1,1,148,145,0.1862439,"\int \frac{1}{\left(2^{2/3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[1/((2^(2/3) + x)*Sqrt[1 + x^3]),x]","\frac{4 i \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{x^3+1}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{x^3+1}}\right)}{3 \sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"((4*I)*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[1 + x^3])","C",0
2,1,148,160,0.1236099,"\int \frac{1}{\left(2^{2/3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[1/((2^(2/3) - x)*Sqrt[1 - x^3]),x]","-\frac{4 i \sqrt{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{1-x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{3 \sqrt{3}}-\frac{2 \sqrt[3]{2} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"((-4*I)*Sqrt[2]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[1 - x^3])","C",0
3,1,146,163,0.1568729,"\int \frac{1}{\left(2^{2/3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[1/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]","-\frac{4 i \sqrt{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{x^3-1}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{x^3-1}}\right)}{3 \sqrt{3}}-\frac{2 \sqrt[3]{2} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"((-4*I)*Sqrt[2]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[-1 + x^3])","C",0
4,1,150,156,0.1127525,"\int \frac{1}{\left(2^{2/3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[1/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]","\frac{4 i \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{-x^3-1}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{-x^3-1}}\right)}{3 \sqrt{3}}+\frac{2 \sqrt[3]{2} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"((4*I)*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[-1 - x^3])","C",0
5,1,164,280,0.2066365,"\int \frac{1}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[1/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","-\frac{2 i \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{a+b x^3}}","\frac{2 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{3 \sqrt{3} \sqrt{a} \sqrt[3]{b}}",1,"((-2*I)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[a + b*x^3])","C",1
6,1,166,288,0.2084218,"\int \frac{1}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[1/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 i \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{a-b x^3}}","-\frac{2 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{3 \sqrt{3} \sqrt{a} \sqrt[3]{b}}",1,"((2*I)*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[a - b*x^3])","C",1
7,1,167,297,0.1235504,"\int \frac{1}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[1/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 i \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{b x^3-a}}","-\frac{2 \sqrt[3]{2} \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} \sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{3 \sqrt{3} \sqrt{a} \sqrt[3]{b}}",1,"((2*I)*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[-a + b*x^3])","C",1
8,1,167,293,0.146255,"\int \frac{1}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[1/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","-\frac{2 i \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{-a-b x^3}}","\frac{2 \sqrt[3]{2} \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} \sqrt[3]{b} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{3 \sqrt{3} \sqrt{a} \sqrt[3]{b}}",1,"((-2*I)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[-a - b*x^3])","C",1
9,1,169,249,0.2076602,"\int \frac{1}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx","Integrate[1/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]","-\frac{i 2^{5/6} \sqrt{\frac{\sqrt[3]{2} c+2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt[3]{2} d x}{c}+2^{2/3}} \Pi \left(\frac{i \sqrt[3]{2} \sqrt{3}}{2+\sqrt[3]{-2}};\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)}{\left(2+\sqrt[3]{-2}\right) d \sqrt{c^3+4 d^3 x^3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right)}{3 \sqrt{3} c^{3/2} d}+\frac{2 \sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(c+2^{2/3} d x\right) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt[3]{2} d^2 x^2}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) c+2^{2/3} d x}{\left(1+\sqrt{3}\right) c+2^{2/3} d x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} c d \sqrt{\frac{c \left(c+2^{2/3} d x\right)}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} \sqrt{c^3+4 d^3 x^3}}",1,"((-I)*2^(5/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)])/((2 + (-2)^(1/3))*d*Sqrt[c^3 + 4*d^3*x^3])","C",0
10,1,136,146,0.2096818,"\int \frac{1}{\left(1+\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[1/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{x^3+1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}+\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[1 + x^3])","C",0
11,1,136,164,0.1252956,"\int \frac{1}{\left(1+\sqrt{3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[1/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]","\frac{4 \sqrt{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{1-x^3}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}-\frac{\sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(4*Sqrt[2]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[1 - x^3])","C",0
12,1,134,167,0.1818556,"\int \frac{1}{\left(1+\sqrt{3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[1/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]","\frac{4 \sqrt{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{x^3-1}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(4*Sqrt[2]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[-1 + x^3])","C",0
13,1,138,157,0.1072124,"\int \frac{1}{\left(1+\sqrt{3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[1/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-x^3-1}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[-1 - x^3])","C",0
14,1,128,329,0.0700594,"\int \frac{1}{(3+x) \sqrt{1+x^3}} \, dx","Integrate[1/((3 + x)*Sqrt[1 + x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{7 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(\sqrt{3}+7 i\right) \sqrt{x^3+1}}","\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \tan ^{-1}\left(\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}}}{\sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}}}\right)}{\sqrt{26} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}+\frac{2 \sqrt{26+15 \sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{4 \sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(97-56 \sqrt{3};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(7*I + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((7*I + Sqrt[3])*Sqrt[1 + x^3])","C",0
15,1,128,380,0.0804593,"\int \frac{1}{(3+x) \sqrt{1-x^3}} \, dx","Integrate[1/((3 + x)*Sqrt[1 - x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)}{\left(\sqrt{3}+5 i\right) \sqrt{1-x^3}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \tanh ^{-1}\left(\frac{\sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}}}{2 \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}}}\right)}{2 \sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \left(4+\sqrt{3}\right) \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{1}{169} \left(553+304 \sqrt{3}\right);\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(-4*Sqrt[2]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])])/((5*I + Sqrt[3])*Sqrt[1 - x^3])","C",0
16,1,126,374,0.0731841,"\int \frac{1}{(3+x) \sqrt{-1+x^3}} \, dx","Integrate[1/((3 + x)*Sqrt[-1 + x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)}{\left(\sqrt{3}+5 i\right) \sqrt{x^3-1}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \tanh ^{-1}\left(\frac{\sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}}}{2 \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}}}\right)}{2 \sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{2 \sqrt{62-35 \sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{13 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{1}{169} \left(553+304 \sqrt{3}\right);\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(-4*Sqrt[2]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])])/((5*I + Sqrt[3])*Sqrt[-1 + x^3])","C",0
