1,1,355,415,0.5442692,"\int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^3 \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3*(A + B*x + C*x^2),x]","\frac{105 d \sin ^{-1}(d x) \left(8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right)+\sqrt{1-d^2 x^2} \left(14 A d^2 \left(6 d^4 x \left(10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right)-d^2 f \left(120 e^2+45 e f x+8 f^2 x^2\right)-16 f^3\right)+7 B \left(4 d^6 x^2 \left(20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right)-2 d^4 \left(40 e^3+45 e^2 f x+24 e f^2 x^2+5 f^3 x^3\right)-3 d^2 f^2 (32 e+5 f x)\right)-C \left(-12 d^6 x^3 \left(35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right)+6 d^4 x \left(35 e^3+56 e^2 f x+35 e f^2 x^2+8 f^3 x^3\right)+d^2 f \left(672 e^2+315 e f x+64 f^2 x^2\right)+128 f^3\right)\right)}{1680 d^6}","-\frac{\left(1-d^2 x^2\right)^{3/2} (e+f x)^2 \left(7 d^2 f (2 A f+B e)-C \left(3 d^2 e^2-8 f^2\right)\right)}{70 d^4 f}+\frac{x \sqrt{1-d^2 x^2} \left(8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right)}{16 d^4}+\frac{\left(1-d^2 x^2\right)^{3/2} \left(3 d^2 f x \left(-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right)+8 \left(C \left(3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right)-7 d^2 f \left(2 A f \left(6 d^2 e^2+f^2\right)+B \left(d^2 e^3+6 e f^2\right)\right)\right)\right)}{840 d^6 f}+\frac{\sin ^{-1}(d x) \left(8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right)}{16 d^5}+\frac{\left(1-d^2 x^2\right)^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac{C \left(1-d^2 x^2\right)^{3/2} (e+f x)^4}{7 d^2 f}",1,"(Sqrt[1 - d^2*x^2]*(14*A*d^2*(-16*f^3 - d^2*f*(120*e^2 + 45*e*f*x + 8*f^2*x^2) + 6*d^4*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)) + 7*B*(-3*d^2*f^2*(32*e + 5*f*x) - 2*d^4*(40*e^3 + 45*e^2*f*x + 24*e*f^2*x^2 + 5*f^3*x^3) + 4*d^6*x^2*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3)) - C*(128*f^3 + d^2*f*(672*e^2 + 315*e*f*x + 64*f^2*x^2) + 6*d^4*x*(35*e^3 + 56*e^2*f*x + 35*e*f^2*x^2 + 8*f^3*x^3) - 12*d^6*x^3*(35*e^3 + 84*e^2*f*x + 70*e*f^2*x^2 + 20*f^3*x^3))) + 105*d*(2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*ArcSin[d*x])/(1680*d^6)","A",1
2,1,244,286,0.3489573,"\int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^2 \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2*(A + B*x + C*x^2),x]","\frac{15 \sin ^{-1}(d x) \left(2 d^2 \left(A \left(4 d^2 e^2+f^2\right)+2 B e f\right)+C \left(2 d^2 e^2+f^2\right)\right)+d \sqrt{1-d^2 x^2} \left(10 A d^2 \left(12 d^2 e^2 x+16 e f \left(d^2 x^2-1\right)+3 f^2 x \left(2 d^2 x^2-1\right)\right)+4 B \left(2 d^4 x^2 \left(10 e^2+15 e f x+6 f^2 x^2\right)-d^2 \left(20 e^2+15 e f x+4 f^2 x^2\right)-8 f^2\right)+C \left(30 d^2 e^2 x \left(2 d^2 x^2-1\right)+32 e f \left(3 d^4 x^4-d^2 x^2-2\right)+5 f^2 x \left(8 d^4 x^4-2 d^2 x^2-3\right)\right)\right)}{240 d^5}","\frac{\sin ^{-1}(d x) \left(2 d^2 \left(A \left(4 d^2 e^2+f^2\right)+2 B e f\right)+C \left(2 d^2 e^2+f^2\right)\right)}{16 d^5}+\frac{x \sqrt{1-d^2 x^2} \left(2 d^2 \left(A \left(4 d^2 e^2+f^2\right)+2 B e f\right)+C \left(2 d^2 e^2+f^2\right)\right)}{16 d^4}+\frac{\left(1-d^2 x^2\right)^{3/2} \left(8 \left(C \left(d^2 e^3-4 e f^2\right)-2 f \left(5 A d^2 e f+B \left(d^2 e^2+f^2\right)\right)\right)-3 f x \left(5 f^2 \left(2 A d^2+C\right)-2 d^2 e (C e-2 B f)\right)\right)}{120 d^4 f}+\frac{\left(1-d^2 x^2\right)^{3/2} (e+f x)^2 (C e-2 B f)}{10 d^2 f}-\frac{C \left(1-d^2 x^2\right)^{3/2} (e+f x)^3}{6 d^2 f}",1,"(d*Sqrt[1 - d^2*x^2]*(10*A*d^2*(12*d^2*e^2*x + 16*e*f*(-1 + d^2*x^2) + 3*f^2*x*(-1 + 2*d^2*x^2)) + 4*B*(-8*f^2 - d^2*(20*e^2 + 15*e*f*x + 4*f^2*x^2) + 2*d^4*x^2*(10*e^2 + 15*e*f*x + 6*f^2*x^2)) + C*(30*d^2*e^2*x*(-1 + 2*d^2*x^2) + 32*e*f*(-2 - d^2*x^2 + 3*d^4*x^4) + 5*f^2*x*(-3 - 2*d^2*x^2 + 8*d^4*x^4))) + 15*(C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2)))*ArcSin[d*x])/(240*d^5)","A",1
3,1,141,168,0.2115277,"\int \sqrt{1-d x} \sqrt{1+d x} (e+f x) \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)*(A + B*x + C*x^2),x]","\frac{15 d \sin ^{-1}(d x) \left(4 A d^2 e+B f+C e\right)+\sqrt{1-d^2 x^2} \left(60 A d^4 e x+40 A d^2 f \left(d^2 x^2-1\right)+5 B d^2 \left(8 d^2 e x^2+6 d^2 f x^3-8 e-3 f x\right)+15 C d^2 e x \left(2 d^2 x^2-1\right)+8 C f \left(3 d^4 x^4-d^2 x^2-2\right)\right)}{120 d^4}","\frac{x \sqrt{1-d^2 x^2} \left(4 A d^2 e+B f+C e\right)}{8 d^2}-\frac{\left(1-d^2 x^2\right)^{3/2} \left(4 \left(5 d^2 f (A f+B e)-C \left(3 d^2 e^2-2 f^2\right)\right)-3 d^2 f x (3 C e-5 B f)\right)}{60 d^4 f}+\frac{\sin ^{-1}(d x) \left(4 A d^2 e+B f+C e\right)}{8 d^3}-\frac{C \left(1-d^2 x^2\right)^{3/2} (e+f x)^2}{5 d^2 f}",1,"(Sqrt[1 - d^2*x^2]*(60*A*d^4*e*x + 40*A*d^2*f*(-1 + d^2*x^2) + 15*C*d^2*e*x*(-1 + 2*d^2*x^2) + 5*B*d^2*(-8*e - 3*f*x + 8*d^2*e*x^2 + 6*d^2*f*x^3) + 8*C*f*(-2 - d^2*x^2 + 3*d^4*x^4)) + 15*d*(C*e + 4*A*d^2*e + B*f)*ArcSin[d*x])/(120*d^4)","A",1
4,1,71,95,0.0637056,"\int \sqrt{1-d x} \sqrt{1+d x} \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(A + B*x + C*x^2),x]","\frac{d \sqrt{1-d^2 x^2} \left(12 A d^2 x+8 B d^2 x^2-8 B+6 C d^2 x^3-3 C x\right)+3 \left(4 A d^2+C\right) \sin ^{-1}(d x)}{24 d^3}","\frac{x \sqrt{1-d^2 x^2} \left(4 A d^2+C\right)}{8 d^2}+\frac{\left(4 A d^2+C\right) \sin ^{-1}(d x)}{8 d^3}-\frac{B \left(1-d^2 x^2\right)^{3/2}}{3 d^2}-\frac{C x \left(1-d^2 x^2\right)^{3/2}}{4 d^2}",1,"(d*Sqrt[1 - d^2*x^2]*(-8*B - 3*C*x + 12*A*d^2*x + 8*B*d^2*x^2 + 6*C*d^2*x^3) + 3*(C + 4*A*d^2)*ArcSin[d*x])/(24*d^3)","A",1
5,1,117,122,0.1503854,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]","\frac{\frac{\left(f (A f-B e)+C e^2\right) \tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right)}{\sqrt{d^2 e^2-f^2}}+\frac{\sin ^{-1}(d x) (B f-C e)}{d}-\frac{C f \sqrt{1-d^2 x^2}}{d^2}}{f^2}","\frac{\left(A f^2-B e f+C e^2\right) \tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right)}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}",1,"(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + ((-(C*e) + B*f)*ArcSin[d*x])/d + ((C*e^2 + f*(-(B*e) + A*f))*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/Sqrt[d^2*e^2 - f^2])/f^2","A",1
6,1,211,163,0.4731471,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)^2} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]","\frac{-\frac{f \sqrt{1-d^2 x^2} \left(f (A f-B e)+C e^2\right)}{\left(f^2-d^2 e^2\right) (e+f x)}-\frac{\log \left(\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{\left(f^2-d^2 e^2\right)^{3/2}}+\frac{\log (e+f x) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{\left(f^2-d^2 e^2\right)^{3/2}}+\frac{C \sin ^{-1}(d x)}{d}}{f^2}","\frac{\sqrt{1-d^2 x^2} \left(A f^2-B e f+C e^2\right)}{f \left(d^2 e^2-f^2\right) (e+f x)}-\frac{\tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{f^2 \left(d^2 e^2-f^2\right)^{3/2}}+\frac{C \sin ^{-1}(d x)}{d f^2}",1,"(-((f*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 - d^2*x^2])/((-(d^2*e^2) + f^2)*(e + f*x))) + (C*ArcSin[d*x])/d + ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*Log[e + f*x])/(-(d^2*e^2) + f^2)^(3/2) - ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)^(3/2))/f^2","A",1
7,1,273,248,0.4158645,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)^3} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3),x]","\frac{1}{2} \left(-\frac{\log \left(\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right) \left(d^2 \left(A \left(2 d^2 e^2+f^2\right)-3 B e f\right)+C \left(d^2 e^2+2 f^2\right)\right)}{\left(f^2-d^2 e^2\right)^{5/2}}+\frac{\log (e+f x) \left(d^2 \left(A \left(2 d^2 e^2+f^2\right)-3 B e f\right)+C \left(d^2 e^2+2 f^2\right)\right)}{\left(f^2-d^2 e^2\right)^{5/2}}-\frac{\sqrt{1-d^2 x^2} \left(-A d^2 e f (4 e+3 f x)+A f^3+B d^2 e^2 (2 e+f x)+B f^2 (e+2 f x)+C e \left(d^2 e^2 x-3 e f-4 f^2 x\right)\right)}{\left(f^2-d^2 e^2\right)^2 (e+f x)^2}\right)","\frac{\sqrt{1-d^2 x^2} \left(A f^2-B e f+C e^2\right)}{2 f \left(d^2 e^2-f^2\right) (e+f x)^2}+\frac{\tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right) \left(C \left(d^2 e^2+2 f^2\right)-d^2 \left(3 B e f-A \left(2 d^2 e^2+f^2\right)\right)\right)}{2 \left(d^2 e^2-f^2\right)^{5/2}}-\frac{\sqrt{1-d^2 x^2} \left(-3 A d^2 e f^2+B d^2 e^2 f+2 B f^3+C d^2 e^3-4 C e f^2\right)}{2 f \left(d^2 e^2-f^2\right)^2 (e+f x)}",1,"(-((Sqrt[1 - d^2*x^2]*(A*f^3 + B*d^2*e^2*(2*e + f*x) + B*f^2*(e + 2*f*x) - A*d^2*e*f*(4*e + 3*f*x) + C*e*(-3*e*f + d^2*e^2*x - 4*f^2*x)))/((-(d^2*e^2) + f^2)^2*(e + f*x)^2)) + ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2*d^2*e^2 + f^2)))*Log[e + f*x])/(-(d^2*e^2) + f^2)^(5/2) - ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2*d^2*e^2 + f^2)))*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)^(5/2))/2","A",1
8,1,241,340,0.3924111,"\int \frac{(e+f x)^3 \left(A+B x+C x^2\right)}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[((e + f*x)^3*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{15 d \sin ^{-1}(d x) \left(8 A d^4 e^3+12 A d^2 e f^2+12 B d^2 e^2 f+3 B f^3+4 C d^2 e^3+9 C e f^2\right)-\sqrt{1-d^2 x^2} \left(20 A d^2 f \left(d^2 \left(18 e^2+9 e f x+2 f^2 x^2\right)+4 f^2\right)+15 B \left(2 d^4 \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right)+d^2 f^2 (16 e+3 f x)\right)+C \left(6 d^4 x \left(10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right)+d^2 f \left(240 e^2+135 e f x+32 f^2 x^2\right)+64 f^3\right)\right)}{120 d^6}","-\frac{\sqrt{1-d^2 x^2} (e+f x)^2 \left(4 f^2 \left(5 A d^2+4 C\right)-3 d^2 e (C e-5 B f)\right)}{60 d^4 f}+\frac{\sqrt{1-d^2 x^2} \left(d^2 f x \left(-100 A d^2 e f^2-30 B d^2 e^2 f-45 B f^3+6 C d^2 e^3-71 C e f^2\right)+4 \left(C \left(3 d^4 e^4-52 d^2 e^2 f^2-16 f^4\right)-5 d^2 f \left(4 A f \left(4 d^2 e^2+f^2\right)+3 B \left(d^2 e^3+4 e f^2\right)\right)\right)\right)}{120 d^6 f}+\frac{\sin ^{-1}(d x) \left(8 A d^4 e^3+12 A d^2 e f^2+12 B d^2 e^2 f+3 B f^3+4 C d^2 e^3+9 C e f^2\right)}{8 d^5}+\frac{\sqrt{1-d^2 x^2} (e+f x)^3 (C e-5 B f)}{20 d^2 f}-\frac{C \sqrt{1-d^2 x^2} (e+f x)^4}{5 d^2 f}",1,"(-(Sqrt[1 - d^2*x^2]*(20*A*d^2*f*(4*f^2 + d^2*(18*e^2 + 9*e*f*x + 2*f^2*x^2)) + 15*B*(d^2*f^2*(16*e + 3*f*x) + 2*d^4*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)) + C*(64*f^3 + d^2*f*(240*e^2 + 135*e*f*x + 32*f^2*x^2) + 6*d^4*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)))) + 15*d*(4*C*d^2*e^3 + 8*A*d^4*e^3 + 12*B*d^2*e^2*f + 9*C*e*f^2 + 12*A*d^2*e*f^2 + 3*B*f^3)*ArcSin[d*x])/(120*d^6)","A",1
9,1,160,228,0.220915,"\int \frac{(e+f x)^2 \left(A+B x+C x^2\right)}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{3 \sin ^{-1}(d x) \left(4 d^2 \left(A \left(2 d^2 e^2+f^2\right)+2 B e f\right)+C \left(4 d^2 e^2+3 f^2\right)\right)-d \sqrt{1-d^2 x^2} \left(12 A d^2 f (4 e+f x)+8 B \left(d^2 \left(3 e^2+3 e f x+f^2 x^2\right)+2 f^2\right)+C \left(12 d^2 e^2 x+16 e f \left(d^2 x^2+2\right)+3 f^2 x \left(2 d^2 x^2+3\right)\right)\right)}{24 d^5}","\frac{\sin ^{-1}(d x) \left(4 d^2 \left(A \left(2 d^2 e^2+f^2\right)+2 B e f\right)+C \left(4 d^2 e^2+3 f^2\right)\right)}{8 d^5}+\frac{\sqrt{1-d^2 x^2} \left(4 \left(C \left(d^2 e^3-8 e f^2\right)-4 f \left(3 A d^2 e f+B \left(d^2 e^2+f^2\right)\right)\right)-f x \left(3 f^2 \left(4 A d^2+3 C\right)-2 d^2 e (C e-4 B f)\right)\right)}{24 d^4 f}+\frac{\sqrt{1-d^2 x^2} (e+f x)^2 (C e-4 B f)}{12 d^2 f}-\frac{C \sqrt{1-d^2 x^2} (e+f x)^3}{4 d^2 f}",1,"(-(d*Sqrt[1 - d^2*x^2]*(12*A*d^2*f*(4*e + f*x) + C*(12*d^2*e^2*x + 16*e*f*(2 + d^2*x^2) + 3*f^2*x*(3 + 2*d^2*x^2)) + 8*B*(2*f^2 + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)))) + 3*(C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*e^2 + f^2)))*ArcSin[d*x])/(24*d^5)","A",1
10,1,88,130,0.1044076,"\int \frac{(e+f x) \left(A+B x+C x^2\right)}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[((e + f*x)*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{3 d \sin ^{-1}(d x) \left(2 A d^2 e+B f+C e\right)-\sqrt{1-d^2 x^2} \left(6 A d^2 f+3 B d^2 (2 e+f x)+C \left(3 d^2 e x+2 d^2 f x^2+4 f\right)\right)}{6 d^4}","-\frac{\sqrt{1-d^2 x^2} \left(2 \left(3 d^2 f (A f+B e)-C \left(d^2 e^2-2 f^2\right)\right)-d^2 f x (C e-3 B f)\right)}{6 d^4 f}+\frac{\sin ^{-1}(d x) \left(2 A d^2 e+B f+C e\right)}{2 d^3}-\frac{C \sqrt{1-d^2 x^2} (e+f x)^2}{3 d^2 f}",1,"(-(Sqrt[1 - d^2*x^2]*(6*A*d^2*f + 3*B*d^2*(2*e + f*x) + C*(4*f + 3*d^2*e*x + 2*d^2*f*x^2))) + 3*d*(C*e + 2*A*d^2*e + B*f)*ArcSin[d*x])/(6*d^4)","A",1
11,1,45,63,0.0361709,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{\left(2 A d^2+C\right) \sin ^{-1}(d x)-d \sqrt{1-d^2 x^2} (2 B+C x)}{2 d^3}","\frac{\left(2 A d^2+C\right) \sin ^{-1}(d x)}{2 d^3}-\frac{B \sqrt{1-d^2 x^2}}{d^2}-\frac{C x \sqrt{1-d^2 x^2}}{2 d^2}",1,"(-(d*(2*B + C*x)*Sqrt[1 - d^2*x^2]) + (C + 2*A*d^2)*ArcSin[d*x])/(2*d^3)","A",1
12,1,117,122,0.1307602,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]","\frac{\frac{\left(f (A f-B e)+C e^2\right) \tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right)}{\sqrt{d^2 e^2-f^2}}+\frac{\sin ^{-1}(d x) (B f-C e)}{d}-\frac{C f \sqrt{1-d^2 x^2}}{d^2}}{f^2}","\frac{\left(A f^2-B e f+C e^2\right) \tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right)}{f^2 \sqrt{d^2 e^2-f^2}}-\frac{\sin ^{-1}(d x) (C e-B f)}{d f^2}-\frac{C \sqrt{1-d^2 x^2}}{d^2 f}",1,"(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + ((-(C*e) + B*f)*ArcSin[d*x])/d + ((C*e^2 + f*(-(B*e) + A*f))*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - d^2*x^2])])/Sqrt[d^2*e^2 - f^2])/f^2","A",1
13,1,211,163,0.4314404,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)^2} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2),x]","\frac{-\frac{f \sqrt{1-d^2 x^2} \left(f (A f-B e)+C e^2\right)}{\left(f^2-d^2 e^2\right) (e+f x)}-\frac{\log \left(\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{\left(f^2-d^2 e^2\right)^{3/2}}+\frac{\log (e+f x) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{\left(f^2-d^2 e^2\right)^{3/2}}+\frac{C \sin ^{-1}(d x)}{d}}{f^2}","\frac{\sqrt{1-d^2 x^2} \left(A f^2-B e f+C e^2\right)}{f \left(d^2 e^2-f^2\right) (e+f x)}-\frac{\tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right) \left(-A d^2 e f^2+B f^3+C d^2 e^3-2 C e f^2\right)}{f^2 \left(d^2 e^2-f^2\right)^{3/2}}+\frac{C \sin ^{-1}(d x)}{d f^2}",1,"(-((f*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 - d^2*x^2])/((-(d^2*e^2) + f^2)*(e + f*x))) + (C*ArcSin[d*x])/d + ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*Log[e + f*x])/(-(d^2*e^2) + f^2)^(3/2) - ((C*d^2*e^3 - 2*C*e*f^2 - A*d^2*e*f^2 + B*f^3)*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)^(3/2))/f^2","A",1
14,1,273,248,0.3841503,"\int \frac{A+B x+C x^2}{\sqrt{1-d x} \sqrt{1+d x} (e+f x)^3} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3),x]","\frac{1}{2} \left(-\frac{\log \left(\sqrt{1-d^2 x^2} \sqrt{f^2-d^2 e^2}+d^2 e x+f\right) \left(d^2 \left(A \left(2 d^2 e^2+f^2\right)-3 B e f\right)+C \left(d^2 e^2+2 f^2\right)\right)}{\left(f^2-d^2 e^2\right)^{5/2}}+\frac{\log (e+f x) \left(d^2 \left(A \left(2 d^2 e^2+f^2\right)-3 B e f\right)+C \left(d^2 e^2+2 f^2\right)\right)}{\left(f^2-d^2 e^2\right)^{5/2}}-\frac{\sqrt{1-d^2 x^2} \left(-A d^2 e f (4 e+3 f x)+A f^3+B d^2 e^2 (2 e+f x)+B f^2 (e+2 f x)+C e \left(d^2 e^2 x-3 e f-4 f^2 x\right)\right)}{\left(f^2-d^2 e^2\right)^2 (e+f x)^2}\right)","\frac{\sqrt{1-d^2 x^2} \left(A f^2-B e f+C e^2\right)}{2 f \left(d^2 e^2-f^2\right) (e+f x)^2}+\frac{\tan ^{-1}\left(\frac{d^2 e x+f}{\sqrt{1-d^2 x^2} \sqrt{d^2 e^2-f^2}}\right) \left(C \left(d^2 e^2+2 f^2\right)-d^2 \left(3 B e f-A \left(2 d^2 e^2+f^2\right)\right)\right)}{2 \left(d^2 e^2-f^2\right)^{5/2}}-\frac{\sqrt{1-d^2 x^2} \left(-3 A d^2 e f^2+B d^2 e^2 f+2 B f^3+C d^2 e^3-4 C e f^2\right)}{2 f \left(d^2 e^2-f^2\right)^2 (e+f x)}",1,"(-((Sqrt[1 - d^2*x^2]*(A*f^3 + B*d^2*e^2*(2*e + f*x) + B*f^2*(e + 2*f*x) - A*d^2*e*f*(4*e + 3*f*x) + C*e*(-3*e*f + d^2*e^2*x - 4*f^2*x)))/((-(d^2*e^2) + f^2)^2*(e + f*x)^2)) + ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2*d^2*e^2 + f^2)))*Log[e + f*x])/(-(d^2*e^2) + f^2)^(5/2) - ((C*(d^2*e^2 + 2*f^2) + d^2*(-3*B*e*f + A*(2*d^2*e^2 + f^2)))*Log[f + d^2*e*x + Sqrt[-(d^2*e^2) + f^2]*Sqrt[1 - d^2*x^2]])/(-(d^2*e^2) + f^2)^(5/2))/2","A",1
15,1,57,79,0.0679419,"\int \frac{x \left(a+b x+c x^2\right)}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(x*(a + b*x + c*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{3 b d \sin ^{-1}(d x)-\sqrt{1-d^2 x^2} \left(3 d^2 (2 a+b x)+2 c \left(d^2 x^2+2\right)\right)}{6 d^4}","-\frac{\sqrt{1-d^2 x^2} \left(2 \left(3 a d^2+2 c\right)+3 b d^2 x\right)}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2}",1,"(-(Sqrt[1 - d^2*x^2]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2))) + 3*b*d*ArcSin[d*x])/(6*d^4)","A",1
16,1,45,63,0.0340548,"\int \frac{a+b x+c x^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","\frac{\left(2 a d^2+c\right) \sin ^{-1}(d x)-d \sqrt{1-d^2 x^2} (2 b+c x)}{2 d^3}","\frac{\left(2 a d^2+c\right) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2}",1,"(-(d*(2*b + c*x)*Sqrt[1 - d^2*x^2]) + (c + 2*a*d^2)*ArcSin[d*x])/(2*d^3)","A",1
17,1,48,48,0.0547244,"\int \frac{a+b x+c x^2}{x \sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x*Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","-a \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2}","-a \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2}",1,"-((c*Sqrt[1 - d^2*x^2])/d^2) + (b*ArcSin[d*x])/d - a*ArcTanh[Sqrt[1 - d^2*x^2]]","A",1
