b^2*h + 5*a^2*b*j + 2*a^3*k) + b^8*m - b^6*c*(b*l + 8*a*m) + b^4*c^2*(b^2*k + 7*a*b*l + 20*a^2*m) - b^2*c^3*(b^3*j + 6*a*b^2*k + 14*a^2*b*l + 16*a^3*m) + c^4*(b^4*h + 5*a*b^3*j + 9*a^2*b^2*k + 7*a^3*b*l + 2*a^4*m))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 210*(c^7*e - c^6*(b*f + a*g) + c^5*(b^2*g + 2*a*b*h + a^2*j) - c^4*(b^3*h + 3*a*b^2*j + 3*a^2*b*k + a^3*l) - b^7*m + b^5*c*(b*l + 6*a*m) - b^3*c^2*(b^2*k + 5*a*b*l + 10*a^2*m) + b*c^3*(b^3*j + 4*a*b^2*k + 6*a^2*b*l + 4*a^3*m))*Log[a + x*(b + c*x)])/(420*c^8)","A",1
374,1,85,208,0.3051914,"\int \left(1+4 x-7 x^2\right)^3 \left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2} \, dx","Integrate[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]","\frac{-5 \sqrt{5 x^2+2 x+3} \left(67528125000 x^9+248031875000 x^8-497593468750 x^7-34674656250 x^6+225922362500 x^5+56757413000 x^4+17642392275 x^3-78839046795 x^2-57768004650 x+93436408944\right)-68055105006 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{9843750000}","\frac{98060877 \left(5 x^2+2 x+3\right)^{3/2} x^2}{4375000}+\frac{1045360143 \left(5 x^2+2 x+3\right)^{3/2} x}{43750000}-\frac{1968340667 \left(5 x^2+2 x+3\right)^{3/2}}{131250000}-\frac{77159983 (5 x+1) \sqrt{5 x^2+2 x+3}}{31250000}-\frac{343}{50} \left(5 x^2+2 x+3\right)^{3/2} x^7-\frac{50519 \left(5 x^2+2 x+3\right)^{3/2} x^6}{2250}+\frac{190939 \left(5 x^2+2 x+3\right)^{3/2} x^5}{3000}-\frac{888751 \left(5 x^2+2 x+3\right)^{3/2} x^4}{105000}-\frac{90960857 \left(5 x^2+2 x+3\right)^{3/2} x^3}{1575000}-\frac{540119881 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{15625000 \sqrt{5}}",1,"(-5*Sqrt[3 + 2*x + 5*x^2]*(93436408944 - 57768004650*x - 78839046795*x^2 + 17642392275*x^3 + 56757413000*x^4 + 225922362500*x^5 - 34674656250*x^6 - 497593468750*x^7 + 248031875000*x^8 + 67528125000*x^9) - 68055105006*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/9843750000","A",1
375,1,75,166,0.1892783,"\int \left(1+4 x-7 x^2\right)^2 \left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2} \, dx","Integrate[(1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]","\frac{5 \sqrt{5 x^2+2 x+3} \left(22968750 x^7+101906250 x^6-107112500 x^5-65693000 x^4+15583725 x^3+23531995 x^2+44333650 x-4588584\right)-105912366 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{18750000}","-\frac{77509 \left(5 x^2+2 x+3\right)^{3/2} x^2}{25000}+\frac{1781669 \left(5 x^2+2 x+3\right)^{3/2} x}{250000}+\frac{198439 \left(5 x^2+2 x+3\right)^{3/2}}{750000}-\frac{2521723 (5 x+1) \sqrt{5 x^2+2 x+3}}{1250000}+\frac{49}{40} \left(5 x^2+2 x+3\right)^{3/2} x^5+\frac{989}{200} \left(5 x^2+2 x+3\right)^{3/2} x^4-\frac{25277 \left(5 x^2+2 x+3\right)^{3/2} x^3}{3000}-\frac{17652061 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{625000 \sqrt{5}}",1,"(5*Sqrt[3 + 2*x + 5*x^2]*(-4588584 + 44333650*x + 23531995*x^2 + 15583725*x^3 - 65693000*x^4 - 107112500*x^5 + 101906250*x^6 + 22968750*x^7) - 105912366*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/18750000","A",1
376,1,65,124,0.0998852,"\int \left(1+4 x-7 x^2\right) \left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2} \, dx","Integrate[(1 + 4*x - 7*x^2)*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]","\frac{5 \sqrt{5 x^2+2 x+3} \left(-43750 x^5-234250 x^4+48225 x^3+129895 x^2+105400 x+103386\right)-194586 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{187500}","-\frac{289}{250} \left(5 x^2+2 x+3\right)^{3/2} x^2+\frac{2149 \left(5 x^2+2 x+3\right)^{3/2} x}{2500}+\frac{7819 \left(5 x^2+2 x+3\right)^{3/2}}{7500}-\frac{4633 (5 x+1) \sqrt{5 x^2+2 x+3}}{12500}-\frac{7}{30} \left(5 x^2+2 x+3\right)^{3/2} x^3-\frac{32431 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{6250 \sqrt{5}}",1,"(5*Sqrt[3 + 2*x + 5*x^2]*(103386 + 105400*x + 129895*x^2 + 48225*x^3 - 234250*x^4 - 43750*x^5) - 194586*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/187500","A",1
377,1,189,187,0.8945771,"\int \frac{\left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx","Integrate[((2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2),x]","\frac{-385 \sqrt{5 x^2+2 x+3} (35 x+397)-75 \sqrt{250-34 \sqrt{11}} \left(61 \sqrt{11}-143\right) \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)+75 \sqrt{250+34 \sqrt{11}} \left(143+61 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)-181126 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{188650}","-\frac{1}{490} \sqrt{5 x^2+2 x+3} (35 x+397)-\frac{3}{343} \sqrt{\frac{1}{11} \left(497041-146555 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)+\frac{3}{343} \sqrt{\frac{1}{11} \left(497041+146555 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)-\frac{8233 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1715 \sqrt{5}}",1,"(-385*(397 + 35*x)*Sqrt[3 + 2*x + 5*x^2] - 181126*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 75*Sqrt[250 - 34*Sqrt[11]]*(-143 + 61*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] + 75*Sqrt[250 + 34*Sqrt[11]]*(143 + 61*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])])/188650","A",1