17,1,130,340,0.0674733,"\int \frac{1}{(3+x) \sqrt{-1-x^3}} \, dx","Integrate[1/((3 + x)*Sqrt[-1 - x^3]),x]","-\frac{4 \sqrt{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{7 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)}{\left(\sqrt{3}+7 i\right) \sqrt{-x^3-1}}","\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \tan ^{-1}\left(\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}}}{\sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}}}\right)}{\sqrt{26} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}+\frac{2 (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{4 \sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(97-56 \sqrt{3};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(7*I + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((7*I + Sqrt[3])*Sqrt[-1 - x^3])","C",0
18,0,0,139,0.0764264,"\int \frac{1}{(c+d x) \sqrt[3]{-c^3+d^3 x^3}} \, dx","Integrate[1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]","\int \frac{1}{(c+d x) \sqrt[3]{-c^3+d^3 x^3}} \, dx","-\frac{3 \log \left(2^{2/3} d \sqrt[3]{d^3 x^3-c^3}+d (c-d x)\right)}{4 \sqrt[3]{2} c d}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{\sqrt[3]{2} (c-d x)}{\sqrt[3]{d^3 x^3-c^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} c d}+\frac{\log \left((c-d x) (c+d x)^2\right)}{4 \sqrt[3]{2} c d}",1,"Integrate[1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)), x]","F",-1
19,0,0,186,0.084296,"\int \frac{1}{(c+d x) \sqrt[3]{2 c^3+d^3 x^3}} \, dx","Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)),x]","\int \frac{1}{(c+d x) \sqrt[3]{2 c^3+d^3 x^3}} \, dx","-\frac{\log \left(\sqrt[3]{2 c^3+d^3 x^3}-d x\right)}{4 c d}+\frac{3 \log \left(d (2 c+d x)-d \sqrt[3]{2 c^3+d^3 x^3}\right)}{4 c d}+\frac{\tan ^{-1}\left(\frac{\frac{2 d x}{\sqrt[3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 (2 c+d x)}{\sqrt[3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right)}{2 c d}-\frac{\log (c+d x)}{2 c d}",1,"Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x]","F",-1
20,0,0,187,0.0635146,"\int \frac{1}{(c+d x) \left(2 c^3+d^3 x^3\right)^{2/3}} \, dx","Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(2/3)),x]","\int \frac{1}{(c+d x) \left(2 c^3+d^3 x^3\right)^{2/3}} \, dx","-\frac{\log (c+d x)}{2 c^2 d}-\frac{\log \left(d x-\sqrt[3]{2 c^3+d^3 x^3}\right)}{4 c^2 d}+\frac{3 \log \left(d (2 c+d x)-d \sqrt[3]{2 c^3+d^3 x^3}\right)}{4 c^2 d}-\frac{\tan ^{-1}\left(\frac{\frac{2 d x}{\sqrt[3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right)}{2 \sqrt{3} c^2 d}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 (2 c+d x)}{\sqrt[3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right)}{2 c^2 d}",1,"Integrate[1/((c + d*x)*(2*c^3 + d^3*x^3)^(2/3)), x]","F",-1
21,0,0,147,0.0730397,"\int \frac{1}{\left(1+\sqrt[3]{2} x\right) \left(1+x^3\right)^{2/3}} \, dx","Integrate[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)),x]","\int \frac{1}{\left(1+\sqrt[3]{2} x\right) \left(1+x^3\right)^{2/3}} \, dx","-\frac{\log \left(x-\sqrt[3]{x^3+1}\right)}{2\ 2^{2/3}}+\frac{3 \log \left(-\sqrt[3]{2} \sqrt[3]{x^3+1}+\sqrt[3]{2} x+2\right)}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left(\frac{\frac{2 x}{\sqrt[3]{x^3+1}}+1}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \left(x+2^{2/3}\right)}{\sqrt[3]{x^3+1}}+1}{\sqrt{3}}\right)}{2^{2/3}}-\frac{\log \left(\sqrt[3]{2} x+1\right)}{2^{2/3}}",1,"Integrate[1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)), x]","F",-1
22,0,0,159,0.0848328,"\int \frac{1}{\left(1-\sqrt[3]{2} x\right) \left(1-x^3\right)^{2/3}} \, dx","Integrate[1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)),x]","\int \frac{1}{\left(1-\sqrt[3]{2} x\right) \left(1-x^3\right)^{2/3}} \, dx","\frac{\log \left(-\sqrt[3]{1-x^3}-x\right)}{2\ 2^{2/3}}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{1-x^3}+\sqrt[3]{2} x-2\right)}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{2\ 2^{2/3}-2 x}{\sqrt[3]{1-x^3}}+1}{\sqrt{3}}\right)}{2^{2/3}}+\frac{\tan ^{-1}\left(\frac{1-\frac{2 x}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3} \sqrt{3}}+\frac{\log \left(1-\sqrt[3]{2} x\right)}{2^{2/3}}",1,"Integrate[1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)), x]","F",-1
23,1,163,387,0.1963459,"\int (c+d x)^4 \sqrt[3]{a+b x^3} \, dx","Integrate[(c + d*x)^4*(a + b*x^3)^(1/3),x]","\frac{\sqrt[3]{a+b x^3} \left(6 b c^4 x \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x^2 \left(12 b c^3-a d^3\right) \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+d^2 \left(\left(a+b x^3\right) \sqrt[3]{\frac{b x^3}{a}+1} \left(9 c^2+d^2 x^2\right)+6 b c d x^4 \, _2F_1\left(-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)\right)\right)}{6 b \sqrt[3]{\frac{b x^3}{a}+1}}","\frac{a^2 d^4 \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{18 b^{5/3}}+\frac{a^2 d^4 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} b^{5/3}}-\frac{2 a c^3 d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{3 b^{2/3}}-\frac{4 a c^3 d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{2/3}}+\frac{a c^4 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \left(a+b x^3\right)^{2/3}}+\frac{3 a c^2 d^2 \sqrt[3]{a+b x^3}}{2 b}+\frac{1}{30} \sqrt[3]{a+b x^3} \left(15 c^4 x+40 c^3 d x^2+45 c^2 d^2 x^3+24 c d^3 x^4+5 d^4 x^5\right)+\frac{a c d^3 x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{5 \left(a+b x^3\right)^{2/3}}+\frac{a d^4 x^2 \sqrt[3]{a+b x^3}}{18 b}",1,"((a + b*x^3)^(1/3)*(6*b*c^4*x*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)] + d*(12*b*c^3 - a*d^3)*x^2*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^3)/a)] + d^2*((9*c^2 + d^2*x^2)*(a + b*x^3)*(1 + (b*x^3)/a)^(1/3) + 6*b*c*d*x^4*Hypergeometric2F1[-1/3, 4/3, 7/3, -((b*x^3)/a)])))/(6*b*(1 + (b*x^3)/a)^(1/3))","A",1
24,1,142,242,0.1349224,"\int (c+d x)^3 \sqrt[3]{a+b x^3} \, dx","Integrate[(c + d*x)^3*(a + b*x^3)^(1/3),x]","\frac{\sqrt[3]{a+b x^3} \left(4 b c^3 x \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(6 b c^2 x^2 \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+d \left(3 c \left(a+b x^3\right) \sqrt[3]{\frac{b x^3}{a}+1}+b d x^4 \, _2F_1\left(-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)\right)\right)\right)}{4 b \sqrt[3]{\frac{b x^3}{a}+1}}","-\frac{a c^2 d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 b^{2/3}}-\frac{a c^2 d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3}}+\frac{a x \left(\frac{b x^3}{a}+1\right)^{2/3} \left(5 b c^3-a d^3\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{10 b \left(a+b x^3\right)^{2/3}}+\frac{1}{20} \sqrt[3]{a+b x^3} \left(10 c^3 x+20 c^2 d x^2+15 c d^2 x^3+4 d^3 x^4\right)+\frac{3 a c d^2 \sqrt[3]{a+b x^3}}{4 b}+\frac{a d^3 x \sqrt[3]{a+b x^3}}{10 b}",1,"((a + b*x^3)^(1/3)*(4*b*c^3*x*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)] + d*(6*b*c^2*x^2*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^3)/a)] + d*(3*c*(a + b*x^3)*(1 + (b*x^3)/a)^(1/3) + b*d*x^4*Hypergeometric2F1[-1/3, 4/3, 7/3, -((b*x^3)/a)]))))/(4*b*(1 + (b*x^3)/a)^(1/3))","A",1
25,1,111,192,0.07971,"\int (c+d x)^2 \sqrt[3]{a+b x^3} \, dx","Integrate[(c + d*x)^2*(a + b*x^3)^(1/3),x]","\frac{\sqrt[3]{a+b x^3} \left(4 b c^2 x \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(4 b c x^2 \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+d \left(a+b x^3\right) \sqrt[3]{\frac{b x^3}{a}+1}\right)\right)}{4 b \sqrt[3]{\frac{b x^3}{a}+1}}","-\frac{a c d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{3 b^{2/3}}-\frac{2 a c d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{2/3}}+\frac{1}{12} \sqrt[3]{a+b x^3} \left(6 c^2 x+8 c d x^2+3 d^2 x^3\right)+\frac{a c^2 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \left(a+b x^3\right)^{2/3}}+\frac{a d^2 \sqrt[3]{a+b x^3}}{4 b}",1,"((a + b*x^3)^(1/3)*(4*b*c^2*x*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)] + d*(d*(a + b*x^3)*(1 + (b*x^3)/a)^(1/3) + 4*b*c*x^2*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^3)/a)])))/(4*b*(1 + (b*x^3)/a)^(1/3))","A",1