18,1,48,48,0.0608897,"\int \frac{a+b x+c x^2}{x^2 \sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x^2*Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","-\frac{a \sqrt{1-d^2 x^2}}{x}-b \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)+\frac{c \sin ^{-1}(d x)}{d}","-\frac{a \sqrt{1-d^2 x^2}}{x}-b \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)+\frac{c \sin ^{-1}(d x)}{d}",1,"-((a*Sqrt[1 - d^2*x^2])/x) + (c*ArcSin[d*x])/d - b*ArcTanh[Sqrt[1 - d^2*x^2]]","A",1
19,1,56,71,0.0490209,"\int \frac{a+b x+c x^2}{x^3 \sqrt{1-d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x^3*Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]","-\frac{\sqrt{1-d^2 x^2} (a+2 b x)}{2 x^2}-\frac{1}{2} \left(a d^2+2 c\right) \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)","-\frac{1}{2} \left(a d^2+2 c\right) \tanh ^{-1}\left(\sqrt{1-d^2 x^2}\right)-\frac{a \sqrt{1-d^2 x^2}}{2 x^2}-\frac{b \sqrt{1-d^2 x^2}}{x}",1,"-1/2*((a + 2*b*x)*Sqrt[1 - d^2*x^2])/x^2 - ((2*c + a*d^2)*ArcTanh[Sqrt[1 - d^2*x^2]])/2","A",1
20,1,427,591,1.4632506,"\int \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3 \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3*(A + B*x + C*x^2),x]","\frac{\sqrt{c (a-b x)} \left(210 a^{5/2} b \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right) \left(a^4 f^2 (B f+3 C e)+A \left(6 a^2 b^2 e f^2+8 b^4 e^3\right)+2 a^2 b^2 e^2 (3 B f+C e)\right)+\left(a^2-b^2 x^2\right) \left(128 a^6 C f^3+a^4 b^2 f \left(7 f (32 A f+96 B e+15 B f x)+C \left(672 e^2+315 e f x+64 f^2 x^2\right)\right)+2 a^2 b^4 \left(7 A f \left(120 e^2+45 e f x+8 f^2 x^2\right)+7 B \left(40 e^3+45 e^2 f x+24 e f^2 x^2+5 f^3 x^3\right)+3 C x \left(35 e^3+56 e^2 f x+35 e f^2 x^2+8 f^3 x^3\right)\right)-4 b^6 x \left(21 A \left(10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right)+x \left(7 B \left(20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right)+3 C x \left(35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right)\right)\right)\right)\right)}{1680 b^6 (b x-a) \sqrt{a+b x}}","-\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^2 \sqrt{a c-b c x} \left(8 a^2 C f^2-b^2 \left(3 C e^2-7 f (2 A f+B e)\right)\right)}{70 b^4 f}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left(A \left(6 a^2 b^2 e f^2+8 b^4 e^3\right)+a^2 \left(a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right)\right)}{16 b^4}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(A \left(6 a^2 b^2 e f^2+8 b^4 e^3\right)+a^2 \left(a^2 f^2 (B f+3 C e)+2 b^2 e^2 (3 B f+C e)\right)\right)}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^3 \sqrt{a c-b c x} (3 C e-7 B f)}{42 b^2 f}-\frac{C \sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^4 \sqrt{a c-b c x}}{7 b^2 f}-\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) \sqrt{a c-b c x} \left(3 b^2 f x \left(a^2 f^2 (35 B f+41 C e)-2 b^2 e \left(3 C e^2-7 f (7 A f+B e)\right)\right)+8 \left(8 a^4 C f^4+2 a^2 b^2 f^2 \left(7 f (A f+3 B e)+15 C e^2\right)+b^4 \left(-e^2\right) \left(3 C e^2-7 f (12 A f+B e)\right)\right)\right)}{840 b^6 f}",1,"(Sqrt[c*(a - b*x)]*((a^2 - b^2*x^2)*(128*a^6*C*f^3 + a^4*b^2*f*(7*f*(96*B*e + 32*A*f + 15*B*f*x) + C*(672*e^2 + 315*e*f*x + 64*f^2*x^2)) + 2*a^2*b^4*(7*A*f*(120*e^2 + 45*e*f*x + 8*f^2*x^2) + 7*B*(40*e^3 + 45*e^2*f*x + 24*e*f^2*x^2 + 5*f^3*x^3) + 3*C*x*(35*e^3 + 56*e^2*f*x + 35*e*f^2*x^2 + 8*f^3*x^3)) - 4*b^6*x*(21*A*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3) + x*(7*B*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3) + 3*C*x*(35*e^3 + 84*e^2*f*x + 70*e*f^2*x^2 + 20*f^3*x^3)))) + 210*a^(5/2)*b*(a^4*f^2*(3*C*e + B*f) + 2*a^2*b^2*e^2*(C*e + 3*B*f) + A*(8*b^4*e^3 + 6*a^2*b^2*e*f^2))*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]))/(1680*b^6*(-a + b*x)*Sqrt[a + b*x])","A",1
21,1,311,451,1.0160025,"\int \sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2 \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2*(A + B*x + C*x^2),x]","\frac{\sqrt{c (a-b x)} \left(b \left(a^2-b^2 x^2\right) \left(a^4 f (32 B f+64 C e+15 C f x)+2 a^2 b^2 \left(5 A f (16 e+3 f x)+B \left(40 e^2+30 e f x+8 f^2 x^2\right)+C x \left(15 e^2+16 e f x+5 f^2 x^2\right)\right)-4 b^4 x \left(5 A \left(6 e^2+8 e f x+3 f^2 x^2\right)+x \left(2 B \left(10 e^2+15 e f x+6 f^2 x^2\right)+C x \left(15 e^2+24 e f x+10 f^2 x^2\right)\right)\right)\right)+30 a^{5/2} \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right) \left(a^4 C f^2+2 A \left(a^2 b^2 f^2+4 b^4 e^2\right)+2 a^2 b^2 e (2 B f+C e)\right)\right)}{240 b^5 (b x-a) \sqrt{a+b x}}","-\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) \sqrt{a c-b c x} \left(3 f x \left(5 a^2 C f^2-b^2 \left(2 C e^2-2 f (5 A f+2 B e)\right)\right)+8 \left(2 a^2 f^2 (B f+2 C e)-b^2 e \left(C e^2-2 f (5 A f+B e)\right)\right)\right)}{120 b^4 f}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left(2 A \left(a^2 b^2 f^2+4 b^4 e^2\right)+a^2 \left(a^2 C f^2+2 b^2 e (2 B f+C e)\right)\right)}{16 b^4}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(2 A \left(a^2 b^2 f^2+4 b^4 e^2\right)+a^2 \left(a^2 C f^2+2 b^2 e (2 B f+C e)\right)\right)}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^2 \sqrt{a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac{C \sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^3 \sqrt{a c-b c x}}{6 b^2 f}",1,"(Sqrt[c*(a - b*x)]*(b*(a^2 - b^2*x^2)*(a^4*f*(64*C*e + 32*B*f + 15*C*f*x) + 2*a^2*b^2*(5*A*f*(16*e + 3*f*x) + C*x*(15*e^2 + 16*e*f*x + 5*f^2*x^2) + B*(40*e^2 + 30*e*f*x + 8*f^2*x^2)) - 4*b^4*x*(5*A*(6*e^2 + 8*e*f*x + 3*f^2*x^2) + x*(2*B*(10*e^2 + 15*e*f*x + 6*f^2*x^2) + C*x*(15*e^2 + 24*e*f*x + 10*f^2*x^2)))) + 30*a^(5/2)*(a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*f^2))*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]))/(240*b^5*(-a + b*x)*Sqrt[a + b*x])","A",1
22,1,200,300,0.6815052,"\int \sqrt{a+b x} \sqrt{a c-b c x} (e+f x) \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)*(A + B*x + C*x^2),x]","-\frac{c \left(30 a^{5/2} b \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right) \left(a^2 (B f+C e)+4 A b^2 e\right)+\left(a^2-b^2 x^2\right) \left(16 a^4 C f+a^2 b^2 (40 A f+5 B (8 e+3 f x)+C x (15 e+8 f x))-2 b^4 x (10 A (3 e+2 f x)+x (5 B (4 e+3 f x)+3 C x (5 e+4 f x)))\right)\right)}{120 b^4 \sqrt{a+b x} \sqrt{c (a-b x)}}","\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left(a^2 (B f+C e)+4 A b^2 e\right)}{8 b^2}-\frac{\sqrt{a+b x} \left(a^2-b^2 x^2\right) \sqrt{a c-b c x} \left(4 \left(2 a^2 C f^2-b^2 \left(3 C e^2-5 f (A f+B e)\right)\right)-3 b^2 f x (3 C e-5 B f)\right)}{60 b^4 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(a^2 (B f+C e)+4 A b^2 e\right)}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{C \sqrt{a+b x} \left(a^2-b^2 x^2\right) (e+f x)^2 \sqrt{a c-b c x}}{5 b^2 f}",1,"-1/120*(c*((a^2 - b^2*x^2)*(16*a^4*C*f + a^2*b^2*(40*A*f + 5*B*(8*e + 3*f*x) + C*x*(15*e + 8*f*x)) - 2*b^4*x*(10*A*(3*e + 2*f*x) + x*(5*B*(4*e + 3*f*x) + 3*C*x*(5*e + 4*f*x)))) + 30*a^(5/2)*b*(4*A*b^2*e + a^2*(C*e + B*f))*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]))/(b^4*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])","A",1
23,1,142,221,0.4079564,"\int \sqrt{a+b x} \sqrt{a c-b c x} \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(A + B*x + C*x^2),x]","-\frac{c \left(b \left(b^2 x^2-a^2\right) \left(2 b^2 x \left(6 A+4 B x+3 C x^2\right)-a^2 (8 B+3 C x)\right)+6 a^{5/2} \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \left(a^2 C+4 A b^2\right) \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)\right)}{24 b^3 \sqrt{a+b x} \sqrt{c (a-b x)}}","\frac{1}{8} x \sqrt{a+b x} \left(\frac{a^2 C}{b^2}+4 A\right) \sqrt{a c-b c x}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \left(a^2 C+4 A b^2\right) \sqrt{a c-b c x} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{B \sqrt{a+b x} \left(a^2-b^2 x^2\right) \sqrt{a c-b c x}}{3 b^2}-\frac{C x \sqrt{a+b x} \left(a^2-b^2 x^2\right) \sqrt{a c-b c x}}{4 b^2}",1,"-1/24*(c*(b*(-a^2 + b^2*x^2)*(-(a^2*(8*B + 3*C*x)) + 2*b^2*x*(6*A + 4*B*x + 3*C*x^2)) + 6*a^(5/2)*(4*A*b^2 + a^2*C)*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]))/(b^3*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])","A",1
24,1,225,278,0.7675788,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]","\frac{\sqrt{a-b x} \left(\frac{2 \left(f (A f-B e)+C e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) (a C f-b B f+b C e)}{b^2}+\frac{C f \sqrt{a+b x} \left(-\sqrt{a-b x}-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{\frac{b x}{a}+1}}\right)}{b^2}\right)}{f^2 \sqrt{c (a-b x)}}","\frac{\sqrt{a^2 c-b^2 c x^2} \left(A f^2-B e f+C e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{\sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x} \sqrt{b^2 e^2-a^2 f^2}}-\frac{\sqrt{a^2 c-b^2 c x^2} (C e-B f) \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{b \sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left(a^2-b^2 x^2\right)}{b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}",1,"(Sqrt[a - b*x]*((C*f*Sqrt[a + b*x]*(-Sqrt[a - b*x] - (2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/Sqrt[1 + (b*x)/a]))/b^2 + (2*(b*C*e - b*B*f + a*C*f)*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/b^2 + (2*(C*e^2 + f*(-(B*e) + A*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f])))/(f^2*Sqrt[c*(a - b*x)])","A",1
25,1,309,322,0.8523785,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2),x]","\frac{-\frac{2 b^2 e \sqrt{a-b x} \left(f (A f-B e)+C e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{(-a f-b e)^{3/2} (b e-a f)^{3/2}}+\frac{f (b x-a) \sqrt{a+b x} \left(f (A f-B e)+C e^2\right)}{(e+f x) (a f-b e) (a f+b e)}-\frac{2 \sqrt{a-b x} (2 C e-B f) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}-\frac{2 C \sqrt{a-b x} \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right)}{b}}{f^2 \sqrt{c (a-b x)}}","\frac{f \left(a^2-b^2 x^2\right) \left(A+\frac{e (C e-B f)}{f^2}\right)}{\sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)}+\frac{\sqrt{a^2 c-b^2 c x^2} \left(a^2 f^2 (2 C e-B f)-b^2 \left(C e^3-A e f^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{\sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^{3/2}}+\frac{C \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{b \sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x}}",1,"((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)) - (2*C*Sqrt[a - b*x]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/b - (2*(2*C*e - B*f)*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]) - (2*b^2*e*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/((-(b*e) - a*f)^(3/2)*(b*e - a*f)^(3/2)))/(f^2*Sqrt[c*(a - b*x)])","A",1
26,1,492,363,1.7947261,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3),x]","\frac{\frac{b^2 \sqrt{a-b x} \left(f (A f-B e)+C e^2\right) \left(2 (e+f x) \left(a^2 f^2+2 b^2 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)+3 e f \sqrt{a-b x} \sqrt{a+b x} \sqrt{-a f-b e} \sqrt{b e-a f}\right)}{(e+f x) (-a f-b e)^{5/2} (b e-a f)^{5/2}}+\frac{2 f (b x-a) \sqrt{a+b x} (B f-2 C e)}{(e+f x) \left(a^2 f^2-b^2 e^2\right)}+\frac{f (b x-a) \sqrt{a+b x} \left(f (A f-B e)+C e^2\right)}{(e+f x)^2 (a f-b e) (a f+b e)}+\frac{4 b^2 e \sqrt{a-b x} (2 C e-B f) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{(-a f-b e)^{3/2} (b e-a f)^{3/2}}+\frac{4 C \sqrt{a-b x} \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}}{2 f^2 \sqrt{c (a-b x)}}","\frac{\left(a^2-b^2 x^2\right) \left(2 a^2 f^2 (2 C e-B f)-b^2 e \left(f (B e-3 A f)+C e^2\right)\right)}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^2}+\frac{f \left(a^2-b^2 x^2\right) \left(A+\frac{e (C e-B f)}{f^2}\right)}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)}+\frac{\sqrt{a^2 c-b^2 c x^2} \left(A \left(a^2 b^2 f^2+2 b^4 e^2\right)+a^2 \left(2 a^2 C f^2+b^2 e (C e-3 B f)\right)\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^{5/2}}",1,"((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)^2) + (2*f*(-2*C*e + B*f)*(-a + b*x)*Sqrt[a + b*x])/((-(b^2*e^2) + a^2*f^2)*(e + f*x)) + (4*C*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]) + (4*b^2*e*(2*C*e - B*f)*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/((-(b*e) - a*f)^(3/2)*(b*e - a*f)^(3/2)) + (b^2*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*(3*e*f*Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]*Sqrt[a - b*x]*Sqrt[a + b*x] + 2*(2*b^2*e^2 + a^2*f^2)*(e + f*x)*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])]))/((-(b*e) - a*f)^(5/2)*(b*e - a*f)^(5/2)*(e + f*x)))/(2*f^2*Sqrt[c*(a - b*x)])","A",1
27,1,727,501,4.9017689,"\int \frac{(e+f x)^3 \left(A+B x+C x^2\right)}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx","Integrate[((e + f*x)^3*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]","\frac{-120 \sqrt{a-b x} \sqrt{a+b x} (b e-a f)^2 \left(\sqrt{a-b x} \sqrt{\frac{b x}{a}+1}+2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)\right) \left(5 a^2 C f-2 a b (2 B f+C e)+b^2 (3 A f+B e)\right)-20 f \sqrt{a-b x} \sqrt{a+b x} \left(30 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \left(22 a^2+9 a b x+2 b^2 x^2\right)\right) \left(10 a^2 C f^2-4 a b f (B f+3 C e)+b^2 \left(f (A f+3 B e)+3 C e^2\right)\right)-60 \sqrt{a-b x} \sqrt{a+b x} (b e-a f) \left(6 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} (4 a+b x) \sqrt{\frac{b x}{a}+1}\right) \left(10 a^2 C f^2-2 a b f (3 B f+4 C e)+b^2 \left(3 f (A f+B e)+C e^2\right)\right)-5 f^2 \sqrt{a-b x} \sqrt{a+b x} \left(210 a^{7/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \left(160 a^3+81 a^2 b x+32 a b^2 x^2+6 b^3 x^3\right)\right) (-5 a C f+b B f+3 b C e)-3 C f^3 \sqrt{a+b x} \left(630 a^{9/2} \sqrt{a-b x} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+(a-b x) \sqrt{\frac{b x}{a}+1} \left(488 a^4+275 a^3 b x+144 a^2 b^2 x^2+50 a b^3 x^3+8 b^4 x^4\right)\right)-240 \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} (b e-a f)^3 \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) \left(a (a C-b B)+A b^2\right)}{120 b^6 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}}","-\frac{\left(a^2-b^2 x^2\right) (e+f x)^2 \left(16 a^2 C f^2-b^2 \left(3 C e^2-5 f (4 A f+3 B e)\right)\right)}{60 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(4 A \left(3 a^2 b^2 e f^2+2 b^4 e^3\right)+a^2 \left(3 a^2 f^2 (B f+3 C e)+4 b^2 e^2 (3 B f+C e)\right)\right)}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left(a^2-b^2 x^2\right) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left(a^2-b^2 x^2\right) (e+f x)^4}{5 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left(a^2-b^2 x^2\right) \left(b^2 f x \left(a^2 f^2 (45 B f+71 C e)-2 b^2 e \left(3 C e^2-5 f (10 A f+3 B e)\right)\right)+4 \left(16 a^4 C f^4+4 a^2 b^2 f^2 \left(5 f (A f+3 B e)+13 C e^2\right)+b^4 \left(-e^2\right) \left(3 C e^2-5 f (16 A f+3 B e)\right)\right)\right)}{120 b^6 f \sqrt{a+b x} \sqrt{a c-b c x}}",1,"(-120*(b*e - a*f)^2*(5*a^2*C*f + b^2*(B*e + 3*A*f) - 2*a*b*(C*e + 2*B*f))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a] + 2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 60*(b*e - a*f)*(10*a^2*C*f^2 - 2*a*b*f*(4*C*e + 3*B*f) + b^2*(C*e^2 + 3*f*(B*e + A*f)))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*(4*a + b*x)*Sqrt[1 + (b*x)/a] + 6*a^(3/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 20*f*(10*a^2*C*f^2 - 4*a*b*f*(3*C*e + B*f) + b^2*(3*C*e^2 + f*(3*B*e + A*f)))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 5*f^2*(3*b*C*e + b*B*f - 5*a*C*f)*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*(160*a^3 + 81*a^2*b*x + 32*a*b^2*x^2 + 6*b^3*x^3) + 210*a^(7/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 3*C*f^3*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(488*a^4 + 275*a^3*b*x + 144*a^2*b^2*x^2 + 50*a*b^3*x^3 + 8*b^4*x^4) + 630*a^(9/2)*Sqrt[a - b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 240*(A*b^2 + a*(-(b*B) + a*C))*(b*e - a*f)^3*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(120*b^6*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])","A",1
