1,1,806,700,11.8252017,"\int \frac{(a+b x)^2 (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)^2*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","-\frac{2 \left(-\sqrt{\frac{d e}{f}-c} \left(\left(B \left(\left(8 f^2 g^2+7 e f h g+8 e^2 h^2\right) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right)-10 A d f h (d f g+d e h+c f h)\right) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right) (e+f x) (g+h x) d^2+b \sqrt{\frac{d e}{f}-c} f h (c+d x) (e+f x) (g+h x) (-5 A b d f h-10 a B d f h+b B (4 c f h+d (4 f g+4 e h-3 f h x))) d^2-i h \left(-\left(\left(B \left(e \left(4 f^2 g^2+3 e f h g+8 e^2 h^2\right) d^2+c f \left(-4 f^2 g^2+e f h g+3 e^2 h^2\right) d+4 c^2 f^2 h (e h-f g)\right)-5 A d f h (c f (e h-f g)+d e (f g+2 e h))\right) b^2\right)+10 a d f h (-3 A d e f h+B c f (e h-f g)+B d e (f g+2 e h)) b+15 a^2 d^2 f^2 (A f-B e) h^2\right) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right) d-i (d e-c f) h \left(\left(B \left(\left(8 f^2 g^2+7 e f h g+8 e^2 h^2\right) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right)-10 A d f h (d f g+d e h+c f h)\right) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)\right)}{15 d^4 \sqrt{\frac{d e}{f}-c} f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}","-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B (c h (f g-e h)+d g (e h+2 f g)))-\left(b^2 \left(5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left(4 c^2 f h^2 (f g-e h)+c d h \left(-4 e^2 h^2+e f g h+3 f^2 g^2\right)+d^2 g \left(4 e^2 h^2+3 e f g h+8 f^2 g^2\right)\right)\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right) \left(15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))-\left(b^2 \left(10 A d f h (c f h+d e h+d f g)-B \left(8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left(8 e^2 h^2+7 e f g h+8 f^2 g^2\right)\right)\right)\right)\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (7 a B d f h+b (5 A d f h-4 B (c f h+d e h+d f g)))}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}",1,"(-2*(-(d^2*Sqrt[-c + (d*e)/f]*(15*a^2*B*d^2*f^2*h^2 - 10*a*b*d*f*h*(-3*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h)) + b^2*(-10*A*d*f*h*(d*f*g + d*e*h + c*f*h) + B*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(e + f*x)*(g + h*x)) + b*d^2*Sqrt[-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(-5*A*b*d*f*h - 10*a*B*d*f*h + b*B*(4*c*f*h + d*(4*f*g + 4*e*h - 3*f*h*x))) - I*(d*e - c*f)*h*(15*a^2*B*d^2*f^2*h^2 - 10*a*b*d*f*h*(-3*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h)) + b^2*(-10*A*d*f*h*(d*f*g + d*e*h + c*f*h) + B*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] - I*d*h*(15*a^2*d^2*f^2*(-(B*e) + A*f)*h^2 + 10*a*b*d*f*h*(-3*A*d*e*f*h + B*c*f*(-(f*g) + e*h) + B*d*e*(f*g + 2*e*h)) - b^2*(-5*A*d*f*h*(c*f*(-(f*g) + e*h) + d*e*(f*g + 2*e*h)) + B*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
2,1,450,405,5.2759327,"\int \frac{(a+b x) (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{\sqrt{c+d x} \left(\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (3 a d f h (A f-B e)+b (-3 A d e f h+B c f (e h-f g)+B d e (2 e h+f g))) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)}{\sqrt{\frac{d e}{f}-c}}-\frac{2 d^2 (e+f x) (g+h x) (-3 a B d f h-3 A b d f h+2 b B (c f h+d e h+d f g))}{c+d x}+\frac{2 i h \sqrt{c+d x} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right) (3 a B d f h+3 A b d f h-2 b B (c f h+d e h+d f g))}{\sqrt{\frac{d e}{f}-c}}+2 b B d^2 f h (e+f x) (g+h x)\right)}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}}","-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} (3 a d f h (B g-A h)+b (3 A d f g h-B (c h (f g-e h)+d g (e h+2 f g)))) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right) (3 a B d f h+b (3 A d f h-2 B (c f h+d e h+d f g)))}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b B \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}",1,"(Sqrt[c + d*x]*(2*b*B*d^2*f*h*(e + f*x)*(g + h*x) - (2*d^2*(-3*A*b*d*f*h - 3*a*B*d*f*h + 2*b*B*(d*f*g + d*e*h + c*f*h))*(e + f*x)*(g + h*x))/(c + d*x) + ((2*I)*(d*e - c*f)*h*(3*A*b*d*f*h + 3*a*B*d*f*h - 2*b*B*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[-c + (d*e)/f] + ((2*I)*d*h*(3*a*d*f*(-(B*e) + A*f)*h + b*(-3*A*d*e*f*h + B*c*f*(-(f*g) + e*h) + B*d*e*(f*g + 2*e*h)))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
3,1,319,284,1.9486794,"\int \frac{A+B x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","-\frac{2 \left(i d h (c+d x)^{3/2} (B e-A f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)-B d^2 (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c}-i B h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)\right)}{d^2 f h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}}","\frac{2 B \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 (B g-A h) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}",1,"(-2*(-(B*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x)) - I*B*(d*e - c*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + I*d*(B*e - A*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(d^2*Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
4,1,245,313,2.0967254,"\int \frac{A+B x}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 i \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(b (A d-B c) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)+d (a B-A b) \Pi \left(-\frac{b c f-a d f}{b d e-b c f};i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)\right)}{b f \sqrt{g+h x} (a d-b c) \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}}}","\frac{2 B \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{b d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \left(A-\frac{a B}{b}\right) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)}",1,"((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(-(B*c) + A*d)*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + (-(A*b) + a*B)*d*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(b*(-(b*c) + a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[g + h*x])","C",1