26,1,75,155,0.0263219,"\int (c+d x) \sqrt[3]{a+b x^3} \, dx","Integrate[(c + d*x)*(a + b*x^3)^(1/3),x]","\frac{x \sqrt[3]{a+b x^3} \left(2 c \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)\right)}{2 \sqrt[3]{\frac{b x^3}{a}+1}}","-\frac{a d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{6 b^{2/3}}-\frac{a d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{2/3}}+\frac{1}{6} \sqrt[3]{a+b x^3} \left(3 c x+2 d x^2\right)+\frac{a c x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 \left(a+b x^3\right)^{2/3}}",1,"(x*(a + b*x^3)^(1/3)*(2*c*Hypergeometric2F1[-1/3, 1/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[-1/3, 2/3, 5/3, -((b*x^3)/a)]))/(2*(1 + (b*x^3)/a)^(1/3))","A",1
27,0,0,435,0.3312917,"\int \frac{\sqrt[3]{a+b x^3}}{c+d x} \, dx","Integrate[(a + b*x^3)^(1/3)/(c + d*x),x]","\int \frac{\sqrt[3]{a+b x^3}}{c+d x} \, dx","\frac{x \sqrt[3]{a+b x^3} F_1\left(\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{c \sqrt[3]{\frac{b x^3}{a}+1}}+\frac{\sqrt[3]{b c^3-a d^3} \log \left(c^3+d^3 x^3\right)}{3 d^2}-\frac{\sqrt[3]{b c^3-a d^3} \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{2 d^2}-\frac{\sqrt[3]{b c^3-a d^3} \log \left(\sqrt[3]{b c^3-a d^3}+d \sqrt[3]{a+b x^3}\right)}{2 d^2}-\frac{\sqrt[3]{b c^3-a d^3} \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{\sqrt[3]{b c^3-a d^3} \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{a+b x^3}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{\sqrt[3]{b} c \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 d^2}+\frac{\sqrt[3]{b} c \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{\sqrt[3]{a+b x^3}}{d}",1,"Integrate[(a + b*x^3)^(1/3)/(c + d*x), x]","F",-1
28,0,0,818,0.2114828,"\int \frac{\sqrt[3]{a+b x^3}}{(c+d x)^2} \, dx","Integrate[(a + b*x^3)^(1/3)/(c + d*x)^2,x]","\int \frac{\sqrt[3]{a+b x^3}}{(c+d x)^2} \, dx","-\frac{d^3 \sqrt[3]{b x^3+a} F_1\left(\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) x^4}{2 c^5 \sqrt[3]{\frac{b x^3}{a}+1}}-\frac{d \sqrt[3]{b x^3+a} x^2}{c^3+d^3 x^3}+\frac{\sqrt[3]{b x^3+a} F_1\left(\frac{1}{3};-\frac{1}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) x}{c^2 \sqrt[3]{\frac{b x^3}{a}+1}}-\frac{\sqrt[3]{b} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{2 a d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c \left(b c^3-a d^3\right)^{2/3}}+\frac{\left(3 b c^3-2 a d^3\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c d^2 \left(b c^3-a d^3\right)^{2/3}}-\frac{b c^2 \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{b x^3+a}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} d^2 \left(b c^3-a d^3\right)^{2/3}}-\frac{a d \log \left(c^3+d^3 x^3\right)}{9 c \left(b c^3-a d^3\right)^{2/3}}-\frac{\left(3 b c^3-2 a d^3\right) \log \left(c^3+d^3 x^3\right)}{18 c d^2 \left(b c^3-a d^3\right)^{2/3}}-\frac{b c^2 \log \left(c^3+d^3 x^3\right)}{6 d^2 \left(b c^3-a d^3\right)^{2/3}}-\frac{\sqrt[3]{b} \log \left(\sqrt[3]{b} x-\sqrt[3]{b x^3+a}\right)}{2 d^2}+\frac{a d \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right)}{3 c \left(b c^3-a d^3\right)^{2/3}}+\frac{\left(3 b c^3-2 a d^3\right) \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right)}{6 c d^2 \left(b c^3-a d^3\right)^{2/3}}+\frac{b c^2 \log \left(\sqrt[3]{b x^3+a} d+\sqrt[3]{b c^3-a d^3}\right)}{2 d^2 \left(b c^3-a d^3\right)^{2/3}}-\frac{c^2 \sqrt[3]{b x^3+a}}{d \left(c^3+d^3 x^3\right)}",1,"Integrate[(a + b*x^3)^(1/3)/(c + d*x)^2, x]","F",-1
29,1,392,310,0.4635581,"\int \frac{(c+d x)^4}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x)^4/(a + b*x^3)^(1/3),x]","\frac{5 c \left(3 b c^3 \sqrt[3]{a+b x^3} \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-4 a d^3 \sqrt[3]{a+b x^3} \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)+2 \sqrt[3]{a+b x^3} \left(4 a d^3-3 b c^3\right) \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \sqrt[3]{a+b x^3} \left(3 b c^3-4 a d^3\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)+54 a \sqrt[3]{b} c d^2+24 a \sqrt[3]{b} d^3 x+54 b^{4/3} c d^2 x^3+24 b^{4/3} d^3 x^4\right)+180 b^{4/3} c^3 d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)+18 b^{4/3} d^4 x^5 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{5}{3};\frac{8}{3};-\frac{b x^3}{a}\right)}{90 b^{4/3} \sqrt[3]{a+b x^3}}","\frac{2 a c d^3 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{3 b^{4/3}}-\frac{4 a c d^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{4/3}}-\frac{c^4 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}+\frac{c^4 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{2 c^3 d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}+\frac{3 c^2 d^2 \left(a+b x^3\right)^{2/3}}{b}+\frac{4 c d^3 x \left(a+b x^3\right)^{2/3}}{3 b}+\frac{d^4 x^5 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{5}{3};\frac{8}{3};-\frac{b x^3}{a}\right)}{5 \sqrt[3]{a+b x^3}}",1,"(180*b^(4/3)*c^3*d*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)] + 18*b^(4/3)*d^4*x^5*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 5/3, 8/3, -((b*x^3)/a)] + 5*c*(54*a*b^(1/3)*c*d^2 + 24*a*b^(1/3)*d^3*x + 54*b^(4/3)*c*d^2*x^3 + 24*b^(4/3)*d^3*x^4 + 2*Sqrt[3]*(3*b*c^3 - 4*a*d^3)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] + 2*(-3*b*c^3 + 4*a*d^3)*(a + b*x^3)^(1/3)*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + 3*b*c^3*(a + b*x^3)^(1/3)*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)] - 4*a*d^3*(a + b*x^3)^(1/3)*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(90*b^(4/3)*(a + b*x^3)^(1/3))","A",1
30,1,287,255,0.3910056,"\int \frac{(c+d x)^3}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x)^3/(a + b*x^3)^(1/3),x]","\frac{1}{18} \left(\frac{3 b c^3 \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-a d^3 \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)+\left(2 a d^3-6 b c^3\right) \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \left(3 b c^3-a d^3\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)+27 \sqrt[3]{b} c d^2 \left(a+b x^3\right)^{2/3}+6 \sqrt[3]{b} d^3 x \left(a+b x^3\right)^{2/3}}{b^{4/3}}+\frac{27 c^2 d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}\right)","\frac{a d^3 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{6 b^{4/3}}-\frac{a d^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{4/3}}-\frac{c^3 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}+\frac{c^3 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{3 c^2 d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt[3]{a+b x^3}}+\frac{3 c d^2 \left(a+b x^3\right)^{2/3}}{2 b}+\frac{d^3 x \left(a+b x^3\right)^{2/3}}{3 b}",1,"((27*c^2*d*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(a + b*x^3)^(1/3) + (27*b^(1/3)*c*d^2*(a + b*x^3)^(2/3) + 6*b^(1/3)*d^3*x*(a + b*x^3)^(2/3) + 2*Sqrt[3]*(3*b*c^3 - a*d^3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] + (-6*b*c^3 + 2*a*d^3)*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + 3*b*c^3*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)] - a*d^3*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/b^(4/3))/18","A",1
31,1,201,147,0.167974,"\int \frac{(c+d x)^2}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x)^2/(a + b*x^3)^(1/3),x]","\frac{c^2 \log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)}{6 \sqrt[3]{b}}-\frac{c^2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)}{3 \sqrt[3]{b}}+\frac{c^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{c d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}+\frac{d^2 \left(a+b x^3\right)^{2/3}}{2 b}","-\frac{c^2 \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}+\frac{c^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{c d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}+\frac{d^2 \left(a+b x^3\right)^{2/3}}{2 b}",1,"(d^2*(a + b*x^3)^(2/3))/(2*b) + (c^2*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(1/3)) + (c*d*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(a + b*x^3)^(1/3) - (c^2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(3*b^(1/3)) + (c^2*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(6*b^(1/3))","A",1