378,1,354,199,1.2763264,"\int \frac{\left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2}}{\left(1+4 x-7 x^2\right)^2} \, dx","Integrate[((2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2,x]","\frac{\frac{56364 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{2772 \sqrt{5 x^2+2 x+3}}{-7 x^2+4 x+1}+22892 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+2398 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+2 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left(11446 \sqrt{11}-1199\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{5 \sqrt{11} x-17 x+\sqrt{11}-23}\right)-22892 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left(-7 x+\sqrt{11}+2\right)-2398 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left(-7 x+\sqrt{11}+2\right)+968 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{47432}","\frac{3 \sqrt{5 x^2+2 x+3} (61 x+3)}{154 \left(-7 x^2+4 x+1\right)}-\frac{\sqrt{\frac{325022311+39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{2156}+\frac{\sqrt{\frac{325022311-39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{2156}+\frac{1}{49} \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)",1,"((2772*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (56364*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + 968*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] + 2*Sqrt[2/(125 - 17*Sqrt[11])]*(-1199 + 11446*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] - 17*x + 5*Sqrt[11]*x)] - 2398*Sqrt[2/(125 + 17*Sqrt[11])]*Log[2 + Sqrt[11] - 7*x] - 22892*Sqrt[22/(125 + 17*Sqrt[11])]*Log[2 + Sqrt[11] - 7*x] + 2398*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 22892*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/47432","A",1
379,1,334,213,1.3965298,"\int \frac{\left(2+5 x+x^2\right) \sqrt{3+2 x+5 x^2}}{\left(1+4 x-7 x^2\right)^3} \, dx","Integrate[((2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^3,x]","\frac{-\sqrt{\frac{22}{125-17 \sqrt{11}}} \left(126542 \sqrt{11}-1740003\right) \log \left(49 x^2+14 \left(\sqrt{11}-2\right) x-4 \sqrt{11}+15\right)+2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(1740003+126542 \sqrt{11}\right) \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)-2 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(126542 \sqrt{11}-1740003\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-\frac{44 \sqrt{5 x^2+2 x+3} \left(813113 x^3-737577 x^2-106279 x+31807\right)}{\left(-7 x^2+4 x+1\right)^2}-2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(1740003+126542 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)+\sqrt{\frac{22}{125-17 \sqrt{11}}} \left(126542 \sqrt{11}-1740003\right) \log \left(\left(7 x+\sqrt{11}-2\right)^2\right)}{10818368}","-\frac{\sqrt{5 x^2+2 x+3} (272941-813113 x)}{1721104 \left(-7 x^2+4 x+1\right)}+\frac{3 (61 x+3) \sqrt{5 x^2+2 x+3}}{308 \left(-7 x^2+4 x+1\right)^2}-\frac{\sqrt{\frac{6492253020949-11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{491744}+\frac{\sqrt{\frac{6492253020949+11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{491744}",1,"((-44*Sqrt[3 + 2*x + 5*x^2]*(31807 - 106279*x - 737577*x^2 + 813113*x^3))/(1 + 4*x - 7*x^2)^2 - 2*Sqrt[22/(125 - 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 2*Sqrt[22/(125 + 17*Sqrt[11])]*(1740003 + 126542*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + Sqrt[22/(125 - 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] - Sqrt[22/(125 - 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] + 2*Sqrt[22/(125 + 17*Sqrt[11])]*(1740003 + 126542*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/10818368","A",1
380,1,95,231,0.4177575,"\int \left(1+4 x-7 x^2\right)^3 \left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2} \, dx","Integrate[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2),x]","\frac{-4653589321458 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)-5 \sqrt{5 x^2+2 x+3} \left(30950390625000 x^{11}+125007421875000 x^{10}-148393743750000 x^9-30505457500000 x^8-72918247281250 x^7+52106830406250 x^6+85130334087500 x^5-2573089891000 x^4-19041688239675 x^3-15865844408685 x^2-6352777129950 x+10506617068392\right)}{1082812500000}","\frac{2173004363 \left(5 x^2+2 x+3\right)^{5/2} x^2}{173250000}+\frac{837379699 \left(5 x^2+2 x+3\right)^{5/2} x}{72187500}-\frac{6133820867 \left(5 x^2+2 x+3\right)^{5/2}}{1203125000}-\frac{22840599 (5 x+1) \left(5 x^2+2 x+3\right)^{3/2}}{62500000}-\frac{479652579 (5 x+1) \sqrt{5 x^2+2 x+3}}{312500000}-\frac{343}{60} \left(5 