5,1,14548,678,15.8554483,"\int \frac{A+B x}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{\sqrt{f} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{b \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(a^3 (-B) d f h+3 a^2 A b d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right) \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{f} \sqrt{g+h x} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}}",1,"Result too large to show","C",0
6,1,21611,981,18.1560458,"\int \frac{(a+b x)^{3/2} (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)^(3/2)*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} B}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{(b e-a f) \sqrt{b g-a h} (3 a B d f h+b (4 A d f h-B (c f h+3 d (f g+e h)))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{4 b d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{c h-d g} (4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))-(a d f h+b (d f g+d e h+c f h)) (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h)))) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{4 b d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}+\frac{(5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}}",1,"Result too large to show","B",0
7,1,6648,736,15.8522616,"\int \frac{\sqrt{a+b x} (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(Sqrt[a + b*x]*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{B \sqrt{g+h x} (b e-a f) \sqrt{b g-a h} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{b f h \sqrt{c+d x} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{(a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (B (a d f h-b (c f h+d e h+d f g))+2 A b d f h) \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{b d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{B \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{f h \sqrt{c+d x}}-\frac{B \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"Result too large to show","B",0
8,1,586,442,6.3761593,"\int \frac{A+B x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 (a+b x)^{3/2} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left(-\frac{A b (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{(a+b x) (b g-a h) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}-\frac{a B (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{(a+b x) (a h-b g) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}+\frac{B (e h-f g) \sqrt{-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}} \Pi \left(\frac{b (e h-f g)}{(b e-a f) h};\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{h (b e-a f)}\right)}{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}","\frac{2 \sqrt{g+h x} (A b-a B) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{b \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{2 B (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{b h \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}",1,"(2*(a + b*x)^(3/2)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-((A*b*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*g - a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])) - (a*B*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((-(b*g) + a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]) + (B*(-(f*g) + e*h)*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2))]*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*e - a*f)*h)))/(b*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","A",1
9,1,333,606,9.5868129,"\int \frac{A+B x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 (e+f x)^{3/2} (g+h x)^{3/2} (b e-a f) \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left((b g-a h) (B c-A d) \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)+(A b-a B) (d g-c h) E\left(\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)\right)}{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d) (f g-e h)^3 \left(-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}\right)^{3/2}}","\frac{2 \sqrt{g+h x} (B c-A d) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac{2 \sqrt{a+b x} (A b-a B) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*((A*b - a*B)*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))] + (B*c - A*d)*(b*g - a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)*(f*g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))","A",0
10,1,10667,1081,24.0572547,"\int \frac{A+B x}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + B*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left(3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left(3 d (B c-A d) f h a^2+b \left(3 A d^2 (f g+e h)-B \left(f h c^2+2 d (f g+e h) c+d^2 e g\right)\right) a+b^2 \left(3 B c d e g-A \left(-f h c^2+d (f g+e h) c+2 d^2 e g\right)\right)\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac{2 b \left(3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{2 d \left(3 B d f h a^3-b (6 A d f h+B (d f g+d e h+c f h)) a^2-b^2 (B (d e g+c f g+c e h)-4 A (d f g+d e h+c f h)) a+b^3 (3 B c e g-2 A (d e g+c f g+c e h))\right) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}",1,"Result too large to show","B",0
11,1,14893,898,16.0659723,"\int \frac{(a+b x)^{3/2} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)^(3/2)*(d*e + c*f + 2*d*f*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b}{h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{2 d f h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(5 a d f h-b (3 d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{2 f h^2 \sqrt{c+d x}}-\frac{(b e-a f) \sqrt{b g-a h} (3 a d f h+b (c f h-d (3 f g+e h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{2 f h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} b}-\frac{\sqrt{c h-d g} \left(\left(-\left(\left(3 f^2 g^2+e^2 h^2\right) d^2\right)+2 c e f h^2 d-c^2 f^2 h^2\right) b^2+6 a d^2 f^2 g h b-3 a^2 d^2 f^2 h^2\right) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{2 d \sqrt{b c-a d} f h^3 \sqrt{c+d x} \sqrt{e+f x} b}",1,"Result too large to show","B",0