32,1,163,124,0.1219093,"\int \frac{c+d x}{\sqrt[3]{a+b x^3}} \, dx","Integrate[(c + d*x)/(a + b*x^3)^(1/3),x]","\frac{1}{6} \left(\frac{c \left(\log \left(\frac{b^{2/3} x^2}{\left(a+b x^3\right)^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1\right)-2 \log \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)\right)}{\sqrt[3]{b}}+\frac{3 d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{\sqrt[3]{a+b x^3}}\right)","-\frac{c \log \left(\sqrt[3]{a+b x^3}-\sqrt[3]{b} x\right)}{2 \sqrt[3]{b}}+\frac{c \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b}}+\frac{d x^2 \sqrt[3]{\frac{b x^3}{a}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right)}{2 \sqrt[3]{a+b x^3}}",1,"((3*d*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3)/a)])/(a + b*x^3)^(1/3) + (c*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/b^(1/3))/6","A",1
33,0,0,333,0.0640543,"\int \frac{1}{(c+d x) \sqrt[3]{a+b x^3}} \, dx","Integrate[1/((c + d*x)*(a + b*x^3)^(1/3)),x]","\int \frac{1}{(c+d x) \sqrt[3]{a+b x^3}} \, dx","-\frac{d x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{2 c^2 \sqrt[3]{a+b x^3}}+\frac{\log \left(c^3+d^3 x^3\right)}{3 \sqrt[3]{b c^3-a d^3}}-\frac{\log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{b c^3-a d^3}}-\frac{\log \left(\sqrt[3]{b c^3-a d^3}+d \sqrt[3]{a+b x^3}\right)}{2 \sqrt[3]{b c^3-a d^3}}+\frac{\tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b c^3-a d^3}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{a+b x^3}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b c^3-a d^3}}",1,"Integrate[1/((c + d*x)*(a + b*x^3)^(1/3)), x]","F",-1
34,0,0,761,0.3350626,"\int \frac{1}{(c+d x)^2 \sqrt[3]{a+b x^3}} \, dx","Integrate[1/((c + d*x)^2*(a + b*x^3)^(1/3)),x]","\int \frac{1}{(c+d x)^2 \sqrt[3]{a+b x^3}} \, dx","-\frac{d x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},2;\frac{5}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{c^3 \sqrt[3]{a+b x^3}}+\frac{d^4 x^5 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{5 c^6 \sqrt[3]{a+b x^3}}-\frac{c d^3 x \left(a+b x^3\right)^{2/3}}{\left(c^3+d^3 x^3\right) \left(b c^3-a d^3\right)}+\frac{a d^3 \log \left(c^3+d^3 x^3\right)}{9 c \left(b c^3-a d^3\right)^{4/3}}+\frac{\left(3 b c^3-2 a d^3\right) \log \left(c^3+d^3 x^3\right)}{18 c \left(b c^3-a d^3\right)^{4/3}}-\frac{a d^3 \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{3 c \left(b c^3-a d^3\right)^{4/3}}-\frac{\left(3 b c^3-2 a d^3\right) \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{6 c \left(b c^3-a d^3\right)^{4/3}}+\frac{2 a d^3 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c \left(b c^3-a d^3\right)^{4/3}}+\frac{\left(3 b c^3-2 a d^3\right) \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c \left(b c^3-a d^3\right)^{4/3}}+\frac{b c^2 \log \left(c^3+d^3 x^3\right)}{6 \left(b c^3-a d^3\right)^{4/3}}-\frac{b c^2 \log \left(\sqrt[3]{b c^3-a d^3}+d \sqrt[3]{a+b x^3}\right)}{2 \left(b c^3-a d^3\right)^{4/3}}-\frac{b c^2 \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{a+b x^3}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} \left(b c^3-a d^3\right)^{4/3}}+\frac{c^2 d^2 \left(a+b x^3\right)^{2/3}}{\left(c^3+d^3 x^3\right) \left(b c^3-a d^3\right)}",1,"Integrate[1/((c + d*x)^2*(a + b*x^3)^(1/3)), x]","F",-1
35,0,0,1513,0.4768238,"\int \frac{1}{(c+d x)^3 \sqrt[3]{a+b x^3}} \, dx","Integrate[1/((c + d*x)^3*(a + b*x^3)^(1/3)),x]","\int \frac{1}{(c+d x)^3 \sqrt[3]{a+b x^3}} \, dx","\frac{2 a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right) d^6}{9 \sqrt{3} c^2 \left(b c^3-a d^3\right)^{7/3}}+\frac{a^2 \log \left(c^3+d^3 x^3\right) d^6}{27 c^2 \left(b c^3-a d^3\right)^{7/3}}-\frac{a^2 \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right) d^6}{9 c^2 \left(b c^3-a d^3\right)^{7/3}}+\frac{6 x^5 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{5}{3};\frac{1}{3},3;\frac{8}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) d^4}{5 c^7 \sqrt[3]{b x^3+a}}+\frac{7 a \left(3 b c^3-a d^3\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right) d^3}{9 \sqrt{3} c^2 \left(b c^3-a d^3\right)^{7/3}}+\frac{7 a \left(3 b c^3-a d^3\right) \log \left(c^3+d^3 x^3\right) d^3}{54 c^2 \left(b c^3-a d^3\right)^{7/3}}-\frac{7 a \left(3 b c^3-a d^3\right) \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right) d^3}{18 c^2 \left(b c^3-a d^3\right)^{7/3}}-\frac{7 \left(3 b c^3+a d^3\right) x \left(b x^3+a\right)^{2/3} d^3}{18 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}+\frac{\left(3 b c^3-7 a d^3\right) x \left(b x^3+a\right)^{2/3} d^3}{18 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}-\frac{\left(9 b c^3-5 a d^3\right) x \left(b x^3+a\right)^{2/3} d^3}{18 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}-\frac{3 c^3 x \left(b x^3+a\right)^{2/3} d^3}{2 \left(b c^3-a d^3\right) \left(c^3+d^3 x^3\right)^2}+\frac{4 b c^4 \left(b x^3+a\right)^{2/3} d^2}{3 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}-\frac{c \left(b c^3-3 a d^3\right) \left(b x^3+a\right)^{2/3} d^2}{3 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}+\frac{3 c^4 \left(b x^3+a\right)^{2/3} d^2}{2 \left(b c^3-a d^3\right) \left(c^3+d^3 x^3\right)^2}-\frac{3 x^2 \sqrt[3]{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{3},3;\frac{5}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) d}{2 c^4 \sqrt[3]{b x^3+a}}+\frac{\left(9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{9 \sqrt{3} c^2 \left(b c^3-a d^3\right)^{7/3}}-\frac{4 b^2 c^4 \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{b x^3+a}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{3 \sqrt{3} \left(b c^3-a d^3\right)^{7/3}}+\frac{b c \left(b c^3-3 a d^3\right) \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{b x^3+a}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{3 \sqrt{3} \left(b c^3-a d^3\right)^{7/3}}+\frac{\left(9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right) \log \left(c^3+d^3 x^3\right)}{54 c^2 \left(b c^3-a d^3\right)^{7/3}}+\frac{2 b^2 c^4 \log \left(c^3+d^3 x^3\right)}{9 \left(b c^3-a d^3\right)^{7/3}}-\frac{b c \left(b c^3-3 a d^3\right) \log \left(c^3+d^3 x^3\right)}{18 \left(b c^3-a d^3\right)^{7/3}}-\frac{\left(9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right) \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right)}{18 c^2 \left(b c^3-a d^3\right)^{7/3}}-\frac{2 b^2 c^4 \log \left(\sqrt[3]{b x^3+a} d+\sqrt[3]{b c^3-a d^3}\right)}{3 \left(b c^3-a d^3\right)^{7/3}}+\frac{b c \left(b c^3-3 a d^3\right) \log \left(\sqrt[3]{b x^3+a} d+\sqrt[3]{b c^3-a d^3}\right)}{6 \left(b c^3-a d^3\right)^{7/3}}",1,"Integrate[1/((c + d*x)^3*(a + b*x^3)^(1/3)), x]","F",-1
36,1,166,306,0.1444905,"\int \frac{(c+d x)^4}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x)^4/(a + b*x^3)^(2/3),x]","\frac{3 b c^4 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(x^2 \left(6 b c^3-a d^3\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right)+d \left(\left(a+b x^3\right) \left(18 c^2+d^2 x^2\right)+3 b c d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)\right)\right)}{3 b \left(a+b x^3\right)^{2/3}}","-\frac{2 c^3 d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{b^{2/3}}-\frac{4 c^3 d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3}}+\frac{a d^4 \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{3 b^{5/3}}+\frac{2 a d^4 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} b^{5/3}}+\frac{c^4 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(a+b x^3\right)^{2/3}}+\frac{6 c^2 d^2 \sqrt[3]{a+b x^3}}{b}+\frac{c d^3 x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)}{\left(a+b x^3\right)^{2/3}}+\frac{d^4 x^2 \sqrt[3]{a+b x^3}}{3 b}",1,"(3*b*c^4*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)] + d*((6*b*c^3 - a*d^3)*x^2*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)] + d*((18*c^2 + d^2*x^2)*(a + b*x^3) + 3*b*c*d*x^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 4/3, 7/3, -((b*x^3)/a)])))/(3*b*(a + b*x^3)^(2/3))","A",1