x^2+2 x+3\right)^{5/2} x^7-\frac{61103 \left(5 x^2+2 x+3\right)^{5/2} x^6}{3300}+\frac{1031177 \left(5 x^2+2 x+3\right)^{5/2} x^5}{20625}-\frac{796559 \left(5 x^2+2 x+3\right)^{5/2} x^4}{123750}-\frac{190236913 \left(5 x^2+2 x+3\right)^{5/2} x^3}{4950000}-\frac{3357568053 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{156250000 \sqrt{5}}",1,"(-5*Sqrt[3 + 2*x + 5*x^2]*(10506617068392 - 6352777129950*x - 15865844408685*x^2 - 19041688239675*x^3 - 2573089891000*x^4 + 85130334087500*x^5 + 52106830406250*x^6 - 72918247281250*x^7 - 30505457500000*x^8 - 148393743750000*x^9 + 125007421875000*x^10 + 30950390625000*x^11) - 4653589321458*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/1082812500000","A",1
381,1,85,189,0.2509524,"\int \left(1+4 x-7 x^2\right)^2 \left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2} \, dx","Integrate[(1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2),x]","\frac{5 \sqrt{5 x^2+2 x+3} \left(3215625000 x^9+15281875000 x^8-5561281250 x^7-4105593750 x^6-12554262500 x^5-3227597000 x^4+5959365525 x^3+3721040355 x^2+2291675850 x-249003936\right)-4263523614 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{656250000}","-\frac{219271 \left(5 x^2+2 x+3\right)^{5/2} x^2}{105000}+\frac{86721 \left(5 x^2+2 x+3\right)^{5/2} x}{21875}+\frac{505667 \left(5 x^2+2 x+3\right)^{5/2}}{2187500}-\frac{690561 (5 x+1) \left(5 x^2+2 x+3\right)^{3/2}}{1250000}-\frac{14501781 (5 x+1) \sqrt{5 x^2+2 x+3}}{6250000}+\frac{49}{50} \left(5 x^2+2 x+3\right)^{5/2} x^5+\frac{581}{150} \left(5 x^2+2 x+3\right)^{5/2} x^4-\frac{18379 \left(5 x^2+2 x+3\right)^{5/2} x^3}{3000}-\frac{101512467 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{3125000 \sqrt{5}}",1,"(5*Sqrt[3 + 2*x + 5*x^2]*(-249003936 + 2291675850*x + 3721040355*x^2 + 5959365525*x^3 - 3227597000*x^4 - 12554262500*x^5 - 4105593750*x^6 - 5561281250*x^7 + 15281875000*x^8 + 3215625000*x^9) - 4263523614*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/656250000","A",1
382,1,75,147,0.1477642,"\int \left(1+4 x-7 x^2\right) \left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2} \, dx","Integrate[(1 + 4*x - 7*x^2)*(2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2),x]","\frac{-5 \sqrt{5 x^2+2 x+3} \left(22968750 x^7+127406250 x^6+48237500 x^5+28373000 x^4-78608475 x^3-86464445 x^2-36695150 x-22275576\right)-37861026 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{26250000}","-\frac{1163 \left(5 x^2+2 x+3\right)^{5/2} x^2}{1400}+\frac{2809 \left(5 x^2+2 x+3\right)^{5/2} x}{5250}+\frac{149509 \left(5 x^2+2 x+3\right)^{5/2}}{262500}-\frac{18397 (5 x+1) \left(5 x^2+2 x+3\right)^{3/2}}{150000}-\frac{128779 (5 x+1) \sqrt{5 x^2+2 x+3}}{250000}-\frac{7}{40} \left(5 x^2+2 x+3\right)^{5/2} x^3-\frac{901453 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{125000 \sqrt{5}}",1,"(-5*Sqrt[3 + 2*x + 5*x^2]*(-22275576 - 36695150*x - 86464445*x^2 - 78608475*x^3 + 28373000*x^4 + 48237500*x^5 + 127406250*x^6 + 22968750*x^7) - 37861026*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/26250000","A",1
383,1,202,210,0.9379191,"\int \frac{\left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2}}{1+4 x-7 x^2} \, dx","Integrate[((2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2))/(1 + 4*x - 7*x^2),x]","\frac{-5 \left(600 \sqrt{1572625-425459 \sqrt{11}} \left(61 \sqrt{11}-143\right) \tanh ^{-1}\left(\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)-600 \left(143+61 \sqrt{11}\right) \sqrt{1572625+425459 \sqrt{11}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)+77 \sqrt{5 x^2+2 x+3} \left(42875 x^3+344225 x^2+744870 x+1911108\right)\right)-757365114 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{92438500}","-\frac{1}{980} (35 x+267) \left(5 x^2+2 x+3\right)^{3/2}-\frac{3 (196105 x+571621) \sqrt{5 x^2+2 x+3}}{240100}-\frac{6 \sqrt{\frac{2}{11} \left(8098902607-2434122235 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{16807}+\frac{6 \sqrt{\frac{2}{11} \left(8098902607+2434122235 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{16807}-\frac{34425687 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{840350 \sqrt{5}}",1,"(-757365114*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 5*(77*Sqrt[3 + 2*x + 5*x^2]*(1911108 + 744870*x + 344225*x^2 + 42875*x^3) + 600*Sqrt[1572625 - 425459*Sqrt[11]]*(-143 + 61*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + 17*x - 5*Sqrt[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] - 600*(143 + 61*Sqrt[11])*Sqrt[1572625 + 425459*Sqrt[11]]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])]))/92438500","A",0