12,1,6583,472,15.4632379,"\int \frac{\sqrt{a+b x} (d e+c f+2 d f x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(Sqrt[a + b*x]*(d*e + c*f + 2*d*f*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 d (e+f x) (b g-a h)^{3/2} \sqrt{\frac{(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt{\frac{(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} \Pi \left(\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left(\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right)|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right)}{h^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{b e-a f}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{h \sqrt{a+b x}}-\frac{2 \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right)|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{h \sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}}",1,"Result too large to show","B",0
13,1,724,449,7.556364,"\int \frac{d e+c f+2 d f x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(d*e + c*f + 2*d*f*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 \sqrt{a+b x} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left(-b d e h (g+h x) (b e-a f) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)+2 a d f h (g+h x) (b e-a f) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)+b c f h (g+h x) (a f-b e) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)-2 d f (a+b x) (b g-a h) (f g-e h) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}} \sqrt{-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}} \Pi \left(\frac{b (e h-f g)}{(b e-a f) h};\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)\right)}{b h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b e-a f) (b g-a h) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}","\frac{2 \sqrt{g+h x} (-2 a d f+b c f+b d e) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{b \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{4 d f (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{b h \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}",1,"(2*Sqrt[a + b*x]*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-(b*d*e*(b*e - a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))]) + 2*a*d*f*(b*e - a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] + b*c*f*(-(b*e) + a*f)*h*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))] - 2*d*f*(b*g - a*h)*(f*g - e*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2))]*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))]))/(b*(b*e - a*f)*h*(b*g - a*h)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])","A",1
14,1,341,625,9.1770927,"\int \frac{d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(d*e + c*f + 2*d*f*x)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 (e+f x)^{3/2} (g+h x)^{3/2} (b e-a f) \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left((d g-c h) (-2 a d f+b c f+b d e) E\left(\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)-d (b g-a h) (d e-c f) \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)\right)}{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d) (f g-e h)^3 \left(-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}\right)^{3/2}}","-\frac{2 d \sqrt{g+h x} (d e-c f) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (-2 a d f+b c f+b d e)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (-2 a d f+b c f+b d e)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac{2 \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} (-2 a d f+b c f+b d e) \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*((b*d*e + b*c*f - 2*a*d*f)*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))] - d*(d*e - c*f)*(b*g - a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)*(f*g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))","A",1
15,1,10629,1090,20.6461257,"\int \frac{d e+c f+2 d f x}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(d*e + c*f + 2*d*f*x)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b d e+b c f-2 a d f)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left(3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left(e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right) a-b^3 \left(f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right)\right) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 (d e-c f) \left(\left(f h c^2-d f g c+d e h c+2 d^2 e g\right) b^2-a d (d f g+3 d e h+2 c f h) b+3 a^2 d^2 f h\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac{4 b \left(3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left(e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right) a-b^3 \left(f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right)\right) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{4 d \left(3 d^2 f^2 h a^3-b d f (d f g+4 d e h+4 c f h) a^2+b^2 \left(e (f g+2 e h) d^2+c f (f g+3 e h) d+2 c^2 f^2 h\right) a-b^3 \left(f (f g+e h) c^2-d e (f g-e h) c+d^2 e^2 g\right)\right) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}",1,"Result too large to show","B",0