37,1,145,187,0.1032108,"\int \frac{(c+d x)^3}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x)^3/(a + b*x^3)^(2/3),x]","\frac{4 b c^3 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(6 b c^2 x^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right)+d \left(12 c \left(a+b x^3\right)+b d x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right)\right)\right)}{4 b \left(a+b x^3\right)^{2/3}}","-\frac{3 c^2 d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 b^{2/3}}-\frac{\sqrt{3} c^2 d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{b^{2/3}}+\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} \left(2 b c^3-a d^3\right) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{2 b \left(a+b x^3\right)^{2/3}}+\frac{3 c d^2 \sqrt[3]{a+b x^3}}{b}+\frac{d^3 x \sqrt[3]{a+b x^3}}{2 b}",1,"(4*b*c^3*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)] + d*(6*b*c^2*x^2*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)] + d*(12*c*(a + b*x^3) + b*d*x^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 4/3, 7/3, -((b*x^3)/a)])))/(4*b*(a + b*x^3)^(2/3))","A",1
38,1,95,141,0.0449664,"\int \frac{(c+d x)^2}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x)^2/(a + b*x^3)^(2/3),x]","\frac{b c^2 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d \left(b c x^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right)+d \left(a+b x^3\right)\right)}{b \left(a+b x^3\right)^{2/3}}","-\frac{c d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{b^{2/3}}-\frac{2 c d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3}}+\frac{c^2 x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(a+b x^3\right)^{2/3}}+\frac{d^2 \sqrt[3]{a+b x^3}}{b}",1,"(b*c^2*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)] + d*(d*(a + b*x^3) + b*c*x^2*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)]))/(b*(a + b*x^3)^(2/3))","A",1
39,1,78,121,0.0358554,"\int \frac{c+d x}{\left(a+b x^3\right)^{2/3}} \, dx","Integrate[(c + d*x)/(a + b*x^3)^(2/3),x]","\frac{x \left(2 c \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)+d x \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right)\right)}{2 \left(a+b x^3\right)^{2/3}}","-\frac{d \log \left(\sqrt[3]{b} x-\sqrt[3]{a+b x^3}\right)}{2 b^{2/3}}-\frac{d \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} x}{\sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3}}+\frac{c x \left(\frac{b x^3}{a}+1\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right)}{\left(a+b x^3\right)^{2/3}}",1,"(x*(2*c*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[2/3, 1, 5/3, (b*x^3)/(a + b*x^3)]))/(2*(a + b*x^3)^(2/3))","A",1
40,0,0,332,0.0415535,"\int \frac{1}{(c+d x) \left(a+b x^3\right)^{2/3}} \, dx","Integrate[1/((c + d*x)*(a + b*x^3)^(2/3)),x]","\int \frac{1}{(c+d x) \left(a+b x^3\right)^{2/3}} \, dx","\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{c \left(a+b x^3\right)^{2/3}}-\frac{d \log \left(c^3+d^3 x^3\right)}{3 \left(b c^3-a d^3\right)^{2/3}}+\frac{d \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{2 \left(b c^3-a d^3\right)^{2/3}}+\frac{d \log \left(\sqrt[3]{b c^3-a d^3}+d \sqrt[3]{a+b x^3}\right)}{2 \left(b c^3-a d^3\right)^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} \left(b c^3-a d^3\right)^{2/3}}-\frac{d \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{a+b x^3}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} \left(b c^3-a d^3\right)^{2/3}}",1,"Integrate[1/((c + d*x)*(a + b*x^3)^(2/3)), x]","F",-1
41,0,0,760,0.3167658,"\int \frac{1}{(c+d x)^2 \left(a+b x^3\right)^{2/3}} \, dx","Integrate[1/((c + d*x)^2*(a + b*x^3)^(2/3)),x]","\int \frac{1}{(c+d x)^2 \left(a+b x^3\right)^{2/3}} \, dx","-\frac{d^3 x^4 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{2 c^5 \left(a+b x^3\right)^{2/3}}+\frac{x \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{2}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right)}{c^2 \left(a+b x^3\right)^{2/3}}-\frac{d \left(3 b c^3-a d^3\right) \log \left(c^3+d^3 x^3\right)}{9 c \left(b c^3-a d^3\right)^{5/3}}+\frac{d \left(3 b c^3-a d^3\right) \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{3 c \left(b c^3-a d^3\right)^{5/3}}+\frac{2 d \left(3 b c^3-a d^3\right) \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c \left(b c^3-a d^3\right)^{5/3}}-\frac{a d^4 \log \left(c^3+d^3 x^3\right)}{9 c \left(b c^3-a d^3\right)^{5/3}}+\frac{a d^4 \log \left(\frac{x \sqrt[3]{b c^3-a d^3}}{c}-\sqrt[3]{a+b x^3}\right)}{3 c \left(b c^3-a d^3\right)^{5/3}}+\frac{2 a d^4 \tan ^{-1}\left(\frac{\frac{2 x \sqrt[3]{b c^3-a d^3}}{c \sqrt[3]{a+b x^3}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c \left(b c^3-a d^3\right)^{5/3}}+\frac{d^4 x^2 \sqrt[3]{a+b x^3}}{\left(c^3+d^3 x^3\right) \left(b c^3-a d^3\right)}-\frac{b c^2 d \log \left(c^3+d^3 x^3\right)}{3 \left(b c^3-a d^3\right)^{5/3}}+\frac{b c^2 d \log \left(\sqrt[3]{b c^3-a d^3}+d \sqrt[3]{a+b x^3}\right)}{\left(b c^3-a d^3\right)^{5/3}}-\frac{2 b c^2 d \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{a+b x^3}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{\sqrt{3} \left(b c^3-a d^3\right)^{5/3}}+\frac{c^2 d^2 \sqrt[3]{a+b x^3}}{\left(c^3+d^3 x^3\right) \left(b c^3-a d^3\right)}",1,"Integrate[1/((c + d*x)^2*(a + b*x^3)^(2/3)), x]","F",-1
42,0,0,1357,0.4373987,"\int \frac{1}{(c+d x)^3 \left(a+b x^3\right)^{2/3}} \, dx","Integrate[1/((c + d*x)^3*(a + b*x^3)^(2/3)),x]","\int \frac{1}{(c+d x)^3 \left(a+b x^3\right)^{2/3}} \, dx","\frac{d^6 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{7}{3};\frac{2}{3},3;\frac{10}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) x^7}{7 c^9 \left(b x^3+a\right)^{2/3}}-\frac{7 d^3 \left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{4}{3};\frac{2}{3},3;\frac{7}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) x^4}{4 c^6 \left(b x^3+a\right)^{2/3}}+\frac{d^4 \left(3 b c^3+2 a d^3\right) \sqrt[3]{b x^3+a} x^2}{3 c \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}+\frac{d^4 \left(9 b c^3-4 a d^3\right) \sqrt[3]{b x^3+a} x^2}{6 c \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}+\frac{3 c^2 d^4 \sqrt[3]{b x^3+a} x^2}{2 \left(b c^3-a d^3\right) \left(c^3+d^3 x^3\right)^2}+\frac{\left(\frac{b x^3}{a}+1\right)^{2/3} F_1\left(\frac{1}{3};\frac{2}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right) x}{c^3 \left(b x^3+a\right)^{2/3}}+\frac{2 a d^4 \left(6 b c^3-a d^3\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c^2 \left(b c^3-a d^3\right)^{8/3}}+\frac{d \left(9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right) \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b c^3-a d^3} x}{c \sqrt[3]{b x^3+a}}+1}{\sqrt{3}}\right)}{3 \sqrt{3} c^2 \left(b c^3-a d^3\right)^{8/3}}-\frac{10 b^2 c^4 d \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{b x^3+a}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{3 \sqrt{3} \left(b c^3-a d^3\right)^{8/3}}+\frac{b c d \left(b c^3-6 a d^3\right) \tan ^{-1}\left(\frac{1-\frac{2 d \sqrt[3]{b x^3+a}}{\sqrt[3]{b c^3-a d^3}}}{\sqrt{3}}\right)}{3 \sqrt{3} \left(b c^3-a d^3\right)^{8/3}}-\frac{a d^4 \left(6 b c^3-a d^3\right) \log \left(c^3+d^3 x^3\right)}{9 c^2 \left(b c^3-a d^3\right)^{8/3}}-\frac{d \left(9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right) \log \left(c^3+d^3 x^3\right)}{18 c^2 \left(b c^3-a d^3\right)^{8/3}}-\frac{5 b^2 c^4 d \log \left(c^3+d^3 x^3\right)}{9 \left(b c^3-a d^3\right)^{8/3}}+\frac{b c d \left(b c^3-6 a d^3\right) \log \left(c^3+d^3 x^3\right)}{18 \left(b c^3-a d^3\right)^{8/3}}+\frac{a d^4 \left(6 b c^3-a d^3\right) \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right)}{3 c^2 \left(b c^3-a d^3\right)^{8/3}}+\frac{d \left(9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right) \log \left(\frac{\sqrt[3]{b c^3-a d^3} x}{c}-\sqrt[3]{b x^3+a}\right)}{6 c^2 \left(b c^3-a d^3\right)^{8/3}}+\frac{5 b^2 c^4 d \log \left(\sqrt[3]{b x^3+a} d+\sqrt[3]{b c^3-a d^3}\right)}{3 \left(b c^3-a d^3\right)^{8/3}}-\frac{b c d \left(b c^3-6 a d^3\right) \log \left(\sqrt[3]{b x^3+a} d+\sqrt[3]{b c^3-a d^3}\right)}{6 \left(b c^3-a d^3\right)^{8/3}}+\frac{5 b c^4 d^2 \sqrt[3]{b x^3+a}}{3 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}-\frac{c d^2 \left(b c^3-6 a d^3\right) \sqrt[3]{b x^3+a}}{6 \left(b c^3-a d^3\right)^2 \left(c^3+d^3 x^3\right)}+\frac{3 c^4 d^2 \sqrt[3]{b x^3+a}}{2 \left(b c^3-a d^3\right) \left(c^3+d^3 x^3\right)^2}",1,"Integrate[1/((c + d*x)^3*(a + b*x^3)^(2/3)), x]","F",-1