28,1,555,368,2.6840622,"\int \frac{(e+f x)^2 \left(A+B x+C x^2\right)}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx","Integrate[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]","\frac{-24 \sqrt{a-b x} \sqrt{a+b x} (b e-a f) \left(\sqrt{a-b x} \sqrt{\frac{b x}{a}+1}+2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)\right) \left(4 a^2 C f-a b (3 B f+2 C e)+b^2 (2 A f+B e)\right)-12 \sqrt{a-b x} \sqrt{a+b x} \left(6 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} (4 a+b x) \sqrt{\frac{b x}{a}+1}\right) \left(6 a^2 C f^2-3 a b f (B f+2 C e)+b^2 \left(f (A f+2 B e)+C e^2\right)\right)-4 f \sqrt{a-b x} \sqrt{a+b x} \left(30 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \left(22 a^2+9 a b x+2 b^2 x^2\right)\right) (-4 a C f+b B f+2 b C e)-C f^2 \sqrt{a+b x} \left(210 a^{7/2} \sqrt{a-b x} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+(a-b x) \sqrt{\frac{b x}{a}+1} \left(160 a^3+81 a^2 b x+32 a b^2 x^2+6 b^3 x^3\right)\right)-48 \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} (b e-a f)^2 \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) \left(a (a C-b B)+A b^2\right)}{24 b^5 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}}","-\frac{\left(a^2-b^2 x^2\right) \left(f x \left(9 a^2 C f^2-b^2 \left(2 C e^2-4 f (3 A f+2 B e)\right)\right)+4 \left(4 a^2 f^2 (B f+2 C e)-b^2 e \left(C e^2-4 f (3 A f+B e)\right)\right)\right)}{24 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(4 A \left(a^2 b^2 f^2+2 b^4 e^2\right)+a^2 \left(3 a^2 C f^2+4 b^2 e (2 B f+C e)\right)\right)}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left(a^2-b^2 x^2\right) (e+f x)^2 (C e-4 B f)}{12 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left(a^2-b^2 x^2\right) (e+f x)^3}{4 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}",1,"(-24*(b*e - a*f)*(4*a^2*C*f + b^2*(B*e + 2*A*f) - a*b*(2*C*e + 3*B*f))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a] + 2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 12*(6*a^2*C*f^2 - 3*a*b*f*(2*C*e + B*f) + b^2*(C*e^2 + f*(2*B*e + A*f)))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*(4*a + b*x)*Sqrt[1 + (b*x)/a] + 6*a^(3/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 4*f*(2*b*C*e + b*B*f - 4*a*C*f)*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - C*f^2*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(160*a^3 + 81*a^2*b*x + 32*a*b^2*x^2 + 6*b^3*x^3) + 210*a^(7/2)*Sqrt[a - b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) - 48*(A*b^2 + a*(-(b*B) + a*C))*(b*e - a*f)^2*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(24*b^5*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])","A",1
29,1,390,246,1.4292808,"\int \frac{(e+f x) \left(A+B x+C x^2\right)}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx","Integrate[((e + f*x)*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]","-\frac{3 \sqrt{a-b x} \sqrt{a+b x} \left(6 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{a-b x} (4 a+b x) \sqrt{\frac{b x}{a}+1}\right) (-3 a C f+b B f+b C e)+6 \sqrt{a-b x} \sqrt{a+b x} \left(\sqrt{a-b x} \sqrt{\frac{b x}{a}+1}+2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)\right) \left(3 a^2 C f-2 a b (B f+C e)+b^2 (A f+B e)\right)+C f \sqrt{a+b x} \left(30 a^{5/2} \sqrt{a-b x} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)+(a-b x) \sqrt{\frac{b x}{a}+1} \left(22 a^2+9 a b x+2 b^2 x^2\right)\right)+12 \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} (b e-a f) \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) \left(a (a C-b B)+A b^2\right)}{6 b^4 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}}","-\frac{\left(a^2-b^2 x^2\right) \left(2 \left(2 a^2 C f^2-b^2 \left(C e^2-3 f (A f+B e)\right)\right)-b^2 f x (C e-3 B f)\right)}{6 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right) \left(a^2 (B f+C e)+2 A b^2 e\right)}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left(a^2-b^2 x^2\right) (e+f x)^2}{3 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}",1,"-1/6*(6*(3*a^2*C*f + b^2*(B*e + A*f) - 2*a*b*(C*e + B*f))*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*Sqrt[1 + (b*x)/a] + 2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + 3*(b*C*e + b*B*f - 3*a*C*f)*Sqrt[a - b*x]*Sqrt[a + b*x]*(Sqrt[a - b*x]*(4*a + b*x)*Sqrt[1 + (b*x)/a] + 6*a^(3/2)*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + C*f*Sqrt[a + b*x]*((a - b*x)*Sqrt[1 + (b*x)/a]*(22*a^2 + 9*a*b*x + 2*b^2*x^2) + 30*a^(5/2)*Sqrt[a - b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])]) + 12*(A*b^2 + a*(-(b*B) + a*C))*(b*e - a*f)*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/(b^4*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])","A",1
30,1,169,177,0.4373441,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]","-\frac{\sqrt{a-b x} \left(\sqrt{\frac{b x}{a}+1} \left(4 \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) \left(a (a C-b B)+A b^2\right)+b \sqrt{a-b x} \sqrt{a+b x} (2 B+C x)\right)-2 \sqrt{a} \sqrt{a+b x} (a C-2 b B) \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)\right)}{2 b^3 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}}","\frac{\left(a^2 C+2 A b^2\right) \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{B \left(a^2-b^2 x^2\right)}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left(a^2-b^2 x^2\right)}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}",1,"-1/2*(Sqrt[a - b*x]*(-2*Sqrt[a]*(-2*b*B + a*C)*Sqrt[a + b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])] + Sqrt[1 + (b*x)/a]*(b*Sqrt[a - b*x]*Sqrt[a + b*x]*(2*B + C*x) + 4*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])))/(b^3*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])","A",1
31,1,225,278,0.7110101,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]","\frac{\sqrt{a-b x} \left(\frac{2 \left(f (A f-B e)+C e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right) (a C f-b B f+b C e)}{b^2}+\frac{C f \sqrt{a+b x} \left(-\sqrt{a-b x}-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{\frac{b x}{a}+1}}\right)}{b^2}\right)}{f^2 \sqrt{c (a-b x)}}","\frac{\sqrt{a^2 c-b^2 c x^2} \left(A f^2-B e f+C e^2\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{\sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x} \sqrt{b^2 e^2-a^2 f^2}}-\frac{\sqrt{a^2 c-b^2 c x^2} (C e-B f) \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{b \sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left(a^2-b^2 x^2\right)}{b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}",1,"(Sqrt[a - b*x]*((C*f*Sqrt[a + b*x]*(-Sqrt[a - b*x] - (2*Sqrt[a]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])])/Sqrt[1 + (b*x)/a]))/b^2 + (2*(b*C*e - b*B*f + a*C*f)*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/b^2 + (2*(C*e^2 + f*(-(B*e) + A*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f])))/(f^2*Sqrt[c*(a - b*x)])","A",1
32,1,309,322,0.7936481,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^2} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2),x]","\frac{-\frac{2 b^2 e \sqrt{a-b x} \left(f (A f-B e)+C e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{(-a f-b e)^{3/2} (b e-a f)^{3/2}}+\frac{f (b x-a) \sqrt{a+b x} \left(f (A f-B e)+C e^2\right)}{(e+f x) (a f-b e) (a f+b e)}-\frac{2 \sqrt{a-b x} (2 C e-B f) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}-\frac{2 C \sqrt{a-b x} \tan ^{-1}\left(\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right)}{b}}{f^2 \sqrt{c (a-b x)}}","\frac{f \left(a^2-b^2 x^2\right) \left(A+\frac{e (C e-B f)}{f^2}\right)}{\sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)}+\frac{\sqrt{a^2 c-b^2 c x^2} \left(a^2 f^2 (2 C e-B f)-b^2 \left(C e^3-A e f^2\right)\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{\sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^{3/2}}+\frac{C \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left(\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right)}{b \sqrt{c} f^2 \sqrt{a+b x} \sqrt{a c-b c x}}",1,"((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)) - (2*C*Sqrt[a - b*x]*ArcTan[Sqrt[a - b*x]/Sqrt[a + b*x]])/b - (2*(2*C*e - B*f)*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]) - (2*b^2*e*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/((-(b*e) - a*f)^(3/2)*(b*e - a*f)^(3/2)))/(f^2*Sqrt[c*(a - b*x)])","A",1
33,1,492,363,1.3098591,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x} (e+f x)^3} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3),x]","\frac{\frac{b^2 \sqrt{a-b x} \left(f (A f-B e)+C e^2\right) \left(2 (e+f x) \left(a^2 f^2+2 b^2 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)+3 e f \sqrt{a-b x} \sqrt{a+b x} \sqrt{-a f-b e} \sqrt{b e-a f}\right)}{(e+f x) (-a f-b e)^{5/2} (b e-a f)^{5/2}}+\frac{2 f (b x-a) \sqrt{a+b x} (B f-2 C e)}{(e+f x) \left(a^2 f^2-b^2 e^2\right)}+\frac{f (b x-a) \sqrt{a+b x} \left(f (A f-B e)+C e^2\right)}{(e+f x)^2 (a f-b e) (a f+b e)}+\frac{4 b^2 e \sqrt{a-b x} (2 C e-B f) \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{(-a f-b e)^{3/2} (b e-a f)^{3/2}}+\frac{4 C \sqrt{a-b x} \tanh ^{-1}\left(\frac{\sqrt{a-b x} \sqrt{b e-a f}}{\sqrt{a+b x} \sqrt{-a f-b e}}\right)}{\sqrt{-a f-b e} \sqrt{b e-a f}}}{2 f^2 \sqrt{c (a-b x)}}","\frac{\left(a^2-b^2 x^2\right) \left(2 a^2 f^2 (2 C e-B f)-b^2 e \left(f (B e-3 A f)+C e^2\right)\right)}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^2}+\frac{f \left(a^2-b^2 x^2\right) \left(A+\frac{e (C e-B f)}{f^2}\right)}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)}+\frac{\sqrt{a^2 c-b^2 c x^2} \left(A \left(a^2 b^2 f^2+2 b^4 e^2\right)+a^2 \left(2 a^2 C f^2+b^2 e (C e-3 B f)\right)\right) \tan ^{-1}\left(\frac{\sqrt{c} \left(a^2 f+b^2 e x\right)}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right)}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left(b^2 e^2-a^2 f^2\right)^{5/2}}",1,"((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)^2) + (2*f*(-2*C*e + B*f)*(-a + b*x)*Sqrt[a + b*x])/((-(b^2*e^2) + a^2*f^2)*(e + f*x)) + (4*C*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]) + (4*b^2*e*(2*C*e - B*f)*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/((-(b*e) - a*f)^(3/2)*(b*e - a*f)^(3/2)) + (b^2*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*(3*e*f*Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]*Sqrt[a - b*x]*Sqrt[a + b*x] + 2*(2*b^2*e^2 + a^2*f^2)*(e + f*x)*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])]))/((-(b*e) - a*f)^(5/2)*(b*e - a*f)^(5/2)*(e + f*x)))/(2*f^2*Sqrt[c*(a - b*x)])","A",1
34,1,149,87,0.3580735,"\int \frac{x \left(a+b x+c x^2\right)}{\sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(x*(a + b*x + c*x^2))/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{\sqrt{-(d x-1)^2} \sqrt{d x+1} \left(3 d^2 (2 a+b x)+2 c \left(d^2 x^2+2\right)\right)+6 \sqrt{d x-1} \sin ^{-1}\left(\frac{\sqrt{1-d x}}{\sqrt{2}}\right) (d (2 a d-b)+2 c)-12 \sqrt{1-d x} \tanh ^{-1}\left(\sqrt{\frac{d x-1}{d x+1}}\right) (d (a d-b)+c)}{6 d^4 \sqrt{1-d x}}","\frac{\sqrt{d x-1} \sqrt{d x+1} \left(2 \left(3 a d^2+2 c\right)+3 b d^2 x\right)}{6 d^4}+\frac{b \cosh ^{-1}(d x)}{2 d^3}+\frac{c x^2 \sqrt{d x-1} \sqrt{d x+1}}{3 d^2}",1,"(Sqrt[-(-1 + d*x)^2]*Sqrt[1 + d*x]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2)) + 6*(2*c + d*(-b + 2*a*d))*Sqrt[-1 + d*x]*ArcSin[Sqrt[1 - d*x]/Sqrt[2]] - 12*(c + d*(-b + a*d))*Sqrt[1 - d*x]*ArcTanh[Sqrt[(-1 + d*x)/(1 + d*x)]])/(6*d^4*Sqrt[1 - d*x])","A",0
35,1,126,52,0.2222165,"\int \frac{a+b x+c x^2}{\sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{4 \sqrt{1-d x} \tanh ^{-1}\left(\sqrt{\frac{d x-1}{d x+1}}\right) (d (a d-b)+c)+d \sqrt{-(d x-1)^2} \sqrt{d x+1} (2 b+c x)+2 \sqrt{d x-1} (2 b d-c) \sin ^{-1}\left(\frac{\sqrt{1-d x}}{\sqrt{2}}\right)}{2 d^3 \sqrt{1-d x}}","\frac{\left(2 a d^2+c\right) \cosh ^{-1}(d x)}{2 d^3}+\frac{\sqrt{d x-1} \sqrt{d x+1} (2 b+c x)}{2 d^2}",1,"(d*(2*b + c*x)*Sqrt[-(-1 + d*x)^2]*Sqrt[1 + d*x] + 2*(-c + 2*b*d)*Sqrt[-1 + d*x]*ArcSin[Sqrt[1 - d*x]/Sqrt[2]] + 4*(c + d*(-b + a*d))*Sqrt[1 - d*x]*ArcTanh[Sqrt[(-1 + d*x)/(1 + d*x)]])/(2*d^3*Sqrt[1 - d*x])","B",0
36,1,128,55,0.4211066,"\int \frac{a+b x+c x^2}{x \sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{\frac{a d^2 \sqrt{d^2 x^2-1} \tan ^{-1}\left(\sqrt{d^2 x^2-1}\right)+c d^2 x^2-2 c \sqrt{1-d^2 x^2} \sin ^{-1}\left(\frac{\sqrt{1-d x}}{\sqrt{2}}\right)-c}{\sqrt{d x-1} \sqrt{d x+1}}-2 (c-b d) \tanh ^{-1}\left(\sqrt{\frac{d x-1}{d x+1}}\right)}{d^2}","a \tan ^{-1}\left(\sqrt{d x-1} \sqrt{d x+1}\right)+\frac{b \cosh ^{-1}(d x)}{d}+\frac{c \sqrt{d x-1} \sqrt{d x+1}}{d^2}",1,"((-c + c*d^2*x^2 - 2*c*Sqrt[1 - d^2*x^2]*ArcSin[Sqrt[1 - d*x]/Sqrt[2]] + a*d^2*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - 2*(c - b*d)*ArcTanh[Sqrt[(-1 + d*x)/(1 + d*x)]])/d^2","B",0
37,1,89,55,0.1823837,"\int \frac{a+b x+c x^2}{x^2 \sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{a \left(d^2 x^2-1\right)+b x \sqrt{d^2 x^2-1} \tan ^{-1}\left(\sqrt{d^2 x^2-1}\right)}{x \sqrt{d x-1} \sqrt{d x+1}}+\frac{2 c \tanh ^{-1}\left(\sqrt{\frac{d x-1}{d x+1}}\right)}{d}","\frac{a \sqrt{d x-1} \sqrt{d x+1}}{x}+b \tan ^{-1}\left(\sqrt{d x-1} \sqrt{d x+1}\right)+\frac{c \cosh ^{-1}(d x)}{d}",1,"(a*(-1 + d^2*x^2) + b*x*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(x*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) + (2*c*ArcTanh[Sqrt[(-1 + d*x)/(1 + d*x)]])/d","A",0
38,1,82,83,0.1248245,"\int \frac{a+b x+c x^2}{x^3 \sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{\left(d^2 x^2-1\right) (a+2 b x)+x^2 \sqrt{d^2 x^2-1} \left(a d^2+2 c\right) \tan ^{-1}\left(\sqrt{d^2 x^2-1}\right)}{2 x^2 \sqrt{d x-1} \sqrt{d x+1}}","\frac{1}{2} \left(a d^2+2 c\right) \tan ^{-1}\left(\sqrt{d x-1} \sqrt{d x+1}\right)+\frac{a \sqrt{d x-1} \sqrt{d x+1}}{2 x^2}+\frac{b \sqrt{d x-1} \sqrt{d x+1}}{x}",1,"((a + 2*b*x)*(-1 + d^2*x^2) + (2*c + a*d^2)*x^2*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(2*x^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x])","A",1
39,1,94,116,0.1241455,"\int \frac{a+b x+c x^2}{x^4 \sqrt{-1+d x} \sqrt{1+d x}} \, dx","Integrate[(a + b*x + c*x^2)/(x^4*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]","\frac{\left(d^2 x^2-1\right) \left(a \left(4 d^2 x^2+2\right)+3 x (b+2 c x)\right)+3 b d^2 x^3 \sqrt{d^2 x^2-1} \tan ^{-1}\left(\sqrt{d^2 x^2-1}\right)}{6 x^3 \sqrt{d x-1} \sqrt{d x+1}}","\frac{\sqrt{d x-1} \sqrt{d x+1} \left(2 a d^2+3 c\right)}{3 x}+\frac{a \sqrt{d x-1} \sqrt{d x+1}}{3 x^3}+\frac{1}{2} b d^2 \tan ^{-1}\left(\sqrt{d x-1} \sqrt{d x+1}\right)+\frac{b \sqrt{d x-1} \sqrt{d x+1}}{2 x^2}",1,"((-1 + d^2*x^2)*(3*x*(b + 2*c*x) + a*(2 + 4*d^2*x^2)) + 3*b*d^2*x^3*Sqrt[-1 + d^2*x^2]*ArcTan[Sqrt[-1 + d^2*x^2]])/(6*x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x])","A",1
40,1,343,199,0.760803,"\int \frac{a+b x+c x^2}{\sqrt{-1+x} \sqrt{1+x} (d+e x)^3} \, dx","Integrate[(a + b*x + c*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x)^3),x]","\frac{-(d+e x) \left(3 d e \sqrt{x-1} \sqrt{x+1} \sqrt{d-e} \sqrt{d+e}-2 \left(2 d^2+e^2\right) (d+e x) \tanh ^{-1}\left(\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{d-e}}{\sqrt{d+e}}\right)\right) \left(e (a e-b d)+c d^2\right)-e \sqrt{x-1} \sqrt{x+1} (d-e)^{3/2} (d+e)^{3/2} \left(e (a e-b d)+c d^2\right)+2 e \sqrt{x-1} \sqrt{x+1} (d-e)^{3/2} (d+e)^{3/2} (d+e x) (2 c d-b e)-4 d (d-e) (d+e) (d+e x)^2 (2 c d-b e) \tanh ^{-1}\left(\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{d-e}}{\sqrt{d+e}}\right)+4 c (d-e)^2 (d+e)^2 (d+e x)^2 \tanh ^{-1}\left(\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{d-e}}{\sqrt{d+e}}\right)}{2 e^2 (d-e)^{5/2} (d+e)^{5/2} (d+e x)^2}","-\frac{\sqrt{x-1} \sqrt{x+1} \left(a e^2-b d e+c d^2\right)}{2 e \left(d^2-e^2\right) (d+e x)^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt{x+1} \sqrt{d+e}}{\sqrt{x-1} \sqrt{d-e}}\right) \left(d^2 (2 a+c)+e^2 (a+2 c)-3 b d e\right)}{(d-e)^{5/2} (d+e)^{5/2}}+\frac{\sqrt{x-1} \sqrt{x+1} \left(-d e^2 (3 a+4 c)+b d^2 e+2 b e^3+c d^3\right)}{2 e \left(d^2-e^2\right)^2 (d+e x)}",1,"(-((d - e)^(3/2)*e*(d + e)^(3/2)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[-1 + x]*Sqrt[1 + x]) + 2*(d - e)^(3/2)*e*(d + e)^(3/2)*(2*c*d - b*e)*Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x) + 4*c*(d - e)^2*(d + e)^2*(d + e*x)^2*ArcTanh[(Sqrt[d - e]*Sqrt[(-1 + x)/(1 + x)])/Sqrt[d + e]] - 4*d*(d - e)*(d + e)*(2*c*d - b*e)*(d + e*x)^2*ArcTanh[(Sqrt[d - e]*Sqrt[(-1 + x)/(1 + x)])/Sqrt[d + e]] - (c*d^2 + e*(-(b*d) + a*e))*(d + e*x)*(3*d*Sqrt[d - e]*e*Sqrt[d + e]*Sqrt[-1 + x]*Sqrt[1 + x] - 2*(2*d^2 + e^2)*(d + e*x)*ArcTanh[(Sqrt[d - e]*Sqrt[(-1 + x)/(1 + x)])/Sqrt[d + e]]))/(2*(d - e)^(5/2)*e^2*(d + e)^(5/2)*(d + e*x)^2)","A",0