16,1,825,721,12.757018,"\int \frac{(a+b x) \left(a b B-a^2 C+b^2 B x+b^2 C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)*(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","-\frac{2 \left(b \sqrt{\frac{d e}{f}-c} \left(-\left(\left(C \left(\left(8 f^2 g^2+7 e f h g+8 e^2 h^2\right) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right)-10 B d f h (d f g+d e h+c f h)\right) b^2\right)+10 a d f h (C (d f g+d e h+c f h)-3 B d f h) b+15 a^2 C d^2 f^2 h^2\right) (e+f x) (g+h x) d^2+b^2 \sqrt{\frac{d e}{f}-c} f h (c+d x) (e+f x) (g+h x) (-5 b B d f h-5 a C d f h+b C (4 c f h+d (4 f g+4 e h-3 f h x))) d^2+i h \left(\left(C \left(e \left(4 f^2 g^2+3 e f h g+8 e^2 h^2\right) d^2+c f \left(-4 f^2 g^2+e f h g+3 e^2 h^2\right) d+4 c^2 f^2 h (e h-f g)\right)-5 B d f h (c f (e h-f g)+d e (f g+2 e h))\right) b^3-5 a d f h (-6 B d e f h+c C f (e h-f g)+C d e (f g+2 e h)) b^2-15 a^2 d^2 f^2 (C e+B f) h^2 b+15 a^3 C d^2 f^3 h^2\right) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right) d+i b (d e-c f) h \left(-\left(\left(C \left(\left(8 f^2 g^2+7 e f h g+8 e^2 h^2\right) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right)-10 B d f h (d f g+d e h+c f h)\right) b^2\right)+10 a d f h (C (d f g+d e h+c f h)-3 B d f h) b+15 a^2 C d^2 f^2 h^2\right) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)\right)}{15 d^4 \sqrt{\frac{d e}{f}-c} f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}","-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-C (c h (f g-e h)+d g (e h+2 f g)))-\left(b^3 \left(5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left(4 c^2 f h^2 (f g-e h)+c d h \left(-4 e^2 h^2+e f g h+3 f^2 g^2\right)+d^2 g \left(4 e^2 h^2+3 e f g h+8 f^2 g^2\right)\right)\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 b \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right) \left(15 a^2 C d^2 f^2 h^2-10 a b d f h (3 B d f h-C (c f h+d e h+d f g))+b^2 \left(10 B d f h (c f h+d e h+d f g)-C \left(8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left(8 e^2 h^2+7 e f g h+8 f^2 g^2\right)\right)\right)\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (2 C (a d f h-2 b (c f h+d e h+d f g))+5 b B d f h)}{15 d^2 f^2 h^2}+\frac{2 b^2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}",1,"(-2*(b*d^2*Sqrt[-c + (d*e)/f]*(15*a^2*C*d^2*f^2*h^2 + 10*a*b*d*f*h*(-3*B*d*f*h + C*(d*f*g + d*e*h + c*f*h)) - b^2*(-10*B*d*f*h*(d*f*g + d*e*h + c*f*h) + C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(e + f*x)*(g + h*x) + b^2*d^2*Sqrt[-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(-5*b*B*d*f*h - 5*a*C*d*f*h + b*C*(4*c*f*h + d*(4*f*g + 4*e*h - 3*f*h*x))) + I*b*(d*e - c*f)*h*(15*a^2*C*d^2*f^2*h^2 + 10*a*b*d*f*h*(-3*B*d*f*h + C*(d*f*g + d*e*h + c*f*h)) - b^2*(-10*B*d*f*h*(d*f*g + d*e*h + c*f*h) + C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + I*d*h*(15*a^3*C*d^2*f^3*h^2 - 15*a^2*b*d^2*f^2*(C*e + B*f)*h^2 - 5*a*b^2*d*f*h*(-6*B*d*e*f*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) + b^3*(-5*B*d*f*h*(c*f*(-(f*g) + e*h) + d*e*(f*g + 2*e*h)) + C*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
17,1,442,410,6.6183169,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{\sqrt{c+d x} \left(\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(-3 a^2 C d f^2 h+3 a b B d f^2 h+b^2 (-3 B d e f h+c C f (e h-f g)+C d e (2 e h+f g))\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)}{\sqrt{\frac{d e}{f}-c}}+\frac{2 b^2 d^2 (e+f x) (g+h x) (3 B d f h-2 C (c f h+d e h+d f g))}{c+d x}+2 i b^2 f h \sqrt{c+d x} \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)+2 b^2 C d^2 f h (e+f x) (g+h x)\right)}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}}","\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(-3 a^2 C d f h^2+3 a b B d f h^2-\left(b^2 (3 B d f g h-C (c h (f g-e h)+d g (e h+2 f g)))\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}",1,"(Sqrt[c + d*x]*(2*b^2*C*d^2*f*h*(e + f*x)*(g + h*x) + (2*b^2*d^2*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*(e + f*x)*(g + h*x))/(c + d*x) + (2*I)*b^2*Sqrt[-c + (d*e)/f]*f*h*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + ((2*I)*d*h*(3*a*b*B*d*f^2*h - 3*a^2*C*d*f^2*h + b^2*(-3*B*d*e*f*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
18,1,326,291,2.4761012,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 \left(-i d h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (a C f-b B f+b C e) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)+b C d^2 (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c}+i b C h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)\right)}{d^2 f h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}}","\frac{2 b C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} (a C h-b B h+b C g) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}",1,"(2*(b*C*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x) + I*b*C*(d*e - c*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] - I*d*(b*C*e - b*B*f + a*C*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(d^2*Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
19,1,249,309,1.6517912,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 i \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(d (2 a C-b B) \Pi \left(-\frac{b c f-a d f}{b d e-b c f};i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)-(a C d-b B d+b c C) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)\right)}{f \sqrt{g+h x} (a d-b c) \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}}}","\frac{2 C \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)}",1,"((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(-((b*c*C - b*B*d + a*C*d)*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]) + (-(b*B) + 2*a*C)*d*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/((-(b*c) + a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[g + h*x])","C",1
20,1,16859,680,16.3156926,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{\sqrt{f} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{\sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right) \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{b \sqrt{f} \sqrt{g+h x} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}}",1,"Result too large to show","C",0