43,1,326,37,0.4391638,"\int \frac{2^{2/3}-2 x}{\left(2^{2/3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(2^(2/3) - 2*x)/((2^(2/3) + x)*Sqrt[1 + x^3]),x]","-\frac{4 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(\sqrt{2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x+\sqrt[3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt[3]{2}+6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{x^3+1}}\right)}{\sqrt{3}}",1,"(-4*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*(6*I + (3*I)*2^(1/3) - 2*Sqrt[3] + 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - (6*I)*Sqrt[3]*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
44,1,327,40,0.3716508,"\int \frac{2^{2/3}+2 x}{\left(2^{2/3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(2^(2/3) + 2*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]","-\frac{4 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}}","-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{\sqrt{3}}",1,"(-4*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (6*I)*Sqrt[3]*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])","C",0
45,1,325,38,0.304294,"\int \frac{2^{2/3}+2 x}{\left(2^{2/3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(2^(2/3) + 2*x)/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]","-\frac{4 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}}","-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{x^3-1}}\right)}{\sqrt{3}}",1,"(-4*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (6*I)*Sqrt[3]*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[-1 + x^3])","C",0
46,1,328,39,0.2907447,"\int \frac{2^{2/3}-2 x}{\left(2^{2/3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(2^(2/3) - 2*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]","-\frac{4 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(\sqrt{2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x+\sqrt[3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt[3]{2}+6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(1+2\ 2^{2/3}-i \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{-x^3-1}}\right)}{\sqrt{3}}",1,"(-4*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*(6*I + (3*I)*2^(1/3) - 2*Sqrt[3] + 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - (6*I)*Sqrt[3]*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(1 + 2*2^(2/3) - I*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
47,1,325,63,1.1158728,"\int \frac{2^{2/3} \sqrt[3]{a}-2 \sqrt[3]{b} x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{2 \sqrt[4]{3} \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}-\frac{3 \sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt{3} \sqrt[3]{b} \sqrt{a+b x^3}}","\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((2*3^(1/4)*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))] - (3*(-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/(Sqrt[3]*b^(1/3)*Sqrt[a + b*x^3])","C",0
48,1,336,65,1.1347835,"\int \frac{2^{2/3} \sqrt[3]{a}+2 \sqrt[3]{b} x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(-\frac{2 \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}+\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{3 b^{2/3} x^2}{a^{2/3}}+\frac{3 \sqrt[3]{b} x}{\sqrt[3]{a}}+3} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt[3]{b} \sqrt{a-b x^3}}","-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))] + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[3 + (3*b^(1/3)*x)/a^(1/3) + (3*b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/(b^(1/3)*Sqrt[a - b*x^3])","C",0
49,1,390,66,0.4727032,"\int \frac{2^{2/3} \sqrt[3]{a}+2 \sqrt[3]{b} x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","-\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(2 \left(\sqrt[3]{-1}+2^{2/3}\right) \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)-\sqrt[3]{-1} 2^{2/3} \sqrt{3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(-2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(2*((-1)^(1/3) + 2^(2/3))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] - (-1)^(1/3)*2^(2/3)*Sqrt[3]*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)]))/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",0
50,1,375,66,0.6814004,"\int \frac{2^{2/3} \sqrt[3]{a}-2 \sqrt[3]{b} x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","-\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\sqrt[3]{-1} 2^{2/3} \sqrt{3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)-\frac{2 \left(\sqrt[3]{-1}+2^{2/3}\right) \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(-2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2*((-1)^(1/3) + 2^(2/3))*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/3^(1/4) + (-1)^(1/3)*2^(2/3)*Sqrt[3]*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)]))/(((-1)^(1/3) + 2^(2/3))*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
51,1,373,49,1.096616,"\int \frac{c-2 d x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx","Integrate[(c - 2*d*x)/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]","\frac{\sqrt[6]{2} \sqrt{\frac{\sqrt[3]{2} c+2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(2 \sqrt{\frac{\sqrt[3]{-2} c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(\sqrt[3]{-1} \left(2+\sqrt[3]{-2}\right) c-2 \left(\sqrt[3]{-1}+2^{2/3}\right) d x\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)-\sqrt[3]{-1} 2^{2/3} \sqrt{3} \left(1+\sqrt[3]{-1}\right) c \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt[3]{2} d x}{c}+2^{2/3}} \Pi \left(\frac{i \sqrt[3]{2} \sqrt{3}}{2+\sqrt[3]{-2}};\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)\right)}{\left(2+\sqrt[3]{-2}\right) d \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{c^3+4 d^3 x^3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right)}{\sqrt{3} \sqrt{c} d}",1,"(2^(1/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*(2*Sqrt[((-2)^(1/3)*c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*((-1)^(1/3)*(2 + (-2)^(1/3))*c - 2*((-1)^(1/3) + 2^(2/3))*d*x)*EllipticF[ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)] - (-1)^(1/3)*2^(2/3)*Sqrt[3]*(1 + (-1)^(1/3))*c*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)]))/((2 + (-2)^(1/3))*d*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 + 4*d^3*x^3])","C",0
52,1,336,158,0.5181883,"\int \frac{2+3 x}{\left(2^{2/3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(2 + 3*x)/((2^(2/3) + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(3 \sqrt{2 i x+\sqrt{3}-i} \left(\left(3 \sqrt[3]{2}+4 i \sqrt{3}+i \sqrt[3]{2} \sqrt{3}\right) x+i \sqrt[3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt[3]{2}-6\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-4 \sqrt{3} \left(\sqrt[3]{2}-3\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{2 \left(2-3\ 2^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{x^3+1}}\right)}{3 \sqrt{3}}+\frac{2 \left(3+2 \sqrt[3]{2}\right) \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 4*Sqrt[3]*(-3 + 2^(1/3))*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
53,1,335,173,0.4800006,"\int \frac{2+3 x}{\left(2^{2/3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(2 + 3*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(4 \sqrt{3} \left(3+\sqrt[3]{2}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-3 i \sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}}","\frac{2 \left(3-2 \sqrt[3]{2}\right) \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2 \left(2+3\ 2^{2/3}\right) \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{3 \sqrt{3}}",1,"(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 4*Sqrt[3]*(3 + 2^(1/3))*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])","C",0
54,1,333,176,0.3305865,"\int \frac{2+3 x}{\left(2^{2/3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(2 + 3*x)/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(4 \sqrt{3} \left(3+\sqrt[3]{2}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-3 i \sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}}","\frac{2 \left(3-2 \sqrt[3]{2}\right) \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{2 \left(2+3\ 2^{2/3}\right) \tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{x^3-1}}\right)}{3 \sqrt{3}}",1,"(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 4*Sqrt[3]*(3 + 2^(1/3))*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[-1 + x^3])","C",0