41,1,3599,1348,7.1314117,"\int (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right) \, dx","Integrate[(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]","\text{Result too large to show}","\frac{C (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^3}{6 b d f}-\frac{(2 a C d f-b (4 B d f-3 C (d e+c f))) (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^2}{20 b d^2 f^2}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left(\left(7 C \left(15 d^3 e^3+17 c d^2 f e^2+17 c^2 d f^2 e+15 c^3 f^3\right)+4 d f \left(50 A d f (d e+c f)-B \left(35 d^2 e^2+38 c d f e+35 c^2 f^2\right)\right)\right) b^3-8 a d f \left(C \left(35 d^2 e^2+38 c d f e+35 c^2 f^2\right)+10 d f (8 A d f-5 B (d e+c f))\right) b^2-8 a^2 d^2 f^2 (16 B d f-7 C (d e+c f)) b+6 d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)+(4 a d f-7 b (d e+c f)) (2 a C d f-b (4 B d f-3 C (d e+c f)))) x b+64 a^3 C d^3 f^3\right)}{960 b d^4 f^4}-\frac{(d e-c f)^2 \left(\left(C \left(21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right)+4 d f \left(2 A d f \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)-B \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)\right)\right) b^2-8 a d f \left(C \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)+2 d f \left(8 A d f (d e+c f)-B \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)\right)\right) b+8 a^2 d^2 f^2 \left(C \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)+8 d f (2 A d f-B (d e+c f))\right)\right) \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right)}{512 d^{11/2} f^{11/2}}+\frac{\left(\left(C \left(21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right)+4 d f \left(2 A d f \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)-B \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)\right)\right) b^2-8 a d f \left(C \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)+2 d f \left(8 A d f (d e+c f)-B \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)\right)\right) b+8 a^2 d^2 f^2 \left(C \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)+8 d f (2 A d f-B (d e+c f))\right)\right) (c+d x)^{3/2} \sqrt{e+f x}}{256 d^5 f^4}+\frac{(d e-c f) \left(\left(C \left(21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right)+4 d f \left(2 A d f \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)-B \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)\right)\right) b^2-8 a d f \left(C \left(7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right)+2 d f \left(8 A d f (d e+c f)-B \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)\right)\right) b+8 a^2 d^2 f^2 \left(C \left(5 d^2 e^2+6 c d f e+5 c^2 f^2\right)+8 d f (2 A d f-B (d e+c f))\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{512 d^5 f^5}",1,"(2*b^2*C*(d*e - c*f)^4*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(11/2)*((63/(128*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^5) + 21/(32*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 63/(80*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 9/(10*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/4 + (63*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(2048*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^5)))/(3*d^5*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(9/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*b*(d*e - c*f)^3*(-4*b*C*e + b*B*f + 2*a*C*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2)*((3*(35/(64*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 35/(48*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/10 + (21*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(512*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4)))/(3*d^4*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)^2*(6*b^2*C*e^2 - 3*b^2*B*e*f - 6*a*b*C*e*f + A*b^2*f^2 + 2*a*b*B*f^2 + a^2*C*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2)*((3*(5/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 5/(6*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/8 + (15*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(256*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3)))/(3*d^3*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(5/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)*(d*e - c*f)*(4*b*C*e^2 - 3*b*B*e*f - 2*a*C*e*f + 2*A*b*f^2 + a*B*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*d^2*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)^2*(C*e^2 - B*e*f + A*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*d*f^4*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)])","B",1
42,1,2722,721,6.6056189,"\int (a+b x) \sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right) \, dx","Integrate[(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]","\text{Result too large to show}","-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left(48 a^2 C d^2 f^2+6 b d f x (6 a C d f-b (10 B d f-7 C (c f+d e)))-10 a b d f (8 B d f-5 C (c f+d e))-\left(b^2 \left(10 d f (8 A d f-5 B (c f+d e))+C \left(35 c^2 f^2+38 c d e f+35 d^2 e^2\right)\right)\right)\right)}{240 b d^3 f^3}-\frac{(d e-c f)^2 \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(2 a d f \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)-b \left(2 d f \left(8 A d f (c f+d e)-B \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)+C \left(7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right)\right)\right)}{128 d^{9/2} f^{9/2}}+\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(2 a d f \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)-b \left(2 d f \left(8 A d f (c f+d e)-B \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)+C \left(7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right)\right)\right)}{64 d^4 f^3}+\frac{\sqrt{c+d x} \sqrt{e+f x} (d e-c f) \left(2 a d f \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)-b \left(2 d f \left(8 A d f (c f+d e)-B \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)+C \left(7 c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+7 d^3 e^3\right)\right)\right)}{128 d^4 f^4}+\frac{C (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{5 b d f}",1,"(2*b*C*(d*e - c*f)^3*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2)*((3*(35/(64*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 35/(48*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/10 + (21*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(512*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4)))/(3*d^4*f^3*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)^2*(-3*b*C*e + b*B*f + a*C*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2)*((3*(5/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 5/(6*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/8 + (15*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(256*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3)))/(3*d^3*f^3*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(5/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)*(3*b*C*e^2 - 2*b*B*e*f - 2*a*C*e*f + A*b*f^2 + a*B*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*d^2*f^3*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)*(C*e^2 - B*e*f + A*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*d*f^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)])","B",1
43,1,306,330,1.7157996,"\int \sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]","\frac{d \sqrt{f} \sqrt{c+d x} (e+f x) \left(8 d f \left(6 A d f (c f+d (e+2 f x))+B \left(-3 c^2 f^2+2 c d f (e+f x)+d^2 \left(-3 e^2+2 e f x+8 f^2 x^2\right)\right)\right)+C \left(15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left(-7 e^2+4 e f x+8 f^2 x^2\right)+d^3 \left(15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right)\right)\right)-3 (d e-c f)^{5/2} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right) \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)}{192 d^4 f^{7/2} \sqrt{e+f x}}","-\frac{(d e-c f)^2 \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)}{64 d^{7/2} f^{7/2}}+\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)}{32 d^3 f^2}+\frac{\sqrt{c+d x} \sqrt{e+f x} (d e-c f) \left(8 d f (2 A d f-B (c f+d e))+C \left(5 c^2 f^2+6 c d e f+5 d^2 e^2\right)\right)}{64 d^3 f^3}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} (-8 B d f+11 c C f+5 C d e)}{24 d^2 f^2}+\frac{C (c+d x)^{5/2} (e+f x)^{3/2}}{4 d^2 f}",1,"(d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(C*(15*c^3*f^3 - c^2*d*f^2*(7*e + 10*f*x) + c*d^2*f*(-7*e^2 + 4*e*f*x + 8*f^2*x^2) + d^3*(15*e^3 - 10*e^2*f*x + 8*e*f^2*x^2 + 48*f^3*x^3)) + 8*d*f*(6*A*d*f*(c*f + d*(e + 2*f*x)) + B*(-3*c^2*f^2 + 2*c*d*f*(e + f*x) + d^2*(-3*e^2 + 2*e*f*x + 8*f^2*x^2)))) - 3*(d*e - c*f)^(5/2)*(C*(5*d^2*e^2 + 6*c*d*e*f + 5*c^2*f^2) + 8*d*f*(2*A*d*f - B*(d*e + c*f)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(192*d^4*f^(7/2)*Sqrt[e + f*x])","A",1
44,1,1936,450,6.2147932,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{a+b x} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x),x]","\frac{2 C (d e-c f) (c+d x)^{3/2} \sqrt{e+f x} \left(\frac{3 (d e-c f)^2 \left(\frac{2 d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}-\frac{2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}}\right)}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}} \sqrt{\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1}}\right) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)^2}{32 d^2 f^2 (c+d x)^2 \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^2}+\frac{1}{2} \left(\frac{1}{\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1}+\frac{3}{4 \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^2}\right)\right) \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^{5/2}}{3 b d^2 f \left(\frac{d}{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}\right)^{3/2} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{2 \left(C a^2-b B a+A b^2\right) \sqrt{c+d x} \sqrt{e+f x} \left(\frac{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}}\right)}{2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)}\right) \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^{3/2}}{b^3 \sqrt{\frac{d}{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{2 (-b C e+b B f-a C f) (c+d x)^{3/2} \sqrt{e+f x} \left(\frac{3 (d e-c f)^2 \left(\frac{2 d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}-\frac{2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}}\right)}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}} \sqrt{\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1}}\right) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)^2}{16 d^2 f^2 (c+d x)^2 \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)}+\frac{3}{4 \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)}\right) \left(\frac{d f (c+d x)}{(d e-c f) \left(\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}\right)}+1\right)^{3/2}}{3 b^2 d f \sqrt{\frac{d}{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}} \sqrt{\frac{d (e+f x)}{d e-c f}}}-\frac{\left(C a^2-b B a+A b^2\right) (a d-b c) \left(\frac{2 \sqrt{f} \sqrt{d e-c f} \sqrt{\frac{d}{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f} \sqrt{\frac{d^2 e}{d e-c f}-\frac{c d f}{d e-c f}}}\right)}{b d^{3/2} \sqrt{e+f x}}-\frac{2 \sqrt{a f-b e} \tanh ^{-1}\left(\frac{\sqrt{a f-b e} \sqrt{c+d x}}{\sqrt{a d-b c} \sqrt{e+f x}}\right)}{b \sqrt{a d-b c}}\right)}{b^3}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left(C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right)-\left(b^3 \left(C (d e-c f)^2 (c f+d e)-2 d f \left(B (d e-c f)^2-4 A d f (c f+d e)\right)\right)\right)\right)}{8 b^4 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{8 b^3 d^2 f^2}-\frac{2 \sqrt{b c-a d} \sqrt{b e-a f} \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right)}{b^4}-\frac{\sqrt{c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f}",1,"(2*(A*b^2 - a*b*B + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*C*(d*e - c*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*b*d^2*f*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*C*e) + b*B*f - a*C*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*b^2*d*f*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) - ((A*b^2 - a*b*B + a^2*C)*(-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(3/2)*Sqrt[e + f*x]) - (2*Sqrt[-(b*e) + a*f]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b^3","B",1
45,1,2532,521,6.3746035,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^2} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^2,x]","\text{Result too large to show}","\frac{\sqrt{c+d x} (e+f x)^{3/2} \left(3 a^2 C d f-a b (2 B d f+c C f+C d e)+b^2 (2 A d f+c C e)\right)}{2 b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(24 a^2 C d^2 f^2-8 a b d f (2 B d f+c C f+C d e)-\left(b^2 \left(C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right)\right)\right)}{4 b^4 d^{3/2} f^{3/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left(12 a^2 C d f^2-a b f (8 B d f+c C f+7 C d e)+b^2 (4 d f (A f+B e)-C e (d e-c f))\right)}{4 b^3 d f (b e-a f)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(6 a^3 C d f-a^2 b (4 B d f+5 C (c f+d e))+a b^2 (2 A d f+3 B c f+3 B d e+4 c C e)-b^3 (A c f+A d e+2 B c e)\right)}{b^4 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left(A b^2-a (b B-a C)\right)}{b (a+b x) (b c-a d) (b e-a f)}",1,"-(((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*C*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*b^2*d*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(b*B - 2*a*C)*(b*c - a*d)*((Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(b*d*Sqrt[e + f*x]) - (Sqrt[-(b*e) + a*f]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b^3 - ((A*b^2 - a*b*B + a^2*C)*((-4*f*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + ((2*a*b*d*f + (b*(-2*a*d*f - b*(d*e + c*f)))/2)*((2*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) - ((-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(3/2)*Sqrt[e + f*x]) - (2*Sqrt[-(b*e) + a*f]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b))/b))/(b^2*(b*c - a*d)*(b*e - a*f))","B",0
46,1,2150,658,6.4434989,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^3} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^3,x]","\text{Result too large to show}","-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(12 a^3 C d f^2-a^2 b f (4 B d f+11 c C f+17 C d e)+a b^2 (B f (3 c f+5 d e)+4 C e (4 c f+d e))-b^3 \left(c \left(-A f^2+4 B e f+4 C e^2\right)+A d e f\right)\right)}{4 b^3 (b c-a d) (b e-a f)^2}+\frac{\sqrt{c+d x} (e+f x)^{3/2} \left(6 a^3 C d f-a^2 b (2 B d f+7 C (c f+d e))+a b^2 (-2 A d f+3 B c f+3 B d e+8 c C e)-b^3 (-A c f-A d e+4 B c e)\right)}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(24 a^4 C d^2 f^2-8 a^3 b d f (B d f+5 C (c f+d e))+3 a^2 b^2 \left(4 B d f (c f+d e)+C \left(5 c^2 f^2+22 c d e f+5 d^2 e^2\right)\right)-3 a b^3 \left(c^2 f (B f+8 C e)+2 c d e (3 B f+4 C e)+B d^2 e^2\right)-b^4 \left(-\left(c^2 \left(-A f^2+4 B e f+8 C e^2\right)\right)-2 c d e (A f+2 B e)+A d^2 e^2\right)\right)}{4 b^4 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} (e+f x)^{3/2} \left(A b^2-a (b B-a C)\right)}{2 b (a+b x)^2 (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) (6 a C d f-b (2 B d f+c C f+C d e))}{b^4 \sqrt{d} \sqrt{f}}",1,"-1/2*((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(b^2*(b*e - a*f)*(a + b*x)^2) - ((b*B - 2*a*C)*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x)) + (2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b^3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*C*(b*c - a*d)*((Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(b*d*Sqrt[e + f*x]) - (Sqrt[-(b*e) + a*f]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b^3 - ((A*b^2 - a*(b*B - a*C))*(d*e - c*f)*((Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(a + b*x)) - ((d*e - c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*Sqrt[-(b*e) + a*f])))/(4*b^2*(b*e - a*f)) - ((b*B - 2*a*C)*((-4*f*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + ((2*a*b*d*f + (b*(-2*a*d*f - b*(d*e + c*f)))/2)*((2*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(b*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) - ((-(b*c) + a*d)*((2*Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(b*d^(3/2)*Sqrt[e + f*x]) - (2*Sqrt[-(b*e) + a*f]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b*Sqrt[-(b*c) + a*d])))/b))/b))/(b^2*(b*c - a*d)*(b*e - a*f))","B",0
47,1,3220,1032,6.7022236,"\int \frac{(a+b x)^2 \sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{e+f x}} \, dx","Integrate[((a + b*x)^2*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]","\text{Result too large to show}","\frac{C (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^3}{5 b d f}-\frac{(4 a C d f+b (9 C d e+7 c C f-10 B d f)) (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^2}{40 b d^2 f^2}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(\left(C \left(315 d^3 e^3+203 c d^2 f e^2+145 c^2 d f^2 e+105 c^3 f^3\right)+10 d f \left(8 A d f (5 d e+3 c f)-B \left(35 d^2 e^2+22 c d f e+15 c^2 f^2\right)\right)\right) b^3+20 a d f \left(8 d f (5 B d e+3 B c f-6 A d f)-C \left(35 d^2 e^2+22 c d f e+15 c^2 f^2\right)\right) b^2+8 a^2 d^2 f^2 (23 C d e+9 c C f-30 B d f) b+4 d f (8 b d f (6 b c C e+3 a C d e+a c C f-10 A b d f)-(7 b d e+5 b c f-4 a d f) (4 a C d f+b (9 C d e+7 c C f-10 B d f))) x b+96 a^3 C d^3 f^3\right)}{960 b d^4 f^4}+\frac{(d e-c f) \left(-\left(\left(C \left(63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right)+2 d f \left(8 A d f \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)-B \left(35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right)\right)\right) b^2\right)+4 a d f \left(C \left(35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right)+8 d f \left(2 A d f (3 d e+c f)-B \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)\right)\right) b+16 a^2 d^2 f^2 \left(2 d f (3 B d e+B c f-4 A d f)-C \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right)}{128 d^{9/2} f^{11/2}}-\frac{\left(-\left(\left(C \left(63 d^4 e^4+28 c d^3 f e^3+18 c^2 d^2 f^2 e^2+12 c^3 d f^3 e+7 c^4 f^4\right)+2 d f \left(8 A d f \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)-B \left(35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right)\right)\right) b^2\right)+4 a d f \left(C \left(35 d^3 e^3+15 c d^2 f e^2+9 c^2 d f^2 e+5 c^3 f^3\right)+8 d f \left(2 A d f (3 d e+c f)-B \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)\right)\right) b+16 a^2 d^2 f^2 \left(2 d f (3 B d e+B c f-4 A d f)-C \left(5 d^2 e^2+2 c d f e+c^2 f^2\right)\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{128 d^4 f^5}",1,"((-(b*e) + a*f)^2*(d*e - c*f)^2*(C*e^2 - B*e*f + A*f^2)*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))]*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(2*d^3*f^6*Sqrt[c + d*x]*Sqrt[e + f*x]) + (2*b^2*C*(d*e - c*f)^3*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2)*((3*(35/(64*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 35/(48*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/10 + (21*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(512*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4)))/(3*d^4*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*b*(d*e - c*f)^2*(-4*b*C*e + b*B*f + 2*a*C*f)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2)*((3*(5/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 5/(6*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1)))/8 + (15*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(256*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3)))/(3*d^3*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(5/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)*(6*b^2*C*e^2 - 3*b^2*B*e*f - 6*a*b*C*e*f + A*b^2*f^2 + 2*a*b*B*f^2 + a^2*C*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/2 + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(32*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2)))/(3*d^2*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)*(4*b*C*e^2 - 3*b*B*e*f - 2*a*C*e*f + 2*A*b*f^2 + a*B*f^2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(3/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (3*(d*e - c*f)^2*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))^2*((2*d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))) - (2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*Sqrt[1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))])))/(16*d^2*f^2*(c + d*x)^2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))))))/(3*d*f^4*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)])","B",1