384,1,354,222,1.8332044,"\int \frac{\left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2}}{\left(1+4 x-7 x^2\right)^2} \, dx","Integrate[((2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2))/(1 + 4*x - 7*x^2)^2,x]","\frac{5 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(743879 \sqrt{11}-1701489\right) \log \left(49 x^2+14 \left(\sqrt{11}-2\right) x-4 \sqrt{11}+15\right)-10 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(1701489+743879 \sqrt{11}\right) \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+10 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(743879 \sqrt{11}-1701489\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)+\frac{770 \sqrt{5 x^2+2 x+3} \left(2695 x^3+34265 x^2-81181 x-12975\right)}{7 x^2-4 x-1}+10 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(1701489+743879 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)-5 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(743879 \sqrt{11}-1701489\right) \log \left(\left(7 x+\sqrt{11}-2\right)^2\right)+8078444 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{5810420}","\frac{3 (61 x+3) \left(5 x^2+2 x+3\right)^{3/2}}{154 \left(-7 x^2+4 x+1\right)}+\frac{(3395 x+5826) \sqrt{5 x^2+2 x+3}}{3773}-\frac{\sqrt{\frac{1}{22} \left(52175400311-13155376531 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{26411}-\frac{\sqrt{\frac{1}{22} \left(52175400311+13155376531 \sqrt{11}\right)} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{26411}+\frac{16691 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{2401 \sqrt{5}}",1,"((770*Sqrt[3 + 2*x + 5*x^2]*(-12975 - 81181*x + 34265*x^2 + 2695*x^3))/(-1 - 4*x + 7*x^2) + 8078444*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] + 10*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] + 10*Sqrt[22/(125 + 17*Sqrt[11])]*(1701489 + 743879*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] - 5*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] + 5*Sqrt[22/(125 - 17*Sqrt[11])]*(-1701489 + 743879*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] - 10*Sqrt[22/(125 + 17*Sqrt[11])]*(1701489 + 743879*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/5810420","A",1
385,1,376,234,2.2388405,"\int \frac{\left(2+5 x+x^2\right) \left(3+2 x+5 x^2\right)^{3/2}}{\left(1+4 x-7 x^2\right)^3} \, dx","Integrate[((2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2))/(1 + 4*x - 7*x^2)^3,x]","\frac{\frac{88 \sqrt{5 x^2+2 x+3} (138372-189161 x)}{7 x^2-4 x-1}+\frac{11616 (5028 x+655) \sqrt{5 x^2+2 x+3}}{\left(-7 x^2+4 x+1\right)^2}-\sqrt{\frac{22}{125-17 \sqrt{11}}} \left(674221 \sqrt{11}-7706073\right) \log \left(49 x^2+14 \left(\sqrt{11}-2\right) x-4 \sqrt{11}+15\right)+2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(7706073+674221 \sqrt{11}\right) \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)-2 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(674221 \sqrt{11}-7706073\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(7706073+674221 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)+\sqrt{\frac{22}{125-17 \sqrt{11}}} \left(674221 \sqrt{11}-7706073\right) \log \left(\left(7 x+\sqrt{11}-2\right)^2\right)-212960 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{14609056}","\frac{3 (61 x+3) \left(5 x^2+2 x+3\right)^{3/2}}{308 \left(-7 x^2+4 x+1\right)^2}-\frac{(9495-37088 x) \sqrt{5 x^2+2 x+3}}{23716 \left(-7 x^2+4 x+1\right)}-\frac{\sqrt{\frac{62294197250171-2085440742055 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{332024}+\frac{\sqrt{\frac{62294197250171+2085440742055 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{332024}-\frac{5}{343} \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)",1,"((11616*(655 + 5028*x)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (88*(138372 - 189161*x)*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) - 212960*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 2*Sqrt[22/(125 - 17*Sqrt[11])]*(-7706073 + 674221*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 2*Sqrt[22/(125 + 17*Sqrt[11])]*(7706073 + 674221*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + Sqrt[22/(125 - 17*Sqrt[11])]*(-7706073 + 674221*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] - Sqrt[22/(125 - 17*Sqrt[11])]*(-7706073 + 674221*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] + 2*Sqrt[22/(125 + 17*Sqrt[11])]*(7706073 + 674221*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/14609056","A",1