55,1,338,169,0.3407808,"\int \frac{2+3 x}{\left(2^{2/3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(2 + 3*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(3 \sqrt{2 i x+\sqrt{3}-i} \left(\left(3 \sqrt[3]{2}+4 i \sqrt{3}+i \sqrt[3]{2} \sqrt{3}\right) x+i \sqrt[3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt[3]{2}-6\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-4 \sqrt{3} \left(\sqrt[3]{2}-3\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","\frac{2 \left(2-3\ 2^{2/3}\right) \tanh ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{-x^3-1}}\right)}{3 \sqrt{3}}+\frac{2 \left(3+2 \sqrt[3]{2}\right) \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 4*Sqrt[3]*(-3 + 2^(1/3))*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
56,1,340,159,0.4747405,"\int \frac{e+f x}{\left(2^{2/3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3) + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(f \sqrt{2 i x+\sqrt{3}-i} \left(\left(3 \sqrt[3]{2}+4 i \sqrt{3}+i \sqrt[3]{2} \sqrt{3}\right) x+i \sqrt[3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt[3]{2}-6\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left(\sqrt[3]{2} e-2 f\right) \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{2 \left(e-2^{2/3} f\right) \tan ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{x^3+1}}\right)}{3 \sqrt{3}}+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(\sqrt[3]{2} e+f\right) F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(f*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 2*Sqrt[3]*(2^(1/3)*e - 2*f)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
57,1,340,175,0.4993571,"\int \frac{e+f x}{\left(2^{2/3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(2 \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left(\sqrt[3]{2} e+2 f\right) \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-i f \sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}}","-\frac{2 \left(e+2^{2/3} f\right) \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{3 \sqrt{3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(\sqrt[3]{2} e-f\right) F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*Sqrt[3]*(2^(1/3)*e + 2*f)*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])","C",0
58,1,338,178,0.3527397,"\int \frac{e+f x}{\left(2^{2/3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(2 \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left(\sqrt[3]{2} e+2 f\right) \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-i f \sqrt{-2 i x+\sqrt{3}-i} \left(\left(-3 i \sqrt[3]{2}+4 \sqrt{3}+\sqrt[3]{2} \sqrt{3}\right) x-\sqrt[3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt[3]{2}-6 i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}}","-\frac{2 \left(e+2^{2/3} f\right) \tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{x^3-1}}\right)}{3 \sqrt{3}}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(\sqrt[3]{2} e-f\right) F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*Sqrt[3] + 2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*Sqrt[3]*(2^(1/3)*e + 2*f)*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[-1 + x^3])","C",0
59,1,342,170,0.4609289,"\int \frac{e+f x}{\left(2^{2/3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt[6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(f \sqrt{2 i x+\sqrt{3}-i} \left(\left(3 \sqrt[3]{2}+4 i \sqrt{3}+i \sqrt[3]{2} \sqrt{3}\right) x+i \sqrt[3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt[3]{2}-6\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left(\sqrt[3]{2} e-2 f\right) \Pi \left(\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\sqrt{3} \left(i+2 i 2^{2/3}+\sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","\frac{2 \left(e-2^{2/3} f\right) \tanh ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{-x^3-1}}\right)}{3 \sqrt{3}}+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(\sqrt[3]{2} e+f\right) F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*2^(1/6)*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(f*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6 - 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - 2*Sqrt[3]*(2^(1/3)*e - 2*f)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
60,1,336,316,1.5410475,"\int \frac{e+f x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2^{2/3} \sqrt[3]{a} f-\sqrt[3]{b} e\right) \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}-\frac{\sqrt[4]{3} f \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{3} b^{2/3} \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{a} f+\sqrt[3]{2} \sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \left(\sqrt[3]{b} e-2^{2/3} \sqrt[3]{a} f\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{3 \sqrt{3} \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((3^(1/4)*f*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]) + ((-1)^(1/3)*(1 + (-1)^(1/3))*(-(b^(1/3)*e) + 2^(2/3)*a^(1/3)*f)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/(Sqrt[3]*b^(2/3)*Sqrt[a + b*x^3])","C",0
61,1,399,324,1.2840688,"\int \frac{e+f x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(-\left(\sqrt[3]{-1}+2^{2/3}\right) f \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2^{2/3} \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{2} \sqrt[3]{b} e-\sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \left(2^{2/3} \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{3 \sqrt{3} \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-(((-1)^(1/3) + 2^(2/3))*f*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)]) + ((-1)^(1/3)*(1 + (-1)^(1/3))*(b^(1/3)*e + 2^(2/3)*a^(1/3)*f)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",0
62,1,400,333,0.4300246,"\int \frac{e+f x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(-\left(\sqrt[3]{-1}+2^{2/3}\right) f \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2^{2/3} \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{2} \sqrt[3]{b} e-\sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2 \left(2^{2/3} \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{3 \sqrt{3} \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-(((-1)^(1/3) + 2^(2/3))*f*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)]) + ((-1)^(1/3)*(1 + (-1)^(1/3))*(b^(1/3)*e + 2^(2/3)*a^(1/3)*f)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",0
63,1,387,329,1.0277295,"\int \frac{e+f x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(e + f*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2^{2/3} \sqrt[3]{a} f-\sqrt[3]{b} e\right) \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}-\frac{\left(\sqrt[3]{-1}+2^{2/3}\right) f \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{a} f+\sqrt[3]{2} \sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{2 \left(\sqrt[3]{b} e-2^{2/3} \sqrt[3]{a} f\right) \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{3 \sqrt{3} \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3) + 2^(2/3))*f*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/3^(1/4)) + ((-1)^(1/3)*(1 + (-1)^(1/3))*(-(b^(1/3)*e) + 2^(2/3)*a^(1/3)*f)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