48,1,478,540,3.5394283,"\int \frac{(a+b x) \sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{e+f x}} \, dx","Integrate[((a + b*x)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]","\frac{d \sqrt{f} \sqrt{c+d x} (e+f x) \left(8 a d f \left(6 d f (4 A d f+B (c f-3 d e+2 d f x))+C \left(-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left(15 e^2-10 e f x+8 f^2 x^2\right)\right)\right)+b \left(8 d f \left(6 A d f (c f-3 d e+2 d f x)+B \left(-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left(15 e^2-10 e f x+8 f^2 x^2\right)\right)\right)+C \left(15 c^3 f^3+c^2 d f^2 (17 e-10 f x)+c d^2 f \left(25 e^2-12 e f x+8 f^2 x^2\right)+d^3 \left(-105 e^3+70 e^2 f x-56 e f^2 x^2+48 f^3 x^3\right)\right)\right)\right)+3 (d e-c f)^{3/2} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right) \left(b \left(8 d f \left(2 A d f (c f+3 d e)-B \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)+C \left(5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right)\right)-8 a d f \left(2 d f (4 A d f-B c f-3 B d e)+C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)\right)}{192 d^4 f^{9/2} \sqrt{e+f x}}","-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left(8 d f (-6 A d f+3 B c f+5 B d e)-C \left(15 c^2 f^2+22 c d e f+35 d^2 e^2\right)\right)\right)}{96 b d^3 f^3}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(8 a d f \left(2 d f (-4 A d f+B c f+3 B d e)-C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)+b \left(8 d f \left(2 A d f (c f+3 d e)-B \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)+C \left(5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right)\right)\right)}{64 d^3 f^4}+\frac{(d e-c f) \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(8 a d f \left(2 d f (-4 A d f+B c f+3 B d e)-C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)+b \left(8 d f \left(2 A d f (c f+3 d e)-B \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)+C \left(5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right)\right)\right)}{64 d^{7/2} f^{9/2}}+\frac{C (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{4 b d f}",1,"(d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(8*a*d*f*(6*d*f*(4*A*d*f + B*(-3*d*e + c*f + 2*d*f*x)) + C*(-3*c^2*f^2 + 2*c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))) + b*(C*(15*c^3*f^3 + c^2*d*f^2*(17*e - 10*f*x) + c*d^2*f*(25*e^2 - 12*e*f*x + 8*f^2*x^2) + d^3*(-105*e^3 + 70*e^2*f*x - 56*e*f^2*x^2 + 48*f^3*x^3)) + 8*d*f*(6*A*d*f*(-3*d*e + c*f + 2*d*f*x) + B*(-3*c^2*f^2 + 2*c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))))) + 3*(d*e - c*f)^(3/2)*(-8*a*d*f*(2*d*f*(-3*B*d*e - B*c*f + 4*A*d*f) + C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(192*d^4*f^(9/2)*Sqrt[e + f*x])","A",1
49,1,225,246,1.0697593,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]","\frac{-d \sqrt{f} \sqrt{c+d x} (e+f x) \left(C \left(3 c^2 f^2-2 c d f (f x-2 e)+d^2 \left(-15 e^2+10 e f x-8 f^2 x^2\right)\right)-6 d f (4 A d f+B (c f-3 d e+2 d f x))\right)-3 (d e-c f)^{3/2} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right) \left(2 d f (4 A d f-B (c f+3 d e))+C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{24 d^3 f^{7/2} \sqrt{e+f x}}","\frac{\sqrt{c+d x} \sqrt{e+f x} \left(2 d f (4 A d f-B (c f+3 d e))+C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{8 d^2 f^3}-\frac{(d e-c f) \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(2 d f (4 A d f-B (c f+3 d e))+C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)}{8 d^{5/2} f^{7/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} (-6 B d f+7 c C f+5 C d e)}{12 d^2 f^2}+\frac{C (c+d x)^{5/2} \sqrt{e+f x}}{3 d^2 f}",1,"(-(d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(-6*d*f*(4*A*d*f + B*(-3*d*e + c*f + 2*d*f*x)) + C*(3*c^2*f^2 - 2*c*d*f*(-2*e + f*x) + d^2*(-15*e^2 + 10*e*f*x - 8*f^2*x^2)))) - 3*(d*e - c*f)^(3/2)*(C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + 2*d*f*(4*A*d*f - B*(3*d*e + c*f)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(24*d^3*f^(7/2)*Sqrt[e + f*x])","A",1
50,1,465,290,3.4488203,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x) \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)*Sqrt[e + f*x]),x]","\frac{\frac{8 \sqrt{d e-c f} \left(a (a C-b B)+A b^2\right) \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{f} \sqrt{e+f x}}-\frac{8 \sqrt{a d-b c} \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a f-b e}}+\frac{4 b \sqrt{e+f x} (a C f-b B f+b C e) \left(\sqrt{c+d x} (d e-c f) \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)-\sqrt{f} (c+d x) \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}\right)}{f^{5/2} \sqrt{c+d x} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{b^2 C \sqrt{e+f x} \left(\sqrt{f} \sqrt{c+d x} (c f+d (e+2 f x))-\frac{(d e-c f)^{3/2} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{\frac{d (e+f x)}{d e-c f}}}\right)}{d f^{5/2}}}{4 b^3}","\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) (2 b d f (4 A b d f-a C (c f+3 d e))+(2 a d f-b c f+b d e) (4 a C d f+b (-4 B d f+c C f+3 C d e)))}{4 b^3 d^{3/2} f^{5/2}}-\frac{2 \sqrt{b c-a d} \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right)}{b^3 \sqrt{b e-a f}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-4 B d f+c C f+3 C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 b d f}",1,"((8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[e + f*x]) + (4*b*(b*C*e - b*B*f + a*C*f)*Sqrt[e + f*x]*(-(Sqrt[f]*Sqrt[d*e - c*f]*(c + d*x)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (d*e - c*f)*Sqrt[c + d*x]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]]))/(f^(5/2)*Sqrt[d*e - c*f]*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (b^2*C*Sqrt[e + f*x]*(Sqrt[f]*Sqrt[c + d*x]*(c*f + d*(e + 2*f*x)) - ((d*e - c*f)^(3/2)*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/Sqrt[(d*(e + f*x))/(d*e - c*f)]))/(d*f^(5/2)) - (8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[-(b*c) + a*d]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[-(b*e) + a*f])/(4*b^3)","A",1
51,1,417,364,2.3961053,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^2 \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^2*Sqrt[e + f*x]),x]","\frac{-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left(a (a C-b B)+A b^2\right)}{(a+b x) (b e-a f)}+\frac{2 b (c f-d e) \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} (a f-b e)^{3/2}}+\frac{4 (b B-2 a C) \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{f} \sqrt{e+f x}}-\frac{4 (b B-2 a C) \sqrt{a d-b c} \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a f-b e}}+\frac{2 b C \sqrt{e+f x} \left(\sqrt{f} \sqrt{c+d x}-\frac{\sqrt{d e-c f} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{\frac{d (e+f x)}{d e-c f}}}\right)}{f^{3/2}}}{2 b^3}","\frac{\sqrt{c+d x} \sqrt{e+f x} \left(2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right)}{b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right)}{b^3 \sqrt{b c-a d} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{b (a+b x) (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) (4 a C d f+b (-2 B d f-c C f+C d e))}{b^3 \sqrt{d} f^{3/2}}",1,"((-2*b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + (4*(b*B - 2*a*C)*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[e + f*x]) + (2*b*C*Sqrt[e + f*x]*(Sqrt[f]*Sqrt[c + d*x] - (Sqrt[d*e - c*f]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/Sqrt[(d*(e + f*x))/(d*e - c*f)]))/f^(3/2) - (4*(b*B - 2*a*C)*Sqrt[-(b*c) + a*d]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[-(b*e) + a*f] + (2*b*(A*b^2 + a*(-(b*B) + a*C))*(-(d*e) + c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2)))/(2*b^3)","A",1
52,1,523,484,5.6745779,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^3 \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^3*Sqrt[e + f*x]),x]","-\frac{\frac{2 b^2 (c+d x)^{3/2} \sqrt{e+f x} \left(a (a C-b B)+A b^2\right)}{(a+b x)^2 (b c-a d) (b e-a f)}+\frac{b \left(a (a C-b B)+A b^2\right) (-4 a d f+3 b c f+b d e) \left(\sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} \sqrt{a f-b e}-(a+b x) (d e-c f) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)\right)}{(a+b x) (a d-b c)^{3/2} (a f-b e)^{5/2}}+\frac{4 b \sqrt{c+d x} \sqrt{e+f x} (b B-2 a C)}{(a+b x) (b e-a f)}-\frac{4 b (b B-2 a C) (c f-d e) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} (a f-b e)^{3/2}}+\frac{8 C \sqrt{a d-b c} \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a f-b e}}-\frac{8 C \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{f} \sqrt{e+f x}}}{4 b^3}","\frac{\sqrt{c+d x} \sqrt{e+f x} \left(4 a^3 C d f-a^2 b C (5 c f+7 d e)+a b^2 (-4 A d f+B c f+3 B d e+8 c C e)-b^3 (-3 A c f-A d e+4 B c e)\right)}{4 b^2 (a+b x) (b c-a d) (b e-a f)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(8 a^4 C d^2 f^2-4 a^3 b C d f (3 c f+5 d e)+3 a^2 b^2 C \left(c^2 f^2+10 c d e f+5 d^2 e^2\right)-a b^3 \left(2 c d \left(2 A f^2-B e f+12 C e^2\right)+d^2 e (3 B e-4 A f)+c^2 f (8 C e-B f)\right)-b^4 \left(-\left(c^2 \left(3 A f^2-4 B e f+8 C e^2\right)\right)-2 c d e (2 B e-A f)+A d^2 e^2\right)\right)}{4 b^3 (b c-a d)^{3/2} (b e-a f)^{5/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{2 b (a+b x)^2 (b c-a d) (b e-a f)}+\frac{2 C \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right)}{b^3 \sqrt{f}}",1,"-1/4*((4*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + (2*b^2*(A*b^2 + a*(-(b*B) + a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (8*C*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[e + f*x]) + (8*C*Sqrt[-(b*c) + a*d]*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[-(b*e) + a*f] - (4*b*(b*B - 2*a*C)*(-(d*e) + c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2)) + (b*(A*b^2 + a*(-(b*B) + a*C))*(b*d*e + 3*b*c*f - 4*a*d*f)*(Sqrt[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] - (d*e - c*f)*(a + b*x)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])]))/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(5/2)*(a + b*x)))/b^3","A",1
53,1,729,685,6.336256,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^4 \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^4*Sqrt[e + f*x]),x]","-\frac{\left(a^2 C-a b B+A b^2\right) \left(-\frac{3 \left(8 a^2 d^2 f^2-4 a b d f (3 c f+d e)+b^2 \left(5 c^2 f^2+2 c d e f+d^2 e^2\right)\right) \left(\frac{\sqrt{c+d x} \sqrt{e+f x}}{(a+b x) (a f-b e)}-\frac{(d e-c f) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} (a f-b e)^{3/2}}\right)}{8 (b c-a d) (b e-a f)}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(\frac{1}{2} b (-6 a d f+5 b c f+3 b d e)-a b d f\right)}{2 (a+b x)^2 (b c-a d) (b e-a f)}\right)}{3 b^2 (b c-a d) (b e-a f)}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{3 b (a+b x)^3 (b c-a d) (b e-a f)}+\frac{(b B-2 a C) (-4 a d f+3 b c f+b d e) \left(\frac{\sqrt{c+d x} \sqrt{e+f x}}{(a+b x) (b e-a f)}+\frac{(d e-c f) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} (a f-b e)^{3/2}}\right)}{4 b^2 (b c-a d) (b e-a f)}-\frac{C \sqrt{c+d x} \sqrt{e+f x}}{b^2 (a+b x) (b e-a f)}-\frac{C (d e-c f) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{b^2 \sqrt{a d-b c} (a f-b e)^{3/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} (b B-2 a C)}{2 b (a+b x)^2 (b c-a d) (b e-a f)}","-\frac{(d e-c f) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(-\left(a^2 \left(2 d f (-4 A d f+B c f+3 B d e)-C \left(c^2 f^2+2 c d e f+5 d^2 e^2\right)\right)\right)+a b \left(-2 c d \left(6 A f^2-7 B e f+6 C e^2\right)+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right)+b^2 \left(c^2 \left(5 A f^2-6 B e f+8 C e^2\right)-2 c d e (B e-A f)+A d^2 e^2\right)\right)}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left(4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right)}{12 b^2 (a+b x)^2 (b c-a d) (b e-a f)^2}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left(4 d f (-2 A d f+B c f+4 B d e)-C \left(3 c^2 f^2+44 c d e f+33 d^2 e^2\right)\right)-a b^3 \left(2 c d \left(13 A f^2-14 B e f+30 C e^2\right)+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right)-b^4 \left(-3 c^2 \left(5 A f^2-6 B e f+8 C e^2\right)-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right)\right)}{24 b^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{3 b (a+b x)^3 (b c-a d) (b e-a f)}",1,"-((C*Sqrt[c + d*x]*Sqrt[e + f*x])/(b^2*(b*e - a*f)*(a + b*x))) - ((A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) - ((b*B - 2*a*C)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (C*(d*e - c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(b^2*Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2)) + ((b*B - 2*a*C)*(b*d*e + 3*b*c*f - 4*a*d*f)*((Sqrt[c + d*x]*Sqrt[e + f*x])/((b*e - a*f)*(a + b*x)) + ((d*e - c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2))))/(4*b^2*(b*c - a*d)*(b*e - a*f)) - ((A*b^2 - a*b*B + a^2*C)*(-1/2*((-(a*b*d*f) + (b*(3*b*d*e + 5*b*c*f - 6*a*d*f))/2)*(c + d*x)^(3/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (3*(8*a^2*d^2*f^2 - 4*a*b*d*f*(d*e + 3*c*f) + b^2*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*((Sqrt[c + d*x]*Sqrt[e + f*x])/((-(b*e) + a*f)*(a + b*x)) - ((d*e - c*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*(-(b*e) + a*f)^(3/2))))/(8*(b*c - a*d)*(b*e - a*f))))/(3*b^2*(b*c - a*d)*(b*e - a*f))","A",1
54,1,2195,718,6.4866432,"\int \frac{(a+b x)^2 \left(A+B x+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[((a + b*x)^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(16 a^2 d^2 f^2 \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)-16 a b d f \left(2 d f \left(4 A d f (c f+d e)-B \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)+C \left(5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right)\right)+b^2 \left(8 d f \left(2 A d f \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)-B \left(5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right)\right)+C \left(35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right)\right)\right)}{64 d^{9/2} f^{9/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left(6 d f (4 A d f-3 B (c f+d e))+C \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)+(4 a d f-5 b (c f+d e)) (2 a C d f-b (8 B d f-7 C (c f+d e))))+b^3 \left(8 d f \left(18 A d f (c f+d e)-B \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)+5 C \left(21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right)\right)\right)}{192 b d^4 f^4}-\frac{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} (2 a C d f-b (8 B d f-7 C (c f+d e)))}{24 b d^2 f^2}+\frac{C (a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}}{4 b d f}",1,"(2*(b*e - a*f)^2*Sqrt[d*e - c*f]*(C*e^2 - f*(B*e - A*f))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(d*f^(9/2)*Sqrt[e + f*x]) + (2*b^2*C*(d*e - c*f)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2)*((35/(16*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^4) + 35/(24*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 7/(6*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/8 + (35*Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(128*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(9/2))))/(d^4*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(7/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*b*(d*e - c*f)^2*(-4*b*C*e + b*B*f + 2*a*C*f)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2)*((15/(8*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^3) + 5/(4*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/6 + (5*Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(16*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(7/2))))/(d^3*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(5/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(d*e - c*f)*(6*b^2*C*e^2 - 3*b^2*B*e*f - 6*a*b*C*e*f + A*b^2*f^2 + 2*a*b*B*f^2 + a^2*C*f^2)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2)*((3/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^2) + (1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(-1))/4 + (3*Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(8*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(5/2))))/(d^2*f^4*(d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)))^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(-(b*e) + a*f)*(4*b*C*e^2 - 3*b*B*e*f - 2*a*C*e*f + 2*A*b*f^2 + a*B*f^2)*Sqrt[c + d*x]*Sqrt[e + f*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2)*(1/(2*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))) + (Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)]*ArcSinh[(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d*e - c*f]*Sqrt[(d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f)])])/(2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*(1 + (d*f*(c + d*x))/((d*e - c*f)*((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))))^(3/2))))/(d*f^4*Sqrt[d/((d^2*e)/(d*e - c*f) - (c*d*f)/(d*e - c*f))]*Sqrt[(d*(e + f*x))/(d*e - c*f)])","B",1