386,1,75,185,0.2520288,"\int \frac{\left(1+4 x-7 x^2\right)^3 \left(2+5 x+x^2\right)}{\sqrt{3+2 x+5 x^2}} \, dx","Integrate[((1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2))/Sqrt[3 + 2*x + 5*x^2],x]","\frac{-5 \sqrt{5 x^2+2 x+3} \left(32156250 x^7+106968750 x^6-326987500 x^5+47807000 x^4+387039975 x^3-203614255 x^2-289653850 x+396379416\right)-465082134 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{18750000}","\frac{40722851 \sqrt{5 x^2+2 x+3} x^2}{750000}+\frac{5793077 \sqrt{5 x^2+2 x+3} x}{75000}-\frac{16515809 \sqrt{5 x^2+2 x+3}}{156250}-\frac{343}{40} \sqrt{5 x^2+2 x+3} x^7-\frac{1141}{40} \sqrt{5 x^2+2 x+3} x^6+\frac{26159}{300} \sqrt{5 x^2+2 x+3} x^5-\frac{47807 \sqrt{5 x^2+2 x+3} x^4}{3750}-\frac{5160533 \sqrt{5 x^2+2 x+3} x^3}{50000}-\frac{77513689 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{625000 \sqrt{5}}",1,"(-5*Sqrt[3 + 2*x + 5*x^2]*(396379416 - 289653850*x - 203614255*x^2 + 387039975*x^3 + 47807000*x^4 - 326987500*x^5 + 106968750*x^6 + 32156250*x^7) - 465082134*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/18750000","A",1
387,1,65,143,0.1426613,"\int \frac{\left(1+4 x-7 x^2\right)^2 \left(2+5 x+x^2\right)}{\sqrt{3+2 x+5 x^2}} \, dx","Integrate[((1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2))/Sqrt[3 + 2*x + 5*x^2],x]","\frac{5 \sqrt{5 x^2+2 x+3} \left(306250 x^5+1282750 x^4-2494425 x^3-1037135 x^2+3607300 x-132318\right)-10314582 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{937500}","-\frac{207427 \sqrt{5 x^2+2 x+3} x^2}{37500}+\frac{36073 \sqrt{5 x^2+2 x+3} x}{1875}-\frac{22053 \sqrt{5 x^2+2 x+3}}{31250}+\frac{49}{30} \sqrt{5 x^2+2 x+3} x^5+\frac{5131}{750} \sqrt{5 x^2+2 x+3} x^4-\frac{33259 \sqrt{5 x^2+2 x+3} x^3}{2500}-\frac{1719097 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{31250 \sqrt{5}}",1,"(5*Sqrt[3 + 2*x + 5*x^2]*(-132318 + 3607300*x - 1037135*x^2 - 2494425*x^3 + 1282750*x^4 + 306250*x^5) - 10314582*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/937500","A",1
388,1,55,101,0.0804577,"\int \frac{\left(1+4 x-7 x^2\right) \left(2+5 x+x^2\right)}{\sqrt{3+2 x+5 x^2}} \, dx","Integrate[((1 + 4*x - 7*x^2)*(2 + 5*x + x^2))/Sqrt[3 + 2*x + 5*x^2],x]","\frac{-5 \sqrt{5 x^2+2 x+3} \left(525 x^3+2855 x^2-2950 x-5556\right)-11406 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{7500}","-\frac{571}{300} \sqrt{5 x^2+2 x+3} x^2+\frac{59}{30} \sqrt{5 x^2+2 x+3} x+\frac{463}{125} \sqrt{5 x^2+2 x+3}-\frac{7}{20} \sqrt{5 x^2+2 x+3} x^3-\frac{1901 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{250 \sqrt{5}}",1,"(-5*Sqrt[3 + 2*x + 5*x^2]*(-5556 - 2950*x + 2855*x^2 + 525*x^3) - 11406*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/7500","A",1
389,1,157,164,0.4667355,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right) \sqrt{3+2 x+5 x^2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]),x]","-\frac{3 \left(\sqrt{4091-1055 \sqrt{11}} \tanh ^{-1}\left(\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)-\sqrt{4091+1055 \sqrt{11}} \tanh ^{-1}\left(\frac{5 \sqrt{11} x+17 x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)\right)}{14 \sqrt{2794}}-\frac{\sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{7 \sqrt{5}}","-\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)-\frac{\sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{7 \sqrt{5}}",1,"-1/7*ArcSinh[(1 + 5*x)/Sqrt[14]]/Sqrt[5] - (3*(Sqrt[4091 - 1055*Sqrt[11]]*ArcTanh[(23 - Sqrt[11] + 17*x - 5*Sqrt[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] - Sqrt[4091 + 1055*Sqrt[11]]*ArcTanh[(23 + Sqrt[11] + 17*x + 5*Sqrt[11]*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])]))/(14*Sqrt[2794])","A",1
390,1,313,178,0.9989288,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right)^2 \sqrt{3+2 x+5 x^2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*Sqrt[3 + 2*x + 5*x^2]),x]","\frac{\frac{48972 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{5280 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+53005 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+40623 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+\sqrt{\frac{2}{125-17 \sqrt{11}}} \left(53005 \sqrt{11}-40623\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-\sqrt{\frac{2}{125+17 \sqrt{11}}} \left(40623+53005 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)}{245872}","-\frac{3 \sqrt{5 x^2+2 x+3} (40-371 x)}{5588 \left(-7 x^2+4 x+1\right)}-\frac{\sqrt{\frac{3027900955+14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{11176}+\frac{\sqrt{\frac{3027900955-14035681 \sqrt{11}}{2794}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{11176}",1,"((48972*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (5280*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) + Sqrt[2/(125 - 17*Sqrt[11])]*(-40623 + 53005*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - Sqrt[2/(125 + 17*Sqrt[11])]*(40623 + 53005*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 40623*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 53005*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/245872","A",1