21,1,21611,980,18.1197936,"\int \frac{\sqrt{a+b x} \left(a b B-a^2 C+b^2 B x+b^2 C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(Sqrt[a + b*x]*(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (4 b B d f h+C (a d f h-3 b (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right) b}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(4 b B d f h+C (a d f h-3 b (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{4 d f^2 h^2 \sqrt{c+d x}}+\frac{(b e-a f) \sqrt{b g-a h} (a C d f h-b (4 B d f h-C (3 d f g+3 d e h+c f h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{4 d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac{\sqrt{c h-d g} \left((a d f h+b (d f g+d e h+c f h)) (4 b B d f h+C (a d f h-3 b (d f g+d e h+c f h)))+4 d f h \left(2 C d f h a^2-b (4 B d f h-C (d f g+d e h+c f h)) a+b^2 C (d e g+c f g+c e h)\right)\right) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{4 d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","B",0
22,1,6667,734,15.8446703,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{C \sqrt{g+h x} (b e-a f) \sqrt{b g-a h} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{f h \sqrt{c+d x} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{(a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a C d f h-b (2 B d f h-C (c f h+d e h+d f g))) \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{b C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{f h \sqrt{c+d x}}-\frac{b C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"Result too large to show","B",0
23,1,583,436,6.8386818,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 (a+b x)^{3/2} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left(-\frac{b B (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{(a+b x) (b g-a h) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}-\frac{2 a C (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{(a+b x) (a h-b g) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}+\frac{C (e h-f g) \sqrt{-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}} \Pi \left(\frac{b (e h-f g)}{(b e-a f) h};\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right)}{h (b e-a f)}\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}","\frac{2 \sqrt{g+h x} (b B-2 a C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{\sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{2 C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{h \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}",1,"(2*(a + b*x)^(3/2)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-((b*B*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*g - a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])) - (2*a*C*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((-(b*g) + a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]) + (C*(-(f*g) + e*h)*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2))]*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*e - a*f)*h)))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","A",1
24,1,340,616,10.479168,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{2 (e+f x)^{3/2} (g+h x)^{3/2} (b e-a f) \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left((b g-a h) (a C d-b B d+b c C) \operatorname{EllipticF}\left(\sin ^{-1}\left(\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right),\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)+b (b B-2 a C) (d g-c h) E\left(\sin ^{-1}\left(\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)\right)}{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d) (f g-e h)^3 \left(-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}\right)^{3/2}}","\frac{2 \sqrt{g+h x} (a C d-b B d+b c C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 b d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac{2 b \sqrt{a+b x} (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*(b*(b*B - 2*a*C)*(d*g - c*h)*EllipticE[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))] + (b*c*C - b*B*d + a*C*d)*(b*g - a*h)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)*(f*g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))","A",0
25,1,10675,1128,25.1745792,"\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 \left(9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left(9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right) b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 d \left(9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left(3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left(3 B (f g+e h) d^2+C \left(-2 f h c^2-d f g c-d e h c+d^2 e g\right)\right) a-b^3 \left(-B f h c^2-d (3 C e g-B f g-B e h) c+2 B d^2 e g\right)\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}",1,"Result too large to show","B",0
26,1,18383,1097,18.1335775,"\int \frac{(a+b x)^2 \left(A+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)^2*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^2}{7 d f h}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac{4 \sqrt{c f-d e} \left(\left(35 A d^2 f^2 (d f g+d e h+c f h) h^2+2 C \left(2 \left(6 f^3 g^3+5 e f^2 h g^2+5 e^2 f h^2 g+6 e^3 h^3\right) d^3+c f h \left(10 f^2 g^2+9 e f h g+10 e^2 h^2\right) d^2+10 c^2 f^2 h^2 (f g+e h) d+12 c^3 f^3 h^3\right)\right) b^2-7 a d f h \left(15 A d^2 f^2 h^2+C \left(\left(8 f^2 g^2+7 e f h g+8 e^2 h^2\right) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right)\right) b+35 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h)\right) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{c f-d e} \left(\left(35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left(g \left(48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right) d^3+2 c h \left(8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right) d^2+c^2 f h^2 \left(17 f^2 g^2+6 e f h g-23 e^2 h^2\right) d+24 c^3 f^2 h^3 (f g-e h)\right)\right) b^2-14 a d f h \left(15 A d^2 f^2 g h^2+C \left(g \left(8 f^2 g^2+3 e f h g+4 e^2 h^2\right) d^2+c h \left(3 f^2 g^2+e f h g-4 e^2 h^2\right) d+4 c^2 f h^2 (f g-e h)\right)\right) b+35 a^2 d^2 f^2 h^2 \left(3 A d f h^2+C (c h (f g-e h)+d g (2 f g+e h))\right)\right) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{(d e-c f) h}{f (d g-c h)}\right)}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-C (5 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^3 f^3 h^3}",1,"Result too large to show","C",1