55,1,379,371,1.9627357,"\int \frac{(a+b x) \left(A+B x+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[((a + b*x)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{\sqrt{e+f x} \left(3 \sqrt{d e-c f} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right) \left(b \left(2 d f \left(4 A d f (c f+d e)-B \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)+C \left(5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right)\right)-2 a d f \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)\right)-\frac{d \sqrt{f} \sqrt{c+d x} (e+f x) \left(6 a d f (4 B d f+C (-3 c f-3 d e+2 d f x))+b \left(6 d f (4 A d f+B (-3 c f-3 d e+2 d f x))+C \left(15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left(15 e^2-10 e f x+8 f^2 x^2\right)\right)\right)\right)}{\sqrt{\frac{d (e+f x)}{d e-c f}}}\right)}{24 d^3 f^{7/2} (c f-d e) \sqrt{\frac{d (e+f x)}{d e-c f}}}","-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(8 a^2 C d^2 f^2+2 b d f x (2 a C d f-b (6 B d f-5 C (c f+d e)))-6 a b d f (4 B d f-3 C (c f+d e))-\left(b^2 \left(6 d f (4 A d f-3 B (c f+d e))+C \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)\right)\right)}{24 b d^3 f^3}+\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(2 a d f \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)-b \left(2 d f \left(4 A d f (c f+d e)-B \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)+C \left(5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right)\right)\right)}{8 d^{7/2} f^{7/2}}+\frac{C (a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}",1,"(Sqrt[e + f*x]*(-((d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(6*a*d*f*(4*B*d*f + C*(-3*d*e - 3*c*f + 2*d*f*x)) + b*(6*d*f*(4*A*d*f + B*(-3*d*e - 3*c*f + 2*d*f*x)) + C*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2)))))/Sqrt[(d*(e + f*x))/(d*e - c*f)]) + 3*Sqrt[d*e - c*f]*(-2*a*d*f*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) + b*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))))*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]]))/(24*d^3*f^(7/2)*(-(d*e) + c*f)*Sqrt[(d*(e + f*x))/(d*e - c*f)])","A",1
56,1,173,164,0.7921211,"\int \frac{A+B x+C x^2}{\sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{\sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right) \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)+d \sqrt{f} \sqrt{c+d x} (e+f x) (4 B d f+C (-3 c f-3 d e+2 d f x))}{4 d^3 f^{5/2} \sqrt{e+f x}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)}{4 d^{5/2} f^{5/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (-4 B d f+5 c C f+3 C d e)}{4 d^2 f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 d^2 f}",1,"(d*Sqrt[f]*Sqrt[c + d*x]*(e + f*x)*(4*B*d*f + C*(-3*d*e - 3*c*f + 2*d*f*x)) + Sqrt[d*e - c*f]*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(4*d^3*f^(5/2)*Sqrt[e + f*x])","A",1
57,1,304,188,0.9382715,"\int \frac{A+B x+C x^2}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{2 \left(\frac{\left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} \sqrt{a f-b e}}-\frac{\sqrt{e+f x} (a C f-b B f+b C e) \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{f^{3/2} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{b C \sqrt{e+f x} \left(\sqrt{f} \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{d e-c f}}+\sqrt{d e-c f} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)\right)}{2 d f^{3/2} \sqrt{\frac{d (e+f x)}{d e-c f}}}\right)}{b^2}","-\frac{2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right)}{b^2 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right) (2 a C d f+b (-2 B d f+c C f+C d e))}{b^2 d^{3/2} f^{3/2}}+\frac{C \sqrt{c+d x} \sqrt{e+f x}}{b d f}",1,"(2*(-(((b*C*e - b*B*f + a*C*f)*Sqrt[e + f*x]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(f^(3/2)*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)])) + (b*C*Sqrt[e + f*x]*(Sqrt[f]*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)] + Sqrt[d*e - c*f]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]]))/(2*d*f^(3/2)*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + ((A*b^2 + a*(-(b*B) + a*C))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*Sqrt[-(b*e) + a*f])))/b^2","A",1
58,1,325,254,1.8631095,"\int \frac{A+B x+C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{-\frac{b \sqrt{c+d x} \sqrt{e+f x} \left(a (a C-b B)+A b^2\right)}{(a+b x) (b c-a d) (b e-a f)}-\frac{\left(a (a C-b B)+A b^2\right) (-2 a d f+b c f+b d e) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac{2 (b B-2 a C) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} \sqrt{a f-b e}}+\frac{2 C \sqrt{e+f x} \sinh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right)}{\sqrt{f} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}}{b^2}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(2 a^3 C d f-3 a^2 b C (c f+d e)+a b^2 (-2 A d f+B c f+B d e+4 c C e)-b^3 (-A c f-A d e+2 B c e)\right)}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{b (a+b x) (b c-a d) (b e-a f)}+\frac{2 C \tanh ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right)}{b^2 \sqrt{d} \sqrt{f}}",1,"(-((b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*C*Sqrt[e + f*x]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]) + (2*(b*B - 2*a*C)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]) - ((A*b^2 + a*(-(b*B) + a*C))*(b*d*e + b*c*f - 2*a*d*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2)))/b^2","A",1
59,1,512,424,2.0902079,"\int \frac{A+B x+C x^2}{(a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{\frac{\left(a (a C-b B)+A b^2\right) \left(\frac{\left(8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac{3 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d) (b e-a f)}\right)}{(b c-a d) (b e-a f)}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left(a (a C-b B)+A b^2\right)}{(a+b x)^2 (b c-a d) (b e-a f)}-\frac{4 b \sqrt{c+d x} \sqrt{e+f x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f)}-\frac{4 (b B-2 a C) (-2 a d f+b c f+b d e) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac{8 C \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{\sqrt{a d-b c} \sqrt{a f-b e}}}{4 b^2}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right) \left(a^2 \left(4 d f (2 A d f-B (c f+d e))+C \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right)+a b \left(-2 c d \left(4 A f^2-7 B e f+4 C e^2\right)+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right)+b^2 \left(c^2 \left(3 A f^2-4 B e f+8 C e^2\right)-2 c d e (2 B e-A f)+3 A d^2 e^2\right)\right)}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left(2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right)}{4 b (a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{2 b (a+b x)^2 (b c-a d) (b e-a f)}",1,"((-2*b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) - (4*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (8*C*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-(b*c) + a*d]*Sqrt[-(b*e) + a*f]) - (4*(b*B - 2*a*C)*(b*d*e + b*c*f - 2*a*d*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2)) + ((A*b^2 + a*(-(b*B) + a*C))*((3*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((8*a^2*d^2*f^2 - 8*a*b*d*f*(d*e + c*f) + b^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2))))/((b*c - a*d)*(b*e - a*f)))/(4*b^2)","A",1
60,1,794,826,6.1074051,"\int \frac{A+B x+C x^2}{(a+b x)^4 \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^4*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","-\frac{\frac{\left(a (a C-b B)+A b^2\right) \left(\frac{b \sqrt{c+d x} \sqrt{e+f x} \left(44 a^2 d^2 f^2-44 a b d f (c f+d e)+b^2 \left(15 c^2 f^2+14 c d e f+15 d^2 e^2\right)\right)}{(a+b x) (b c-a d) (b e-a f)}+\frac{3 \left(8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left(5 c^2 f^2-2 c d e f+5 d^2 e^2\right)\right) (-2 a d f+b c f+b d e) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}-\frac{10 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{(a+b x)^2}\right)}{2 (b c-a d)^2 (b e-a f)^2}-\frac{3 (b B-2 a C) \left(\frac{\left(8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left(3 c^2 f^2+2 c d e f+3 d^2 e^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac{3 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d) (b e-a f)}\right)}{(b c-a d) (b e-a f)}+\frac{4 b \sqrt{c+d x} \sqrt{e+f x} \left(a (a C-b B)+A b^2\right)}{(a+b x)^3 (b c-a d) (b e-a f)}+\frac{6 b \sqrt{c+d x} \sqrt{e+f x} (b B-2 a C)}{(a+b x)^2 (b c-a d) (b e-a f)}+\frac{12 b C \sqrt{c+d x} \sqrt{e+f x}}{(a+b x) (b c-a d) (b e-a f)}+\frac{12 C (-2 a d f+b c f+b d e) \tanh ^{-1}\left(\frac{\sqrt{c+d x} \sqrt{a f-b e}}{\sqrt{e+f x} \sqrt{a d-b c}}\right)}{(a d-b c)^{3/2} (a f-b e)^{3/2}}}{12 b^2}","-\frac{\sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left(-2 d f \left(C \left(3 d^2 e^2+2 c d f e+3 c^2 f^2\right)+4 d f (2 A d f-B (d e+c f))\right) a^3+b \left(C \left(d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right)+4 d f \left(6 A d f (d e+c f)-B \left(d^2 e^2+10 c d f e+c^2 f^2\right)\right)\right) a^2+b^2 \left(-f^2 (4 C e-B f) c^3-d f \left(40 C e^2-23 B f e+18 A f^2\right) c^2-d^2 e \left(4 C e^2-23 B f e+12 A f^2\right) c+d^3 e^2 (B e-18 A f)\right) a+b^3 \left(f \left(8 C e^2-6 B f e+5 A f^2\right) c^3+d e \left(8 C e^2-4 B f e+3 A f^2\right) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right)}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac{\left(4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left(C \left(3 d^2 e^2-34 c d f e+3 c^2 f^2\right)+2 d f (22 A d f-5 B (d e+c f))\right) a^2-b^3 \left(-3 f (4 C e-B f) c^2-2 d \left(6 C e^2-29 B f e+22 A f^2\right) c+d^2 e (3 B e-44 A f)\right) a-b^4 \left(3 \left(8 C e^2-6 B f e+5 A f^2\right) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac{\left(2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}",1,"-1/12*((4*b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3) + (6*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) + (12*b*C*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (12*C*(b*d*e + b*c*f - 2*a*d*f)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2)) - (3*(b*B - 2*a*C)*((3*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((8*a^2*d^2*f^2 - 8*a*b*d*f*(d*e + c*f) + b^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2))))/((b*c - a*d)*(b*e - a*f)) + ((A*b^2 + a*(-(b*B) + a*C))*((-10*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/(a + b*x)^2 + (b*(44*a^2*d^2*f^2 - 44*a*b*d*f*(d*e + c*f) + b^2*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (3*(b*d*e + b*c*f - 2*a*d*f)*(8*a^2*d^2*f^2 - 8*a*b*d*f*(d*e + c*f) + b^2*(5*d^2*e^2 - 2*c*d*e*f + 5*c^2*f^2))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2))))/(2*(b*c - a*d)^2*(b*e - a*f)^2))/b^2","A",1
61,1,11933,1182,17.3012732,"\int \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right) \, dx","Integrate[Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2),x]","\text{Result too large to show}","\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} (e+f x)^{3/2}}{9 b d f}-\frac{2 (2 a C d f-b (3 B d f-2 C (d e+c f))) \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{21 b d^2 f^2}-\frac{2 (7 b d f (b c C e+a C d e+a c C f-3 A b d f)+(a d f-4 b (d e+c f)) (2 a C d f-b (3 B d f-2 C (d e+c f)))) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{105 b^2 d^2 f^3}-\frac{2 \sqrt{a d-b c} \left(\left(2 C \left(8 d^4 e^4-4 c d^3 f e^3-3 c^2 d^2 f^2 e^2-4 c^3 d f^3 e+8 c^4 f^4\right)+3 d f \left(14 A d f \left(d^2 e^2-c d f e+c^2 f^2\right)-B \left(8 d^3 e^3-5 c d^2 f e^2-5 c^2 d f^2 e+8 c^3 f^3\right)\right)\right) b^4-a d f \left(C \left(8 d^3 e^3-6 c d^2 f e^2-6 c^2 d f^2 e+8 c^3 f^3\right)+3 d f \left(14 A d f (d e+c f)-B \left(5 d^2 e^2-6 c d f e+5 c^2 f^2\right)\right)\right) b^3+3 a^2 d^2 f^2 \left(d f (5 B d e+5 B c f+14 A d f)-2 C \left(d^2 e^2-c d f e+c^2 f^2\right)\right) b^2-8 a^3 d^3 f^3 (C d e+c C f+3 B d f) b+16 a^4 C d^4 f^4\right) \sqrt{\frac{b (c+d x)}{b c-a d}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right) \sqrt{e+f x}}{315 b^4 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \left(-\left(\left(C \left(16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right)+3 d f \left(7 A d f (2 d e-c f)-B \left(8 d^2 e^2-c d f e-4 c^2 f^2\right)\right)\right) b^3\right)-3 a d f^2 \left(\left(c^2 C-7 A d^2\right) f+B d (d e-2 c f)\right) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{315 b^3 d^3 f^3}-\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \left(-\left(\left(C \left(16 d^3 e^3-3 c^2 d f^2 e-8 c^3 f^3\right)+3 d f \left(7 A d f (2 d e-c f)-B \left(8 d^2 e^2-c d f e-4 c^2 f^2\right)\right)\right) b^3\right)-3 a d f^2 \left(\left(c^2 C-7 A d^2\right) f+B d (d e-2 c f)\right) b^2+3 a^2 d^2 f^2 (C d e-c C f-4 B d f) b+8 a^3 C d^3 f^3\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{315 b^4 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","C",1
62,1,917,774,13.3891221,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{\sqrt{a+b x}} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/Sqrt[a + b*x],x]","-\frac{2 \left(\sqrt{\frac{b c}{d}-a} \left(\left(C \left(-8 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e-8 c^3 f^3\right)-7 d f \left(5 A d f (d e+c f)-2 B \left(d^2 e^2-c d f e+c^2 f^2\right)\right)\right) b^3+a d f \left(7 d f (3 B d e+3 B c f+10 A d f)+C \left(-9 d^2 e^2+8 c d f e-9 c^2 f^2\right)\right) b^2-8 a^2 d^2 f^2 (7 B d f+2 C (d e+c f)) b+48 a^3 C d^3 f^3\right) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left(\left(C \left(\left(4 e^2-3 f x e-15 f^2 x^2\right) d^2-c f (2 e+3 f x) d+4 c^2 f^2\right)-7 d f (B c f+5 A d f+B d (e+3 f x))\right) b^2+a d f (28 B d f+C (5 d e+5 c f+18 d f x)) b-24 a^2 C d^2 f^2\right) b^2-i (b c-a d) f (d e-c f) \left(\left(7 d f (B d e-2 B c f+5 A d f)-C \left(4 d^2 e^2+c d f e-8 c^2 f^2\right)\right) b^2+a d f (-5 C d e+13 c C f-28 B d f) b+24 a^2 C d^2 f^2\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right) b+i (b c-a d) f \left(\left(C \left(-8 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e-8 c^3 f^3\right)-7 d f \left(5 A d f (d e+c f)-2 B \left(d^2 e^2-c d f e+c^2 f^2\right)\right)\right) b^3+a d f \left(7 d f (3 B d e+3 B c f+10 A d f)+C \left(-9 d^2 e^2+8 c d f e-9 c^2 f^2\right)\right) b^2-8 a^2 d^2 f^2 (7 B d f+2 C (d e+c f)) b+48 a^3 C d^3 f^3\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)\right)}{105 b^5 \sqrt{\frac{b c}{d}-a} d^3 f^3 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}","-\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)-\left(b^2 \left(7 d f (-5 A d f-B c f+2 B d e)-C \left(-4 c^2 f^2-c d e f+8 d^2 e^2\right)\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))-(3 a c f+a d e+b c e) (6 a C d f-b (7 B d f-4 C (c f+d e))))+2 \left(\frac{b d e}{2}-f (a d+b c)\right) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))}{105 b^3 d^2 f^2}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} (6 a C d f-b (7 B d f-4 C (c f+d e)))}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}",1,"(-2*(b^2*Sqrt[-a + (b*c)/d]*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f + 2*C*(d*e + c*f)) + a*b^2*d*f*(7*d*f*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(-9*d^2*e^2 + 8*c*d*e*f - 9*c^2*f^2)) + b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(d^2*e^2 - c*d*e*f + c^2*f^2))))*(c + d*x)*(e + f*x) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(-24*a^2*C*d^2*f^2 + a*b*d*f*(28*B*d*f + C*(5*d*e + 5*c*f + 18*d*f*x)) + b^2*(-7*d*f*(B*c*f + 5*A*d*f + B*d*(e + 3*f*x)) + C*(4*c^2*f^2 - c*d*f*(2*e + 3*f*x) + d^2*(4*e^2 - 3*e*f*x - 15*f^2*x^2)))) + I*(b*c - a*d)*f*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f + 2*C*(d*e + c*f)) + a*b^2*d*f*(7*d*f*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(-9*d^2*e^2 + 8*c*d*e*f - 9*c^2*f^2)) + b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(d^2*e^2 - c*d*e*f + c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*(b*c - a*d)*f*(d*e - c*f)*(24*a^2*C*d^2*f^2 + a*b*d*f*(-5*C*d*e + 13*c*C*f - 28*B*d*f) + b^2*(7*d*f*(B*d*e - 2*B*c*f + 5*A*d*f) - C*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(105*b^5*Sqrt[-a + (b*c)/d]*d^3*f^3*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