391,1,371,227,1.1748452,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right)^3 \sqrt{3+2 x+5 x^2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]","\frac{\frac{732651920 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{547311072 \sqrt{5 x^2+2 x+3} x}{\left(-7 x^2+4 x+1\right)^2}-\frac{59009280 \sqrt{5 x^2+2 x+3}}{\left(-7 x^2+4 x+1\right)^2}+\frac{252417704 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+551183234 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+390963650 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+14 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left(39370231 \sqrt{11}-27925975\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-14 \sqrt{\frac{2}{125+17 \sqrt{11}}} \left(27925975+39370231 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)}{5495730944}","-\frac{7 \sqrt{5 x^2+2 x+3} (409769-1189370 x)}{62451488 \left(-7 x^2+4 x+1\right)}-\frac{3 (40-371 x) \sqrt{5 x^2+2 x+3}}{11176 \left(-7 x^2+4 x+1\right)^2}-\frac{7 \left(39370231-2538725 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{124902976 \sqrt{22 \left(125-17 \sqrt{11}\right)}}+\frac{7 \left(39370231+2538725 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{124902976 \sqrt{22 \left(125+17 \sqrt{11}\right)}}",1,"((-59009280*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (547311072*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (732651920*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (252417704*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) + 14*Sqrt[2/(125 - 17*Sqrt[11])]*(-27925975 + 39370231*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[2/(125 + 17*Sqrt[11])]*(27925975 + 39370231*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 390963650*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 551183234*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/5495730944","A",1
392,1,75,166,0.4330244,"\int \frac{\left(1+4 x-7 x^2\right)^3 \left(2+5 x+x^2\right)}{\left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[((1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2))/(3 + 2*x + 5*x^2)^(3/2),x]","\frac{2102001594 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)-\frac{5 \left(75031250 x^7+256821250 x^6-897612625 x^5+174819575 x^4+1795638985 x^3-2135143465 x^2+1045703388 x-3155769618\right)}{\sqrt{5 x^2+2 x+3}}}{32812500}","-\frac{2583293 \sqrt{5 x^2+2 x+3} x^2}{187500}-\frac{3192602 \sqrt{5 x^2+2 x+3} x}{46875}+\frac{15715799 \sqrt{5 x^2+2 x+3}}{156250}+\frac{16 (6122807-5338217 x)}{546875 \sqrt{5 x^2+2 x+3}}-\frac{343}{150} \sqrt{5 x^2+2 x+3} x^5-\frac{25921 \sqrt{5 x^2+2 x+3} x^4}{3750}+\frac{393659 \sqrt{5 x^2+2 x+3} x^3}{12500}+\frac{50047657 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{156250 \sqrt{5}}",1,"((-5*(-3155769618 + 1045703388*x - 2135143465*x^2 + 1795638985*x^3 + 174819575*x^4 - 897612625*x^5 + 256821250*x^6 + 75031250*x^7))/Sqrt[3 + 2*x + 5*x^2] + 2102001594*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/32812500","A",1
393,1,65,124,0.2657852,"\int \frac{\left(1+4 x-7 x^2\right)^2 \left(2+5 x+x^2\right)}{\left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[((1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2))/(3 + 2*x + 5*x^2)^(3/2),x]","\frac{\frac{5 \left(42875 x^5+194775 x^4-515655 x^3-280805 x^2-1298674 x-168536\right)}{\sqrt{5 x^2+2 x+3}}+1254162 \sqrt{5} \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{87500}","\frac{203}{100} \sqrt{5 x^2+2 x+3} x^2-\frac{8749 \sqrt{5 x^2+2 x+3} x}{1250}-\frac{5086 \sqrt{5 x^2+2 x+3}}{3125}-\frac{8 (136602 x+12983)}{21875 \sqrt{5 x^2+2 x+3}}+\frac{49}{100} \sqrt{5 x^2+2 x+3} x^3+\frac{89583 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{1250 \sqrt{5}}",1,"((5*(-168536 - 1298674*x - 280805*x^2 - 515655*x^3 + 194775*x^4 + 42875*x^5))/Sqrt[3 + 2*x + 5*x^2] + 1254162*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/87500","A",1
394,1,55,82,0.1496056,"\int \frac{\left(1+4 x-7 x^2\right) \left(2+5 x+x^2\right)}{\left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[((1 + 4*x - 7*x^2)*(2 + 5*x + x^2))/(3 + 2*x + 5*x^2)^(3/2),x]","\frac{149 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{25 \sqrt{5}}-\frac{245 x^3+1925 x^2+2837 x+2953}{350 \sqrt{5 x^2+2 x+3}}","-\frac{7}{50} \sqrt{5 x^2+2 x+3} x-\frac{261}{250} \sqrt{5 x^2+2 x+3}-\frac{2 (2449 x+2321)}{875 \sqrt{5 x^2+2 x+3}}+\frac{149 \sinh ^{-1}\left(\frac{5 x+1}{\sqrt{14}}\right)}{25 \sqrt{5}}",1,"-1/350*(2953 + 2837*x + 1925*x^2 + 245*x^3)/Sqrt[3 + 2*x + 5*x^2] + (149*ArcSinh[(1 + 5*x)/Sqrt[14]])/(25*Sqrt[5])","A",1