27,1,686,611,9.3848346,"\int \frac{(a+b x) \left(A+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","-\frac{2 \left(-i d h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(5 a d f h \left(3 A d f^2 h+c C f (e h-f g)+C d e (2 e h+f g)\right)-b \left(15 A d^2 e f^2 h^2+C \left(4 c^2 f^2 h (e h-f g)+c d f \left(3 e^2 h^2+e f g h-4 f^2 g^2\right)+d^2 e \left(8 e^2 h^2+3 e f g h+4 f^2 g^2\right)\right)\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)+d^2 (e+f x) (g+h x) \left(-\sqrt{\frac{d e}{f}-c}\right) \left(-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left(8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left(8 e^2 h^2+7 e f g h+8 f^2 g^2\right)\right)\right)-i h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right) \left(-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left(8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left(8 e^2 h^2+7 e f g h+8 f^2 g^2\right)\right)\right)+C d^2 f h (c+d x) (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c} (-5 a d f h+4 b c f h+b d (4 e h+4 f g-3 f h x))\right)}{15 d^4 f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}}","\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(5 a d f h \left(3 A d f h^2+C (c h (f g-e h)+d g (e h+2 f g))\right)-b \left(15 A d^2 f^2 g h^2+C \left(4 c^2 f h^2 (f g-e h)+c d h \left(-4 e^2 h^2+e f g h+3 f^2 g^2\right)+d^2 g \left(4 e^2 h^2+3 e f g h+8 f^2 g^2\right)\right)\right)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right) \left(10 a C d f h (c f h+d e h+d f g)-b \left(15 A d^2 f^2 h^2+C \left(8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left(8 e^2 h^2+7 e f g h+8 f^2 g^2\right)\right)\right)\right)}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{4 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a d f h-2 b (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}",1,"(-2*(-(d^2*Sqrt[-c + (d*e)/f]*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*(e + f*x)*(g + h*x)) + C*d^2*Sqrt[-c + (d*e)/f]*f*h*(c + d*x)*(e + f*x)*(g + h*x)*(4*b*c*f*h - 5*a*d*f*h + b*d*(4*f*g + 4*e*h - 3*f*h*x)) - I*(d*e - c*f)*h*(15*A*b*d^2*f^2*h^2 - 10*a*C*d*f*h*(d*f*g + d*e*h + c*f*h) + b*C*(8*c^2*f^2*h^2 + 7*c*d*f*h*(f*g + e*h) + d^2*(8*f^2*g^2 + 7*e*f*g*h + 8*e^2*h^2)))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] - I*d*h*(5*a*d*f*h*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h)) - b*(15*A*d^2*e*f^2*h^2 + C*(4*c^2*f^2*h*(-(f*g) + e*h) + c*d*f*(-4*f^2*g^2 + e*f*g*h + 3*e^2*h^2) + d^2*e*(4*f^2*g^2 + 3*e*f*g*h + 8*e^2*h^2))))*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(15*d^4*Sqrt[-c + (d*e)/f]*f^3*h^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
28,1,390,368,4.6491747,"\int \frac{A+C x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{\sqrt{c+d x} \left(\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(3 A d f^2 h+c C f (e h-f g)+C d e (2 e h+f g)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right),\frac{d f g-c f h}{d e h-c f h}\right)}{\sqrt{\frac{d e}{f}-c}}-\frac{4 C d^2 (e+f x) (g+h x) (c f h+d e h+d f g)}{c+d x}-4 i C f h \sqrt{c+d x} \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (c f h+d e h+d f g) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right)|\frac{d f g-c f h}{d e h-c f h}\right)+2 C d^2 f h (e+f x) (g+h x)\right)}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}}","\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(3 A d f h^2+C (c h (f g-e h)+d g (e h+2 f g))\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}-\frac{4 C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}",1,"(Sqrt[c + d*x]*(2*C*d^2*f*h*(e + f*x)*(g + h*x) - (4*C*d^2*(d*f*g + d*e*h + c*f*h)*(e + f*x)*(g + h*x))/(c + d*x) - (4*I)*C*Sqrt[-c + (d*e)/f]*f*h*(d*f*g + d*e*h + c*f*h)*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + ((2*I)*d*h*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h))*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
29,1,1036,465,8.1098547,"\int \frac{A+C x^2}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","-\frac{2 \left(-b^2 C f \sqrt{\frac{d g}{h}-c} h c^3+b^2 C d e \sqrt{\frac{d g}{h}-c} h c^2+a b C d f \sqrt{\frac{d g}{h}-c} h c^2+2 b^2 C f \sqrt{\frac{d g}{h}-c} h (c+d x) c^2+b^2 C d f g \sqrt{\frac{d g}{h}-c} c^2-b^2 C f \sqrt{\frac{d g}{h}-c} h (c+d x)^2 c-a b C d^2 e \sqrt{\frac{d g}{h}-c} h c-b^2 C d e \sqrt{\frac{d g}{h}-c} h (c+d x) c-2 a b C d f \sqrt{\frac{d g}{h}-c} h (c+d x) c-b^2 C d f g \sqrt{\frac{d g}{h}-c} (c+d x) c-b^2 C d^2 e g \sqrt{\frac{d g}{h}-c} c-a b C d^2 f g \sqrt{\frac{d g}{h}-c} c+a b C d f \sqrt{\frac{d g}{h}-c} h (c+d x)^2+a b C d^2 e \sqrt{\frac{d g}{h}-c} h (c+d x)+a b C d^2 f g \sqrt{\frac{d g}{h}-c} (c+d x)+i b C (b c-a d) f (c h-d g) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right)|\frac{d e h-c f h}{d f g-c f h}\right)+i b d f (b c C g-a C d g+a c C h+A b d h) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right),\frac{d e h-c f h}{d f g-c f h}\right)-i A b^2 d^2 f h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left(-\frac{b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right)|\frac{d e h-c f h}{d f g-c f h}\right)-i a^2 C d^2 f h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \Pi \left(-\frac{b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right)|\frac{d e h-c f h}{d f g-c f h}\right)+a b C d^3 e g \sqrt{\frac{d g}{h}-c}\right)}{b^2 d^2 (b c-a d) f \sqrt{\frac{d g}{h}-c} h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}","-\frac{2 C (a h+b g) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{b^2 