63,1,633,706,8.1251705,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^{3/2}} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(3/2),x]","-\frac{2 \left(-i b f (a+b x)^{3/2} (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(24 a^2 C d^2 f-a b d (20 B d f+7 c C f+C d e)+b^2 \left(15 A d^2 f+c d (5 B f+C e)-2 c^2 C f\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+b^2 (c+d x) (e+f x) \left(-\sqrt{\frac{b c}{d}-a}\right) \left(48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left(5 d f (6 A d f+B c f+B d e)-2 C \left(c^2 f^2-c d e f+d^2 e^2\right)\right)\right)-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left(5 d f (6 A d f+B c f+B d e)-2 C \left(c^2 f^2-c d e f+d^2 e^2\right)\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)+b^2 d f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(15 d f \left(a (a C-b B)+A b^2\right)-(a+b x) (b (5 B d f+c C f+C d e)-9 a C d f)-3 b C d f x (a+b x)\right)\right)}{15 b^5 d^2 f^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a}}","-\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} \left(6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right)}{5 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left(5 d f (6 A d f+B c f+B d e)-2 C \left(c^2 f^2-c d e f+d^2 e^2\right)\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left(24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right)}{15 b^3 d f (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left(A b^2-a (b B-a C)\right)}{b \sqrt{a+b x} (b c-a d) (b e-a f)}",1,"(-2*(-(b^2*Sqrt[-a + (b*c)/d]*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*x)*(e + f*x)*(15*(A*b^2 + a*(-(b*B) + a*C))*d*f - (-9*a*C*d*f + b*(C*d*e + c*C*f + 5*B*d*f))*(a + b*x) - 3*b*C*d*f*x*(a + b*x)) - I*(b*c - a*d)*f*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*f*(d*e - c*f)*(24*a^2*C*d^2*f - a*b*d*(C*d*e + 7*c*C*f + 20*B*d*f) + b^2*(-2*c^2*C*f + 15*A*d^2*f + c*d*(C*e + 5*B*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(15*b^5*Sqrt[-a + (b*c)/d]*d^2*f^2*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
64,1,938,687,13.2999097,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^{5/2}} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(5/2),x]","\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left(\frac{2 C}{3 b^3}-\frac{2 \left(-8 C d f a^3+7 b C d e a^2+7 b c C f a^2+5 b B d f a^2-6 b^2 c C e a-4 b^2 B d e a-4 b^2 B c f a-2 A b^2 d f a+3 b^3 B c e+A b^3 d e+A b^3 c f\right)}{3 b^3 (b c-a d) (b e-a f) (a+b x)}-\frac{2 \left(C a^2-b B a+A b^2\right)}{3 b^3 (a+b x)^2}\right)-\frac{2 (a+b x)^{3/2} \left(-\sqrt{\frac{b c}{d}-a} \left(-16 C d^2 f^2 a^3+8 b d f (B d f+2 C (d e+c f)) a^2-b^2 \left(d f (7 B d e+7 B c f+2 A d f)+C \left(d^2 e^2+16 c d f e+c^2 f^2\right)\right) a+b^3 \left(C e f c^2+d \left(C e^2+6 B f e+A f^2\right) c+A d^2 e f\right)\right) \left(\frac{b c}{a+b x}+d-\frac{a d}{a+b x}\right) \left(\frac{b e}{a+b x}+f-\frac{a f}{a+b x}\right)+\frac{i (a d-b c) f \left(-16 C d^2 f^2 a^3+8 b d f (B d f+2 C (d e+c f)) a^2-b^2 \left(d f (7 B d e+7 B c f+2 A d f)+C \left(d^2 e^2+16 c d f e+c^2 f^2\right)\right) a+b^3 \left(C e f c^2+d \left(C e^2+6 B f e+A f^2\right) c+A d^2 e f\right)\right) \sqrt{-\frac{a}{a+b x}+\frac{b c}{d (a+b x)}+1} \sqrt{-\frac{a}{a+b x}+\frac{b e}{f (a+b x)}+1} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)}{\sqrt{a+b x}}+\frac{i b (a d-b c) f (d e-c f) \left(8 C d f a^2-b (7 C d e+c C f+4 B d f) a+b^2 (c C e+3 B d e+A d f)\right) \sqrt{-\frac{a}{a+b x}+\frac{b c}{d (a+b x)}+1} \sqrt{-\frac{a}{a+b x}+\frac{b e}{f (a+b x)}+1} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)}{\sqrt{a+b x}}\right)}{3 b^5 \sqrt{\frac{b c}{d}-a} d (b c-a d) f (b e-a f) \sqrt{c+\frac{(a+b x) \left(d-\frac{a d}{a+b x}\right)}{b}} \sqrt{e+\frac{(a+b x) \left(f-\frac{a f}{a+b x}\right)}{b}}}","\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left(8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right)}{3 b^3 (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left(d f (2 A d f+7 B c f+7 B d e)+C \left(c^2 f^2+16 c d e f+d^2 e^2\right)\right)-b^3 \left(c d \left(A f^2+6 B e f+C e^2\right)+A d^2 e f+c^2 C e f\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left(A b^2-a (b B-a C)\right)}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}-\frac{2 \sqrt{c+d x} (e+f x)^{3/2} (b B-2 a C)}{b^2 \sqrt{a+b x} (b e-a f)}",1,"Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*((2*C)/(3*b^3) - (2*(A*b^2 - a*b*B + a^2*C))/(3*b^3*(a + b*x)^2) - (2*(3*b^3*B*c*e - 6*a*b^2*c*C*e + A*b^3*d*e - 4*a*b^2*B*d*e + 7*a^2*b*C*d*e + A*b^3*c*f - 4*a*b^2*B*c*f + 7*a^2*b*c*C*f - 2*a*A*b^2*d*f + 5*a^2*b*B*d*f - 8*a^3*C*d*f))/(3*b^3*(b*c - a*d)*(b*e - a*f)*(a + b*x))) - (2*(a + b*x)^(3/2)*(-(Sqrt[-a + (b*c)/d]*(-16*a^3*C*d^2*f^2 + 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C*e^2 + 6*B*e*f + A*f^2)) - a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e*f + c^2*f^2)))*(d + (b*c)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))) + (I*(-(b*c) + a*d)*f*(-16*a^3*C*d^2*f^2 + 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C*e^2 + 6*B*e*f + A*f^2)) - a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e*f + c^2*f^2)))*Sqrt[1 - a/(a + b*x) + (b*c)/(d*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/Sqrt[a + b*x] + (I*b*(-(b*c) + a*d)*f*(d*e - c*f)*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*d*e + A*d*f) - a*b*(7*C*d*e + c*C*f + 4*B*d*f))*Sqrt[1 - a/(a + b*x) + (b*c)/(d*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/Sqrt[a + b*x]))/(3*b^5*Sqrt[-a + (b*c)/d]*d*(b*c - a*d)*f*(b*e - a*f)*Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a + b*x)*(f - (a*f)/(a + b*x)))/b])","C",1
65,1,9529,964,16.4209492,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^{7/2}} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left(6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right) \sqrt{c+d x} (e+f x)^{3/2}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left(48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left(2 C \left(19 d^2 e^2+81 c d f e+19 c^2 f^2\right)-d f (2 A d f-13 B (d e+c f))\right) a^2-b^3 \left(f (70 C e+3 B f) c^2+2 d \left(35 C e^2+11 B f e-A f^2\right) c+d^2 e (3 B e-2 A f)\right) a-b^4 \left(-\left(\left(30 C e^2+5 B f e-2 A f^2\right) c^2\right)-d e (5 B e+2 A f) c+2 A d^2 e^2\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right) \sqrt{e+f x}}{15 b^4 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \left(24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left(15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right) a-b^3 \left(15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt{a+b x}}+\frac{2 (d e-c f) \left(24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left(15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right) a-b^3 \left(15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^4 \sqrt{d} (a d-b c)^{3/2} (b e-a f) \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","C",0
66,1,15719,1716,18.5557183,"\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left(A+B x+C x^2\right)}{(a+b x)^{9/2}} \, dx","Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac{2 \left(6 C d f a^3+b (B d f-10 C (d e+c f)) a^2+b^2 (14 c C e+3 B d e+3 B c f-8 A d f) a-b^3 (7 B c e-4 A (d e+c f))\right) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left(48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left(C \left(103 d^2 e^2+344 c d f e+103 c^2 f^2\right)+d f (6 A d f-19 B (d e+c f))\right) a^3-3 b^3 \left(C \left(5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right)+d f \left(3 A d f (d e+c f)-B \left(3 d^2 e^2+16 c d f e+3 c^2 f^2\right)\right)\right) a^2-b^4 \left(-6 f^2 (7 C e-B f) c^3-d f \left(238 C e^2-19 f (B e-A f)\right) c^2-d^2 e \left(42 C e^2-f (19 B e+20 A f)\right) c+d^3 e^2 (6 B e-19 A f)\right) a-b^5 \left(f \left(35 C e^2-14 B f e+8 A f^2\right) c^3+d e \left(35 C e^2+14 B f e-5 A f^2\right) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right) \sqrt{e+f x}}{105 b^4 (a d-b c)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \left(48 C d^3 f^3 a^5+8 b d^2 f^2 (B d f-16 C (d e+c f)) a^4+b^2 d f \left(C \left(103 d^2 e^2+344 c d f e+103 c^2 f^2\right)+d f (6 A d f-19 B (d e+c f))\right) a^3-3 b^3 \left(C \left(5 d^3 e^3+94 c d^2 f e^2+94 c^2 d f^2 e+5 c^3 f^3\right)+d f \left(3 A d f (d e+c f)-B \left(3 d^2 e^2+16 c d f e+3 c^2 f^2\right)\right)\right) a^2-b^4 \left(-6 f^2 (7 C e-B f) c^3-d f \left(238 C e^2-19 f (B e-A f)\right) c^2-d^2 e \left(42 C e^2-f (19 B e+20 A f)\right) c+d^3 e^2 (6 B e-19 A f)\right) a-b^5 \left(f \left(35 C e^2-14 B f e+8 A f^2\right) c^3+d e \left(35 C e^2+14 B f e-5 A f^2\right) c^2-d^2 e^2 (14 B e+5 A f) c+8 A d^3 e^3\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left(24 C d^2 f^2 a^4-b d f (61 C d e+43 c C f-4 B d f) a^3-3 b^2 \left(d f (3 B d e+2 B c f-A d f)-C \left(15 d^2 e^2+37 c d f e+5 c^2 f^2\right)\right) a^2-3 b^3 \left(2 f (7 C e-B f) c^2+d \left(28 C e^2-5 B f e+5 A f^2\right) c+d^2 e (B e-3 A f)\right) a-b^4 \left(-\left(\left(35 C e^2-14 B f e+8 A f^2\right) c^2\right)-d e (7 B e-A f) c+4 A d^2 e^2\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} (d e-c f) \left(24 C d^2 f^2 a^4-b d f (43 C d e+61 c C f-4 B d f) a^3-3 b^2 \left(d f (2 B d e+3 B c f-A d f)-C \left(5 d^2 e^2+37 c d f e+15 c^2 f^2\right)\right) a^2+3 b^3 \left(-f (28 C e+B f) c^2-d \left(14 C e^2-5 B f e-3 A f^2\right) c+d^2 e (2 B e-5 A f)\right) a+b^4 \left(\left(35 C e^2+7 B f e-4 A f^2\right) c^2-d e (14 B e+A f) c+8 A d^2 e^2\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{105 b^4 (a d-b c)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","C",0
67,1,12483,1235,17.6345852,"\int \frac{(a+b x)^{3/2} \sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{e+f x}} \, dx","Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]","\text{Result too large to show}","\frac{2 C (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^{5/2}}{9 b d f}-\frac{2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^{3/2}}{63 b d^2 f^2}-\frac{2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) (c+d x)^{3/2} \sqrt{e+f x} \sqrt{a+b x}}{315 b d^3 f^3}-\frac{2 \left(5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left(\frac{a d f}{2}-b (2 d e+c f)\right) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{945 b^2 d^3 f^4}+\frac{2 \sqrt{a d-b c} \left(\left(C \left(128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right)+3 d f \left(7 A d f \left(8 d^2 e^2-3 c d f e-2 c^2 f^2\right)-B \left(48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right)\right)\right) b^4-a d f \left(2 C \left(92 d^3 e^3-33 c d^2 f e^2-18 c^2 d f^2 e-16 c^3 f^3\right)+3 d f \left(7 A d f (13 d e-7 c f)-B \left(72 d^2 e^2-29 c d f e-19 c^2 f^2\right)\right)\right) b^3-3 a^2 d^2 f^2 \left(3 d f (4 B d e-3 B c f-7 A d f)-C \left(9 d^2 e^2-5 c d f e-3 c^2 f^2\right)\right) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{315 b^3 d^{7/2} f^5 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \left(-\left(\left(C \left(128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right)+3 d f \left(7 A d f (8 d e+c f)-4 B \left(12 d^2 e^2+2 c d f e+c^2 f^2\right)\right)\right) b^3\right)-3 a d f \left(3 d f (16 B d e+3 B c f-21 A d f)-5 C \left(8 d^2 e^2+2 c d f e+c^2 f^2\right)\right) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{315 b^3 d^{7/2} f^5 \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","C",1
68,1,922,766,12.9067515,"\int \frac{\sqrt{a+b x} \sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{e+f x}} \, dx","Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x]*(A + B*x + C*x^2))/Sqrt[e + f*x],x]","\frac{2 \left(\sqrt{\frac{b c}{d}-a} \left(\left(C \left(-48 d^3 e^3+16 c d^2 f e^2+9 c^2 d f^2 e+8 c^3 f^3\right)+7 d f \left(5 A d f (c f-2 d e)+B \left(8 d^2 e^2-3 c d f e-2 c^2 f^2\right)\right)\right) b^3+a d f \left(7 d f (-3 B d e+2 B c f+5 A d f)+C \left(16 d^2 e^2-8 c d f e-5 c^2 f^2\right)\right) b^2+a^2 d^2 f^2 (9 C d e-5 c C f-14 B d f) b+8 a^3 C d^3 f^3\right) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left(\left(7 d f (5 A d f+B (-4 d e+c f+3 d f x))+C \left(3 \left(8 e^2-6 f x e+5 f^2 x^2\right) d^2+c f (3 f x-5 e) d-4 c^2 f^2\right)\right) b^2+a d f (7 B d f+C (-5 d e+2 c f+3 d f x)) b-4 a^2 C d^2 f^2\right) b^2-i (b c-a d) f (d e-c f) \left(-\left(\left(7 d f (-4 B d e-2 B c f+5 A d f)+C \left(24 d^2 e^2+13 c d f e+8 c^2 f^2\right)\right) b^2\right)+a d f (5 C d e+c C f-7 B d f) b+4 a^2 C d^2 f^2\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right) b+i (b c-a d) f \left(\left(C \left(-48 d^3 e^3+16 c d^2 f e^2+9 c^2 d f^2 e+8 c^3 f^3\right)+7 d f \left(5 A d f (c f-2 d e)+B \left(8 d^2 e^2-3 c d f e-2 c^2 f^2\right)\right)\right) b^3+a d f \left(7 d f (-3 B d e+2 B c f+5 A d f)+C \left(16 d^2 e^2-8 c d f e-5 c^2 f^2\right)\right) b^2+a^2 d^2 f^2 (9 C d e-5 c C f-14 B d f) b+8 a^3 C d^3 f^3\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)\right)}{105 b^4 \sqrt{\frac{b c}{d}-a} d^3 f^4 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}","\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)-\left(b^2 \left(7 d f (-10 A d f+B c f+8 B d e)-4 C \left(c^2 f^2+2 c d e f+12 d^2 e^2\right)\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left(\frac{b c f}{2}-d (a f+b e)\right) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{105 b^2 d^2 f^3}-\frac{2 \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}",1,"(2*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*C*d*e - 5*c*C*f - 14*B*d*f) + a*b^2*d*f*(7*d*f*(-3*B*d*e + 2*B*c*f + 5*A*d*f) + C*(16*d^2*e^2 - 8*c*d*e*f - 5*c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 8*c^3*f^3) + 7*d*f*(5*A*d*f*(-2*d*e + c*f) + B*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2))))*(c + d*x)*(e + f*x) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(-4*a^2*C*d^2*f^2 + a*b*d*f*(7*B*d*f + C*(-5*d*e + 2*c*f + 3*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e + c*f + 3*d*f*x)) + C*(-4*c^2*f^2 + c*d*f*(-5*e + 3*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 5*f^2*x^2)))) + I*(b*c - a*d)*f*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*C*d*e - 5*c*C*f - 14*B*d*f) + a*b^2*d*f*(7*d*f*(-3*B*d*e + 2*B*c*f + 5*A*d*f) + C*(16*d^2*e^2 - 8*c*d*e*f - 5*c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 8*c^3*f^3) + 7*d*f*(5*A*d*f*(-2*d*e + c*f) + B*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*(b*c - a*d)*f*(d*e - c*f)*(4*a^2*C*d^2*f^2 + a*b*d*f*(5*C*d*e + c*C*f - 7*B*d*f) - b^2*(7*d*f*(-4*B*d*e - 2*B*c*f + 5*A*d*f) + C*(24*d^2*e^2 + 13*c*d*e*f + 8*c^2*f^2)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(105*b^4*Sqrt[-a + (b*c)/d]*d^3*f^4*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
69,1,562,527,9.6265839,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{\sqrt{a+b x} \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/(Sqrt[a + b*x]*Sqrt[e + f*x]),x]","\frac{2 \sqrt{a+b x} \left(i b d f \sqrt{a+b x} \sqrt{\frac{b c}{d}-a} (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-4 a C d f+5 b B d f-2 b C (c f+2 d e)) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+\frac{b^2 (c+d x) (e+f x) \left(8 a^2 C d^2 f^2+a b d f (-10 B d f-3 c C f+7 C d e)+b^2 \left(5 d f (3 A d f+B c f-2 B d e)+C \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)\right)\right)}{a+b x}+\frac{i f \sqrt{a+b x} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(8 a^2 C d^2 f^2+a b d f (-10 B d f-3 c C f+7 C d e)+b^2 \left(5 d f (3 A d f+B c f-2 B d e)+C \left(-2 c^2 f^2-3 c d e f+8 d^2 e^2\right)\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)}{\sqrt{\frac{b c}{d}-a}}+b^2 d f (c+d x) (e+f x) (-4 a C d f+5 b B d f+b C (c f-4 d e+3 d f x))\right)}{15 b^4 d^2 f^3 \sqrt{c+d x} \sqrt{e+f x}}","-\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(4 a^2 C d f^2+a b f (-5 B d f-c C f+3 C d e)-\left(b^2 (5 d f (2 B e-3 A f)-C e (c f+8 d e))\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (3 b d f (a c C f+3 a C d e-5 A b d f+b c C e)-(2 a d f-b c f+2 b d e) (4 a C d f+b (-5 B d f+2 c C f+4 C d e))) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-5 B d f+2 c C f+4 C d e))}{15 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f}",1,"(2*Sqrt[a + b*x]*((b^2*(8*a^2*C*d^2*f^2 + a*b*d*f*(7*C*d*e - 3*c*C*f - 10*B*d*f) + b^2*(5*d*f*(-2*B*d*e + B*c*f + 3*A*d*f) + C*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2)))*(c + d*x)*(e + f*x))/(a + b*x) + b^2*d*f*(c + d*x)*(e + f*x)*(5*b*B*d*f - 4*a*C*d*f + b*C*(-4*d*e + c*f + 3*d*f*x)) + (I*(b*c - a*d)*f*(8*a^2*C*d^2*f^2 + a*b*d*f*(7*C*d*e - 3*c*C*f - 10*B*d*f) + b^2*(5*d*f*(-2*B*d*e + B*c*f + 3*A*d*f) + C*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2)))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/Sqrt[-a + (b*c)/d] + I*b*Sqrt[-a + (b*c)/d]*d*f*(d*e - c*f)*(5*b*B*d*f - 4*a*C*d*f - 2*b*C*(2*d*e + c*f))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(15*b^4*d^2*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