395,1,174,166,1.1347887,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right) \left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)*(3 + 2*x + 5*x^2)^(3/2)),x]","\frac{\frac{2794 (605 x-131)}{\sqrt{5 x^2+2 x+3}}-21 \sqrt{127 \left(125+17 \sqrt{11}\right)} \left(53 \sqrt{11}-66\right) \tanh ^{-1}\left(\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)+21 \sqrt{127 \left(125-17 \sqrt{11}\right)} \left(66+53 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right)}{9935464}","-\frac{131-605 x}{3556 \sqrt{5 x^2+2 x+3}}-\frac{3 \sqrt{\frac{281693-25015 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{1016}+\frac{3 \sqrt{\frac{281693+25015 \sqrt{11}}{1397}} \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{1016}",1,"((2794*(-131 + 605*x))/Sqrt[3 + 2*x + 5*x^2] - 21*Sqrt[127*(125 + 17*Sqrt[11])]*(-66 + 53*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + 17*x - 5*Sqrt[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] + 21*Sqrt[127*(125 - 17*Sqrt[11])]*(66 + 53*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])])/9935464","A",1
396,1,351,215,1.1384285,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right)^2 \left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*(3 + 2*x + 5*x^2)^(3/2)),x]","\frac{\frac{5084772 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{24422640 x}{7 \sqrt{5 x^2+2 x+3}}+\frac{12968296}{7 \sqrt{5 x^2+2 x+3}}+\frac{1672044 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+7581602 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+792176 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)+14 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left(541543 \sqrt{11}-56584\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-14 \sqrt{\frac{2}{125+17 \sqrt{11}}} \left(56584+541543 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)}{124902976}","-\frac{3 (40-371 x)}{5588 \left(-7 x^2+4 x+1\right) \sqrt{5 x^2+2 x+3}}-\frac{22755 x+76567}{19870928 \sqrt{5 x^2+2 x+3}}-\frac{7 \left(541543-5144 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{2838704 \sqrt{22 \left(125-17 \sqrt{11}\right)}}+\frac{7 \left(541543+5144 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{2838704 \sqrt{22 \left(125+17 \sqrt{11}\right)}}",1,"(12968296/(7*Sqrt[3 + 2*x + 5*x^2]) + (24422640*x)/(7*Sqrt[3 + 2*x + 5*x^2]) + (5084772*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (1672044*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) + 14*Sqrt[2/(125 - 17*Sqrt[11])]*(-56584 + 541543*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[2/(125 + 17*Sqrt[11])]*(56584 + 541543*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 792176*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 7581602*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/124902976","A",1
397,1,381,250,1.5855471,"\int \frac{2+5 x+x^2}{\left(1+4 x-7 x^2\right)^3 \left(3+2 x+5 x^2\right)^{3/2}} \, dx","Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*(3 + 2*x + 5*x^2)^(3/2)),x]","\frac{\frac{44 \sqrt{5 x^2+2 x+3} (507770113-1167248019 x)}{7 x^2-4 x-1}+\frac{737616 (38521 x-12667) \sqrt{5 x^2+2 x+3}}{\left(-7 x^2+4 x+1\right)^2}+\frac{21296 (501205 x+1702037)}{7 \sqrt{5 x^2+2 x+3}}-7 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(84865895 \sqrt{11}-2792860024\right) \log \left(49 x^2+14 \left(\sqrt{11}-2\right) x-4 \sqrt{11}+15\right)+14 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(2792860024+84865895 \sqrt{11}\right) \log \left(\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left(55+17 \sqrt{11}\right) x+23 \sqrt{11}+11\right)-14 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(84865895 \sqrt{11}-2792860024\right) \tanh ^{-1}\left(\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left(5 \sqrt{11}-17\right) x+\sqrt{11}-23}\right)-14 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left(2792860024+84865895 \sqrt{11}\right) \log \left(-7 x+\sqrt{11}+2\right)+7 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left(84865895 \sqrt{11}-2792860024\right) \log \left(\left(7 x+\sqrt{11}-2\right)^2\right)}{1395915659776}","-\frac{2701733-9148874 x}{62451488 \left(-7 x^2+4 x+1\right) \sqrt{5 x^2+2 x+3}}-\frac{5 (1118731375 x+461370781)}{222077491328 \sqrt{5 x^2+2 x+3}}-\frac{3 (40-371 x)}{11176 \left(-7 x^2+4 x+1\right)^2 \sqrt{5 x^2+2 x+3}}-\frac{7 \left(2792860024-84865895 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17-5 \sqrt{11}\right) x-\sqrt{11}+23}{\sqrt{2 \left(125-17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{31725355904 \sqrt{22 \left(125-17 \sqrt{11}\right)}}+\frac{7 \left(2792860024+84865895 \sqrt{11}\right) \tanh ^{-1}\left(\frac{\left(17+5 \sqrt{11}\right) x+\sqrt{11}+23}{\sqrt{2 \left(125+17 \sqrt{11}\right)} \sqrt{5 x^2+2 x+3}}\right)}{31725355904 \sqrt{22 \left(125+17 \sqrt{11}\right)}}",1,"((21296*(1702037 + 501205*x))/(7*Sqrt[3 + 2*x + 5*x^2]) + (737616*(-12667 + 38521*x)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (44*(507770113 - 1167248019*x)*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) - 14*Sqrt[22/(125 - 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[22/(125 + 17*Sqrt[11])]*(2792860024 + 84865895*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 7*Sqrt[22/(125 - 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] - 7*Sqrt[22/(125 - 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] + 14*Sqrt[22/(125 + 17*Sqrt[11])]*(2792860024 + 84865895*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/1395915659776","A",1