d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \left(\frac{a^2 C}{b^2}+A\right) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)}+\frac{2 C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{b d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}",1,"(-2*(-(b^2*c*C*d^2*e*g*Sqrt[-c + (d*g)/h]) + a*b*C*d^3*e*g*Sqrt[-c + (d*g)/h] + b^2*c^2*C*d*f*g*Sqrt[-c + (d*g)/h] - a*b*c*C*d^2*f*g*Sqrt[-c + (d*g)/h] + b^2*c^2*C*d*e*Sqrt[-c + (d*g)/h]*h - a*b*c*C*d^2*e*Sqrt[-c + (d*g)/h]*h - b^2*c^3*C*f*Sqrt[-c + (d*g)/h]*h + a*b*c^2*C*d*f*Sqrt[-c + (d*g)/h]*h - b^2*c*C*d*f*g*Sqrt[-c + (d*g)/h]*(c + d*x) + a*b*C*d^2*f*g*Sqrt[-c + (d*g)/h]*(c + d*x) - b^2*c*C*d*e*Sqrt[-c + (d*g)/h]*h*(c + d*x) + a*b*C*d^2*e*Sqrt[-c + (d*g)/h]*h*(c + d*x) + 2*b^2*c^2*C*f*Sqrt[-c + (d*g)/h]*h*(c + d*x) - 2*a*b*c*C*d*f*Sqrt[-c + (d*g)/h]*h*(c + d*x) - b^2*c*C*f*Sqrt[-c + (d*g)/h]*h*(c + d*x)^2 + a*b*C*d*f*Sqrt[-c + (d*g)/h]*h*(c + d*x)^2 + I*b*C*(b*c - a*d)*f*(-(d*g) + c*h)*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] + I*b*d*f*(b*c*C*g - a*C*d*g + a*c*C*h + A*b*d*h)*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - I*A*b^2*d^2*f*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - I*a^2*C*d^2*f*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)]))/(b^2*d^2*(b*c - a*d)*f*Sqrt[-c + (d*g)/h]*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
30,1,637,738,15.2018589,"\int \frac{A+C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\frac{b^2 d (c+d x) (e+f x) (g+h x) \left(a^2 C+A b^2\right) (a d-b c) \sqrt{\frac{d g}{h}-c}+(a+b x) \left(b d^2 (e+f x) (g+h x) \left(a^2 C+A b^2\right) (b c-a d) \sqrt{\frac{d g}{h}-c}-i (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left(-b (b e-a f) \left(a^2 C d (c h-d g)-2 a b h \left(A d^2+c^2 C\right)+b^2 \left(A c d h+A d^2 g+2 c^2 C g\right)\right) \operatorname{EllipticF}\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right),\frac{d e h-c f h}{d f g-c f h}\right)+b f \left(a^2 C+A b^2\right) (b c-a d) (c h-d g) E\left(i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right)|\frac{d e h-c f h}{d f g-c f h}\right)-d \left(a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)+3 a^2 b^2 (C (c e h+c f g+d e g)-A d f h)+2 a b^3 (A c f h+A d e h+A d f g-2 c C e g)-A b^4 (c e h+c f g+d e g)\right) \Pi \left(-\frac{b c h-a d h}{b d g-b c h};i \sinh ^{-1}\left(\frac{\sqrt{\frac{d g}{h}-c}}{\sqrt{c+d x}}\right)|\frac{d e h-c f h}{d f g-c f h}\right)\right)\right)}{b^2 d (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h) \sqrt{\frac{d g}{h}-c}}","\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right),\frac{h (d e-c f)}{f (d g-c h)}\right)}{b^2 d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}-\frac{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left(a^2 C+A b^2\right)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{f} \sqrt{g+h x} \left(\frac{a^2 C}{b}+A b\right) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left(a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right) \Pi \left(-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right)|\frac{(d e-c f) h}{f (d g-c h)}\right)}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}",1,"(b^2*(A*b^2 + a^2*C)*d*(-(b*c) + a*d)*Sqrt[-c + (d*g)/h]*(c + d*x)*(e + f*x)*(g + h*x) + (a + b*x)*(b*(A*b^2 + a^2*C)*d^2*(b*c - a*d)*Sqrt[-c + (d*g)/h]*(e + f*x)*(g + h*x) - I*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(A*b^2 + a^2*C)*(b*c - a*d)*f*(-(d*g) + c*h)*EllipticE[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - b*(b*e - a*f)*(-2*a*b*(c^2*C + A*d^2)*h + a^2*C*d*(-(d*g) + c*h) + b^2*(2*c^2*C*g + A*d^2*g + A*c*d*h))*EllipticF[I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)] - d*(a^4*C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*C*(d*f*g + d*e*h + c*f*h) + 2*a*b^3*(-2*c*C*e*g + A*d*f*g + A*d*e*h + A*c*f*h) + 3*a^2*b^2*(-(A*d*f*h) + C*(d*e*g + c*f*g + c*e*h)))*EllipticPi[-((b*c*h - a*d*h)/(b*d*g - b*c*h)), I*ArcSinh[Sqrt[-c + (d*g)/h]/Sqrt[c + d*x]], (d*e*h - c*f*h)/(d*f*g - c*f*h)])))/(b^2*d*(b*c - a*d)^2*(b*e - a*f)*Sqrt[-c + (d*g)/h]*(b*g - a*h)*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])","C",1
31,1,38402,1395,23.826045,"\int \frac{(a+b x)^{3/2} \left(A+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[((a + b*x)^(3/2)*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^{3/2}}{3 d f h}-\frac{\sqrt{c h-d g} \left(4 b d f h \left(C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left(2 C (d f g+d e h+c f h) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right)\right)+(a d f h+b (d f g+d e h+c f h)) (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h))))\right) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right) (a+b x)}{24 b^2 d^3 \sqrt{b c-a d} f^3 h^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right) \sqrt{a+b x}}{24 b d^3 f^3 h^3 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{12 d^2 f^2 h^2}+\frac{(C (3 a d f h-5 b (d f g+d e h+c f h)) (a d f h-3 b (d f g+d e h+c f h))+8 b d f h (3 A b d f h-C (2 b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{24 b d^2 f^3 h^3 \sqrt{c+d x}}+\frac{(b e-a f) \sqrt{b g-a h} \left(-\left(\left(24 A d^2 f^2 h^2+C \left(\left(15 f^2 g^2+14 e f h g+15 e^2 h^2\right) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right)\right) b^2\right)+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{24 b^2 d^2 f^3 h^3 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}",1,"Result too large to show","B",0
32,1,16699,937,17.3211605,"\int \frac{\sqrt{a+b x} \left(A+C x^2\right)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(Sqrt[a + b*x]*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right) C}{4 b d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(b e-a f) \sqrt{b g-a h} (a d f h+b (c f h+3 d (f g+e h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right) C}{4 b^2 d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} C}{2 d f h}+\frac{(a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} C}{4 b d f^2 h^2 \sqrt{c+d x}}-\frac{\sqrt{c h-d g} (C (a d f h-3 b (d f g+d e h+c f h)) (a d f h+b (d f g+d e h+c f h))-4 b d f h (2 A b d f h-C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{4 b^2 d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}",1,"Result too large to show","B",0