70,1,551,540,6.7354886,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^{3/2} \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(3/2)*Sqrt[e + f*x]),x]","-\frac{2 \left(-i b f (a+b x)^{3/2} (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(4 a^2 C d f-a b (3 B d f+c C f+C d e)+b^2 (3 A d f+c C e)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(-8 a^2 C d f^2+a b f (6 B d f+c C f+3 C d e)+b^2 (C e (2 d e-c f)-3 d f (A f+B e))\right)-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(8 a^2 C d f^2-a b f (6 B d f+c C f+3 C d e)+b^2 (3 d f (A f+B e)+C e (c f-2 d e))\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)+b^2 d f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(3 f \left(a (a C-b B)+A b^2\right)-C (a+b x) (b e-a f)\right)\right)}{3 b^4 d f^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b e-a f)}","\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (4 a C f-3 b B f+2 b C e) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left(4 a^2 C d f-a b (3 B d f+c C f+C d e)+b^2 (3 A d f+c C e)\right)}{3 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(8 a^2 C d f^2-a b f (6 B d f+c C f+3 C d e)+b^2 (3 d f (A f+B e)-C e (2 d e-c f))\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{3 b^3 \sqrt{d} f^2 \sqrt{c+d x} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{b \sqrt{a+b x} (b c-a d) (b e-a f)}",1,"(-2*(b^2*Sqrt[-a + (b*c)/d]*(-8*a^2*C*d*f^2 + a*b*f*(3*C*d*e + c*C*f + 6*B*d*f) + b^2*(-3*d*f*(B*e + A*f) + C*e*(2*d*e - c*f)))*(c + d*x)*(e + f*x) + b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*x)*(e + f*x)*(3*(A*b^2 + a*(-(b*B) + a*C))*f - C*(b*e - a*f)*(a + b*x)) - I*(b*c - a*d)*f*(8*a^2*C*d*f^2 - a*b*f*(3*C*d*e + c*C*f + 6*B*d*f) + b^2*(3*d*f*(B*e + A*f) + C*e*(-2*d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*f*(d*e - c*f)*(4*a^2*C*d*f + b^2*(c*C*e + 3*A*d*f) - a*b*(C*d*e + c*C*f + 3*B*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(3*b^4*Sqrt[-a + (b*c)/d]*d*f^2*(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
71,1,724,597,11.7566191,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^{5/2} \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]","-\frac{2 \left(b^2 f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left((a+b x) \left(-5 a^3 C d f+a^2 b (2 B d f+4 c C f+7 C d e)-a b^2 (-A d f+B c f+4 B d e+6 c C e)+b^3 (-2 A c f+A d e+3 B c e)\right)+(b c-a d) (b e-a f) \left(a (a C-b B)+A b^2\right)\right)+(a+b x) \left(i b f (a+b x)^{3/2} (b c-a d) (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(-4 a^2 C f+a b (B f+6 C e)+b^2 (2 A f-3 B e)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left(c f (3 B e-2 A f)+A d e f+3 c C e^2\right)\right)+i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left(c f (3 B e-2 A f)+A d e f+3 c C e^2\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)\right)\right)}{3 b^4 f (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d) (b e-a f)^2}","\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left(4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right)}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left(c \left(-2 A f^2+3 B e f+3 C e^2\right)+A d e f\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}",1,"(-2*(b^2*Sqrt[-a + (b*c)/d]*f*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-5*a^3*C*d*f + b^3*(3*B*c*e + A*d*e - 2*A*c*f) - a*b^2*(6*c*C*e + 4*B*d*e + B*c*f - A*d*f) + a^2*b*(7*C*d*e + 4*c*C*f + 2*B*d*f))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d*e*f + c*f*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + I*b*(b*c - a*d)*f*(d*e - c*f)*(-4*a^2*C*f + b^2*(-3*B*e + 2*A*f) + a*b*(6*C*e + B*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b^4*Sqrt[-a + (b*c)/d]*(b*c - a*d)*f*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
72,1,9186,1034,16.0939984,"\int \frac{\sqrt{c+d x} \left(A+B x+C x^2\right)}{(a+b x)^{7/2} \sqrt{e+f x}} \, dx","Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(7/2)*Sqrt[e + f*x]),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{e+f x} (c+d x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \left(8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left(d f (7 B d e+2 B c f-3 A d f)-C \left(23 d^2 e^2+37 c d f e+3 c^2 f^2\right)\right) a^2-b^3 \left(2 f (5 C e-B f) c^2+d \left(40 C e^2-13 f (B e-A f)\right) c+d^2 e (3 B e-7 A f)\right) a-b^4 \left(-\left(\left(15 C e^2-10 B f e+8 A f^2\right) c^2\right)-d e (5 B e-3 A f) c+2 A d^2 e^2\right)\right) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left(4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left(8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left(d f (7 B d e+2 B c f-3 A d f)-C \left(23 d^2 e^2+37 c d f e+3 c^2 f^2\right)\right) a^2-b^3 \left(2 f (5 C e-B f) c^2+d \left(40 C e^2-13 f (B e-A f)\right) c+d^2 e (3 B e-7 A f)\right) a-b^4 \left(-\left(\left(15 C e^2-10 B f e+8 A f^2\right) c^2\right)-d e (5 B e-3 A f) c+2 A d^2 e^2\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^3 (a d-b c)^{3/2} (b e-a f)^3 \sqrt{\frac{b (e+f x)}{b e-a f}} \sqrt{c+d x}}+\frac{2 \sqrt{d} (d e-c f) \left(4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^3 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{e+f x} \sqrt{c+d x}}",1,"Result too large to show","C",0
73,1,1000,838,13.8721231,"\int \frac{(a+b x)^{3/2} \left(A+B x+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{2 \left(-\sqrt{\frac{b c}{d}-a} \left(\left(8 C \left(6 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e+6 c^3 f^3\right)+7 d f \left(10 A d f (d e+c f)-B \left(8 d^2 e^2+7 c d f e+8 c^2 f^2\right)\right)\right) b^3-a d f \left(C \left(72 d^2 e^2+62 c d f e+72 c^2 f^2\right)+7 d f (20 A d f-13 B (d e+c f))\right) b^2+3 a^2 d^2 f^2 (4 C (d e+c f)-7 B d f) b+6 a^3 C d^3 f^3\right) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left(\left(7 d f (5 A d f+B (-4 d e-4 c f+3 d f x))+C \left(3 \left(8 e^2-6 f x e+5 f^2 x^2\right) d^2+c f (23 e-18 f x) d+24 c^2 f^2\right)\right) b^2+3 a d f (14 B d f+C (-11 d e-11 c f+8 d f x)) b+3 a^2 C d^2 f^2\right) b^2+i (b c-a d) f \left(\left(C \left(24 d^3 e^3+17 c d^2 f e^2+16 c^2 d f^2 e+48 c^3 f^3\right)+7 d f \left(5 A d f (d e+2 c f)-B \left(4 d^2 e^2+3 c d f e+8 c^2 f^2\right)\right)\right) b^2-3 a d f \left(7 d f (-2 B d e-3 B c f+5 A d f)+C \left(11 d^2 e^2+8 c d f e+16 c^2 f^2\right)\right) b+3 a^2 C d^2 f^2 (d e-c f)\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right) b-i (b c-a d) f \left(\left(8 C \left(6 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e+6 c^3 f^3\right)+7 d f \left(10 A d f (d e+c f)-B \left(8 d^2 e^2+7 c d f e+8 c^2 f^2\right)\right)\right) b^3-a d f \left(C \left(72 d^2 e^2+62 c d f e+72 c^2 f^2\right)+7 d f (20 A d f-13 B (d e+c f))\right) b^2+3 a^2 d^2 f^2 (4 C (d e+c f)-7 B d f) b+6 a^3 C d^3 f^3\right) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)\right)}{105 b^3 \sqrt{\frac{b c}{d}-a} d^4 f^4 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}","\frac{2 C \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{5/2}}{7 b d f}-\frac{2 (2 a C d f-b (7 B d f-6 C (d e+c f))) \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{105 b d^3 f^3}-\frac{2 \sqrt{a d-b c} \left(3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 b c e+a d e+a c f) (2 a C d f-b (7 B d f-6 C (d e+c f))))+2 \left(\frac{a d f}{2}-b (d e+c f)\right) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f))))\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (b e-a f) \left(-\left(\left(C \left(48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right)+7 d f \left(5 A d f (2 d e+c f)-B \left(8 d^2 e^2+3 c d f e+4 c^2 f^2\right)\right)\right) b^2\right)-3 a d f \left(7 d f (3 B d e+2 B c f-5 A d f)-C \left(16 d^2 e^2+8 c d f e+11 c^2 f^2\right)\right) b+3 a^2 C d^2 f^2 (d e-c f)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}",1,"(2*(-(b^2*Sqrt[-a + (b*c)/d]*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(3*a^2*C*d^2*f^2 + 3*a*b*d*f*(14*B*d*f + C*(-11*d*e - 11*c*f + 8*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e - 4*c*f + 3*d*f*x)) + C*(24*c^2*f^2 + c*d*f*(23*e - 18*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 5*f^2*x^2)))) - I*(b*c - a*d)*f*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + I*b*(b*c - a*d)*f*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(-2*B*d*e - 3*B*c*f + 5*A*d*f) + C*(11*d^2*e^2 + 8*c*d*e*f + 16*c^2*f^2)) + b^2*(C*(24*d^3*e^3 + 17*c*d^2*e^2*f + 16*c^2*d*e*f^2 + 48*c^3*f^3) + 7*d*f*(5*A*d*f*(d*e + 2*c*f) - B*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(105*b^3*Sqrt[-a + (b*c)/d]*d^4*f^4*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
74,1,615,528,8.0331104,"\int \frac{\sqrt{a+b x} \left(A+B x+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(Sqrt[a + b*x]*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]","-\frac{2 \left(i b f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(a C d f (c f-d e)+5 b d f (3 A d f-B (2 c f+d e))+b C \left(8 c^2 f^2+3 c d e f+4 d^2 e^2\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(2 a^2 C d^2 f^2+a b d f (3 C (c f+d e)-5 B d f)-\left(b^2 \left(5 d f (3 A d f-2 B (c f+d e))+C \left(8 c^2 f^2+7 c d e f+8 d^2 e^2\right)\right)\right)\right)+i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(2 a^2 C d^2 f^2+a b d f (3 C (c f+d e)-5 B d f)-\left(b^2 \left(5 d f (3 A d f-2 B (c f+d e))+C \left(8 c^2 f^2+7 c d e f+8 d^2 e^2\right)\right)\right)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)-b^2 d f (a+b x) (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} (a C d f+5 b B d f+b C (-4 c f-4 d e+3 d f x))\right)}{15 b^3 d^3 f^3 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a}}","-\frac{2 \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(a C d f (d e-c f)-b \left(5 d f (-3 A d f+B c f+2 B d e)-C \left(4 c^2 f^2+3 c d e f+8 d^2 e^2\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{15 b^2 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (3 b d f (a c C f+a C d e-5 A b d f+3 b c C e)+(a d f-2 b (c f+d e)) (2 a C d f-b (5 B d f-4 C (c f+d e)))) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^2 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (2 a C d f-b (5 B d f-4 C (c f+d e)))}{15 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{5 b d f}",1,"(-2*(b^2*Sqrt[-a + (b*c)/d]*(2*a^2*C*d^2*f^2 + a*b*d*f*(-5*B*d*f + 3*C*(d*e + c*f)) - b^2*(C*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2) + 5*d*f*(3*A*d*f - 2*B*(d*e + c*f))))*(c + d*x)*(e + f*x) - b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(5*b*B*d*f + a*C*d*f + b*C*(-4*d*e - 4*c*f + 3*d*f*x)) + I*(b*c - a*d)*f*(2*a^2*C*d^2*f^2 + a*b*d*f*(-5*B*d*f + 3*C*(d*e + c*f)) - b^2*(C*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2) + 5*d*f*(3*A*d*f - 2*B*(d*e + c*f))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + I*b*(b*c - a*d)*f*(a*C*d*f*(-(d*e) + c*f) + b*C*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2) + 5*b*d*f*(3*A*d*f - B*(d*e + 2*c*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(15*b^3*Sqrt[-a + (b*c)/d]*d^3*f^3*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
75,1,418,387,6.0715917,"\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{\sqrt{a+b x} \left(\frac{2 i b f \sqrt{a+b x} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(a C d (c f-d e)+b \left(3 A d^2 f+c d (C e-3 B f)+2 c^2 C f\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)}{\sqrt{\frac{b c}{d}-a}}-\frac{2 b^2 (c+d x) (e+f x) (2 a C d f-3 b B d f+2 b C (c f+d e))}{a+b x}+2 i d f \sqrt{a+b x} \sqrt{\frac{b c}{d}-a} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-2 a C d f+3 b B d f-2 b C (c f+d e)) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)+2 b^2 C d f (c+d x) (e+f x)\right)}{3 b^3 d^2 f^2 \sqrt{c+d x} \sqrt{e+f x}}","\frac{2 \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (a C f (d e-c f)-b (3 d f (B e-A f)-C e (c f+2 d e))) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (2 a C d f-b (3 B d f-2 C (c f+d e))) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}",1,"(Sqrt[a + b*x]*(2*b^2*C*d*f*(c + d*x)*(e + f*x) - (2*b^2*(-3*b*B*d*f + 2*a*C*d*f + 2*b*C*(d*e + c*f))*(c + d*x)*(e + f*x))/(a + b*x) + (2*I)*Sqrt[-a + (b*c)/d]*d*f*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + ((2*I)*b*f*(a*C*d*(-(d*e) + c*f) + b*(2*c^2*C*f + 3*A*d^2*f + c*d*(C*e - 3*B*f)))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/Sqrt[-a + (b*c)/d]))/(3*b^3*d^2*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
76,1,477,422,5.442238,"\int \frac{A+B x+C x^2}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\frac{2 \left(\frac{i b (a+b x)^{3/2} (a d-b c) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (a C (d e-c f)+b (A d f-B d e+c C e)) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)}{d \sqrt{\frac{b c}{d}-a}}+\frac{b^2 (c+d x) (e+f x) \left(2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right)}{d f}+\frac{i (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)}{d \sqrt{\frac{b c}{d}-a}}-\left(b^2 (c+d x) (e+f x) \left(a (a C-b B)+A b^2\right)\right)\right)}{b^3 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (b c-a d) (b e-a f)}","-\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (a C (d e-c f)-b (A d f-B c f+c C e)) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}}-\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{b \sqrt{a+b x} (b c-a d) (b e-a f)}",1,"(2*(-(b^2*(A*b^2 + a*(-(b*B) + a*C))*(c + d*x)*(e + f*x)) + (b^2*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))*(c + d*x)*(e + f*x))/(d*f) + (I*(b*c - a*d)*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d) + (I*b*(-(b*c) + a*d)*(a*C*(d*e - c*f) + b*(c*C*e - B*d*e + A*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d)))/(b^3*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
77,1,699,642,10.9080456,"\int \frac{A+B x+C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","-\frac{2 \left(b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left((a+b x) \left(-2 a^3 C d f+a^2 b (4 C (c f+d e)-B d f)-a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (3 B c e-2 A (c f+d e))\right)+(b c-a d) (b e-a f) \left(a (a C-b B)+A b^2\right)\right)+(a+b x) \left(-i b (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(a^2 C f (d e-c f)+a b \left(f (-3 A d f+B c f+2 B d e)-3 C d e^2\right)+b^2 \left(c f (2 A f-3 B e)+A d e f+3 c C e^2\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right),\frac{b d e-a d f}{b c f-a d f}\right)+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left(2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (2 A (c f+d e)-3 B c e)\right)+i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left(2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (2 A (c f+d e)-3 B c e)\right) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right)|\frac{b d e-a d f}{b c f-a d f}\right)\right)\right)}{3 b^3 (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d)^2 (b e-a f)^2}","-\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left(a^2 C d (d e-c f)+a b \left(3 f \left(A d^2+c^2 C\right)-B d (2 c f+d e)\right)-b^2 \left(A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{f (b c-a d)}{d (b e-a f)}\right)}{3 b^2 \sqrt{d} \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}+\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left(2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right)}{3 b \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left(2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right) E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{3 b^2 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}",1,"(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-2*a^3*C*d*f - a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(-(B*d*f) + 4*C*(d*e + c*f)))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*b*(b*c - a*d)*(a^2*C*f*(d*e - c*f) + b^2*(3*c*C*e^2 + A*d*e*f + c*f*(-3*B*e + 2*A*f)) + a*b*(-3*C*d*e^2 + f*(2*B*d*e + B*c*f - 3*A*d*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b^3*Sqrt[-a + (b*c)/d]*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])","C",1
78,1,8844,1116,16.2237061,"\int \frac{A+B x+C x^2}{(a+b x)^{7/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx","Integrate[(A + B*x + C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]","\text{Result too large to show}","-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left(A b^2-a (b B-a C)\right)}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left(2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left(C \left(3 d^2 e^2-13 c d f e+3 c^2 f^2\right)+d f (23 A d f-7 B (d e+c f))\right) a^2-b^3 \left(-2 f (5 C e-B f) c^2-d \left(10 C e^2-33 B f e+23 A f^2\right) c+d^2 e (2 B e-23 A f)\right) a-b^4 \left(\left(15 C e^2-10 B f e+8 A f^2\right) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right)|\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^2 (a d-b c)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} \left(C d f (d e-c f) a^3-3 b \left(d f (2 B d e+3 B c f-5 A d f)-C \left(d^2 e^2+c d f e+3 c^2 f^2\right)\right) a^2+b^2 \left(-f (20 C e-B f) c^2-d \left(10 C e^2-27 B f e+11 A f^2\right) c+d^2 e (2 B e-19 A f)\right) a+b^3 \left(\left(15 C e^2-5 B f e+4 A f^2\right) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right)\right) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right),\frac{(b c-a d) f}{d (b e-a f)}\right)}{15 b^2 (a d-b c)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \left(2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left(C \left(3 d^2 e^2-13 c d f e+3 c^2 f^2\right)+d f (23 A d f-7 B (d e+c f))\right) a^2-b^3 \left(-2 f (5 C e-B f) c^2-d \left(10 C e^2-33 B f e+23 A f^2\right) c+d^2 e (2 B e-23 A f)\right) a-b^4 \left(\left(15 C e^2-10 B f e+8 A f^2\right) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right)\right) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left(2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}",1,"Result too large to show","C",0