398,1,242,166,0.4013798,"\int \left(a+c x^2\right)^p \left(A+C x^2\right) \left(d+f x^2\right)^q \, dx","Integrate[(a + c*x^2)^p*(A + C*x^2)*(d + f*x^2)^q,x]","\frac{1}{3} x \left(a+c x^2\right)^p \left(d+f x^2\right)^q \left(\frac{9 a A d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}{2 x^2 \left(c d p F_1\left(\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+a f q F_1\left(\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)+3 a d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}+C x^2 \left(\frac{c x^2}{a}+1\right)^{-p} \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)","A x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d+f x^2\right)^q \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+\frac{1}{3} C x^3 \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d+f x^2\right)^q \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)",1,"(x*(a + c*x^2)^p*(d + f*x^2)^q*((9*a*A*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)] + 2*x^2*(c*d*p*AppellF1[3/2, 1 - p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)] + a*f*q*AppellF1[3/2, -p, 1 - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])) + (C*x^2*AppellF1[3/2, -p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q)))/3","A",0
399,1,236,167,0.3179336,"\int (A+B x) \left(a+c x^2\right)^p \left(d+f x^2\right)^q \, dx","Integrate[(A + B*x)*(a + c*x^2)^p*(d + f*x^2)^q,x]","\frac{1}{2} x \left(a+c x^2\right)^p \left(d+f x^2\right)^q \left(\frac{6 a A d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}{2 x^2 \left(c d p F_1\left(\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+a f q F_1\left(\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)+3 a d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}+B x \left(\frac{c x^2}{a}+1\right)^{-p} \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(1;-p,-q;2;-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)","A x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d+f x^2\right)^q \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+\frac{B \left(a+c x^2\right)^{p+1} \left(d+f x^2\right)^q \left(\frac{c \left(d+f x^2\right)}{c d-a f}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{f \left(c x^2+a\right)}{c d-a f}\right)}{2 c (p+1)}",1,"(x*(a + c*x^2)^p*(d + f*x^2)^q*((B*x*AppellF1[1, -p, -q, 2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (6*a*A*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)] + 2*x^2*(c*d*p*AppellF1[3/2, 1 - p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)] + a*f*q*AppellF1[3/2, -p, 1 - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)]))))/2","A",0
400,1,302,252,0.5171624,"\int \left(a+c x^2\right)^p \left(A+B x+C x^2\right) \left(d+f x^2\right)^q \, dx","Integrate[(a + c*x^2)^p*(A + B*x + C*x^2)*(d + f*x^2)^q,x]","\frac{1}{6} x \left(a+c x^2\right)^p \left(d+f x^2\right)^q \left(\frac{18 a A d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}{2 x^2 \left(c d p F_1\left(\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+a f q F_1\left(\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)+3 a d F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)}+3 B x \left(\frac{c x^2}{a}+1\right)^{-p} \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(1;-p,-q;2;-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+2 C x^2 \left(\frac{c x^2}{a}+1\right)^{-p} \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)\right)","A x \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d+f x^2\right)^q \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+\frac{1}{3} C x^3 \left(a+c x^2\right)^p \left(\frac{c x^2}{a}+1\right)^{-p} \left(d+f x^2\right)^q \left(\frac{f x^2}{d}+1\right)^{-q} F_1\left(\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right)+\frac{B \left(a+c x^2\right)^{p+1} \left(d+f x^2\right)^q \left(\frac{c \left(d+f x^2\right)}{c d-a f}\right)^{-q} \, _2F_1\left(p+1,-q;p+2;-\frac{f \left(c x^2+a\right)}{c d-a f}\right)}{2 c (p+1)}",1,"(x*(a + c*x^2)^p*(d + f*x^2)^q*((3*B*x*AppellF1[1, -p, -q, 2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q) + (18*a*A*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)])/(3*a*d*AppellF1[1/2, -p, -q, 3/2, -((c*x^2)/a), -((f*x^2)/d)] + 2*x^2*(c*d*p*AppellF1[3/2, 1 - p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)] + a*f*q*AppellF1[3/2, -p, 1 - q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])) + (2*C*x^2*AppellF1[3/2, -p, -q, 5/2, -((c*x^2)/a), -((f*x^2)/d)])/((1 + (c*x^2)/a)^p*(1 + (f*x^2)/d)^q)))/6","A",0