33,1,6223,757,14.7395851,"\int \frac{A+C x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","\frac{\sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \left(a^2 C f h+a b C (e h+f g)-b^2 (C e g-2 A f h)\right) \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right)}{b^2 f h \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a d f h+b (c f h+d e h+d f g)) \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{b^2 d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{b d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}",1,"Result too large to show","B",0
34,1,2107,867,15.6620566,"\int \frac{A+C x^2}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right) \left(C a^2+A b^2\right)}{b (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left(C a^2+A b^2\right)}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}+\frac{2 d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} \left(C a^2+A b^2\right)}{b (b c-a d) (b e-a f) (b g-a h) \sqrt{c+d x}}-\frac{2 \left(-C d a^2+2 b c C a+A b^2 d\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{b^2 (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{2 C \sqrt{c h-d g} (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left(-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left(\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right)|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right)}{b^2 \sqrt{b c-a d} h \sqrt{c+d x} \sqrt{e+f x}}",1,"(-2*(A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + (2*(((-(A*b^2) - a^2*C)*(a + b*x)^(5/2)*(d + (b*c)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a + b*x)*(f - (a*f)/(a + b*x)))/b]*Sqrt[g + ((a + b*x)*(h - (a*h)/(a + b*x)))/b]) + ((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)*Sqrt[(d + (b*c)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x))]*((A*b^2*Sqrt[-(((b*c - a*d)*(b*g - a*h)*(d/(-(b*c) + a*d) - (a + b*x)^(-1)))/(b*d*g - b*c*h))]*(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))*Sqrt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g) + a*h))]*(-(((b*d*g - b*c*h)*EllipticE[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/((b*c - a*d)*(b*g - a*h))) - (d*EllipticF[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(-(b*c) + a*d)))/(Sqrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*h))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*h)/(a + b*x))]) + (a^2*C*Sqrt[-(((b*c - a*d)*(b*g - a*h)*(d/(-(b*c) + a*d) - (a + b*x)^(-1)))/(b*d*g - b*c*h))]*(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))*Sqrt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g) + a*h))]*(-(((b*d*g - b*c*h)*EllipticE[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/((b*c - a*d)*(b*g - a*h))) - (d*EllipticF[ArcSin[Sqrt[((b*e - a*f)*(h + (b*g)/(a + b*x) - (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(-(b*c) + a*d)))/(Sqrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*h))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*h)/(a + b*x))]) - (2*a*C*Sqrt[(-(d/(-(b*c) + a*d)) + (a + b*x)^(-1))/(-(d/(-(b*c) + a*d)) + h/(-(b*g) + a*h))]*Sqrt[(-(f/(-(b*e) + a*f)) + (a + b*x)^(-1))/(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*h))]*(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(-h - (b*g)/(a + b*x) + (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(Sqrt[(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1))/(f/(-(b*e) + a*f) - h/(-(b*g) + a*h))]*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*h)/(a + b*x))]) - (C*(-(b*g) + a*h)*(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*h))*Sqrt[(-(d/(-(b*c) + a*d)) + (a + b*x)^(-1))/(-(d/(-(b*c) + a*d)) + h/(-(b*g) + a*h))]*Sqrt[((f/(-(b*e) + a*f) - (a + b*x)^(-1))*(-(h/(-(b*g) + a*h)) + (a + b*x)^(-1)))/(-(f/(-(b*e) + a*f)) + h/(-(b*g) + a*h))^2]*EllipticPi[-((-(b*f*g) + b*e*h)/((-(b*e) + a*f)*h)), ArcSin[Sqrt[((-(b*e) + a*f)*(-h - (b*g)/(a + b*x) + (a*h)/(a + b*x)))/(b*(-(f*g) + e*h))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((-(b*e) + a*f)*(-(d*g) + c*h))])/(h*Sqrt[(d + (b*c - a*d)/(a + b*x))*(f + (b*e - a*f)/(a + b*x))*(h + (b*g - a*h)/(a + b*x))])))/(Sqrt[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a + b*x)*(f - (a*f)/(a + b*x)))/b]*Sqrt[g + ((a + b*x)*(h - (a*h)/(a + b*x)))/b])))/(b^3*(-(b*c) + a*d)*(-(b*e) + a*f)*(-(b*g) + a*h))","B",0
35,1,11188,1070,25.2445863,"\int \frac{A+C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx","Integrate[(A + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]","\text{Result too large to show}","-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left(C a^2+A b^2\right)}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left(C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left(\sin ^{-1}\left(\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right)|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right)}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left(-\left(\left(3 A d^2 f h-C \left(-2 f h c^2-d f g c-d e h c+d^2 e g\right)\right) a^2\right)+3 b \left(C c^2+A d^2\right) (f g+e h) a-b^2 \left((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right)\right) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \operatorname{EllipticF}\left(\sin ^{-1}\left(\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right)}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{4 b \left(C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{4 d \left(C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}",1,"Result too large to show","B",0