64,1,380,265,1.4945813,"\int \frac{e+f x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]","\frac{\sqrt[6]{2} \sqrt{\frac{\sqrt[3]{2} c+2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(-f \sqrt{\frac{\sqrt[3]{-2} c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(\sqrt[3]{-1} \left(2+\sqrt[3]{-2}\right) c-2 \left(\sqrt[3]{-1}+2^{2/3}\right) d x\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)+\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt[3]{2} d x}{c}+2^{2/3}} (c f-d e) \Pi \left(\frac{i \sqrt[3]{2} \sqrt{3}}{2+\sqrt[3]{-2}};\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(2+\sqrt[3]{-2}\right) d^2 \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{c^3+4 d^3 x^3}}","\frac{2 (d e-c f) \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right)}{3 \sqrt{3} c^{3/2} d^2}+\frac{\sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(c+2^{2/3} d x\right) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt[3]{2} d^2 x^2}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} (c f+2 d e) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) c+2^{2/3} d x}{\left(1+\sqrt{3}\right) c+2^{2/3} d x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} c d^2 \sqrt{\frac{c \left(c+2^{2/3} d x\right)}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} \sqrt{c^3+4 d^3 x^3}}",1,"(2^(1/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*(-(f*Sqrt[((-2)^(1/3)*c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*((-1)^(1/3)*(2 + (-2)^(1/3))*c - 2*((-1)^(1/3) + 2^(2/3))*d*x)*EllipticF[ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)]) + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)])/Sqrt[3]))/((2 + (-2)^(1/3))*d^2*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 + 4*d^3*x^3])","C",0
65,1,207,145,0.356621,"\int \frac{x}{\left(2^{2/3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[x/((2^(2/3) + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2-x+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt{x^3+1}}","\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{x^3+1}}\right)}{3 \sqrt{3}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*2^(2/3)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/Sqrt[1 + x^3]","C",0
66,1,209,160,0.3261251,"\int \frac{x}{\left(2^{2/3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[x/((2^(2/3) - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2+x+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt{1-x^3}}","\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{1-x^3}}\right)}{3 \sqrt{3}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*2^(2/3)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/Sqrt[1 - x^3]","C",0
67,1,207,163,0.1893872,"\int \frac{x}{\left(2^{2/3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[x/((2^(2/3) - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2+x+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt{x^3-1}}","\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \left(1-\sqrt[3]{2} x\right)}{\sqrt{x^3-1}}\right)}{3 \sqrt{3}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*2^(2/3)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/Sqrt[-1 + x^3]","C",0
68,1,209,156,0.2329655,"\int \frac{x}{\left(2^{2/3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[x/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2-x+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}\right)}{\sqrt{-x^3-1}}","\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x+1\right)}{\sqrt{-x^3-1}}\right)}{3 \sqrt{3}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*2^(2/3)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/Sqrt[-1 - x^3]","C",0
69,1,324,275,1.0783474,"\int \frac{x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[x/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}+2^{2/3}}-\frac{\sqrt[4]{3} \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{3} b^{2/3} \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{3 \sqrt{3} \sqrt[6]{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((3^(1/4)*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]) + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/(Sqrt[3]*b^(2/3)*Sqrt[a + b*x^3])","C",0
70,1,388,283,0.8878771,"\int \frac{x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[x/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","-\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\left(\sqrt[3]{-1}+2^{2/3}\right) \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)-\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2\ 2^{2/3} \tan ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{3 \sqrt{3} \sqrt[6]{a} b^{2/3}}",1,"(-2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1)^(1/3) + 2^(2/3))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] - ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",0
71,1,389,292,0.3175165,"\int \frac{x}{\left(2^{2/3} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[x/((2^(2/3)*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","-\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\left(\sqrt[3]{-1}+2^{2/3}\right) \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)-\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{3 \sqrt{3} \sqrt[6]{a} b^{2/3}}",1,"(-2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1)^(1/3) + 2^(2/3))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] - ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",0
72,1,375,288,0.7127328,"\int \frac{x}{\left(2^{2/3} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[x/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{i \sqrt{3}}{\sqrt[3]{-1}+2^{2/3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}-\frac{\left(\sqrt[3]{-1}+2^{2/3}\right) \left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3}}\right)}{\left(\sqrt[3]{-1}+2^{2/3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{2} \sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{3 \sqrt{3} \sqrt[6]{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3) + 2^(2/3))*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/3^(1/4)) + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/(((-1)^(1/3) + 2^(2/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
73,1,372,246,1.0690346,"\int \frac{x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx","Integrate[x/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]","\frac{\sqrt[6]{2} \sqrt{\frac{\sqrt[3]{2} c+2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(-\sqrt{\frac{\sqrt[3]{-2} c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(\sqrt[3]{-1} \left(2+\sqrt[3]{-2}\right) c-2 \left(\sqrt[3]{-1}+2^{2/3}\right) d x\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)+\frac{\sqrt[3]{-1} 2^{2/3} \left(1+\sqrt[3]{-1}\right) c \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt[3]{2} d x}{c}+2^{2/3}} \Pi \left(\frac{i \sqrt[3]{2} \sqrt{3}}{2+\sqrt[3]{-2}};\sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}}{\sqrt[6]{2}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(2+\sqrt[3]{-2}\right) d^2 \sqrt{\frac{\sqrt[3]{2} c+2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{c^3+4 d^3 x^3}}","\frac{\sqrt[3]{2} \sqrt{2+\sqrt{3}} \left(c+2^{2/3} d x\right) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt[3]{2} d^2 x^2}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) c+2^{2/3} d x}{\left(1+\sqrt{3}\right) c+2^{2/3} d x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} d^2 \sqrt{\frac{c \left(c+2^{2/3} d x\right)}{\left(\left(1+\sqrt{3}\right) c+2^{2/3} d x\right)^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right)}{3 \sqrt{3} \sqrt{c} d^2}",1,"(2^(1/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*(-(Sqrt[((-2)^(1/3)*c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*((-1)^(1/3)*(2 + (-2)^(1/3))*c - 2*((-1)^(1/3) + 2^(2/3))*d*x)*EllipticF[ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)]) + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*c*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)])/Sqrt[3]))/((2 + (-2)^(1/3))*d^2*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 + 4*d^3*x^3])","C",0
74,1,46,23,0.0088525,"\int \frac{1+x}{(2-x) \sqrt{1+x^3}} \, dx","Integrate[(1 + x)/((2 - x)*Sqrt[1 + x^3]),x]","\frac{1}{3} \log \left(\frac{(x+1)^2}{\sqrt{x^3+1}}+3\right)-\frac{1}{3} \log \left(3-\frac{(x+1)^2}{\sqrt{x^3+1}}\right)","\frac{2}{3} \tanh ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right)",1,"-1/3*Log[3 - (1 + x)^2/Sqrt[1 + x^3]] + Log[3 + (1 + x)^2/Sqrt[1 + x^3]]/3","A",1
75,1,54,27,0.0122634,"\int \frac{1-x}{(2+x) \sqrt{1-x^3}} \, dx","Integrate[(1 - x)/((2 + x)*Sqrt[1 - x^3]),x]","\frac{1}{3} \log \left(3-\frac{(1-x)^2}{\sqrt{1-x^3}}\right)-\frac{1}{3} \log \left(\frac{(1-x)^2}{\sqrt{1-x^3}}+3\right)","-\frac{2}{3} \tanh ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right)",1,"Log[3 - (1 - x)^2/Sqrt[1 - x^3]]/3 - Log[3 + (1 - x)^2/Sqrt[1 - x^3]]/3","A",1
76,1,25,25,0.0081149,"\int \frac{1-x}{(2+x) \sqrt{-1+x^3}} \, dx","Integrate[(1 - x)/((2 + x)*Sqrt[-1 + x^3]),x]","-\frac{2}{3} \tan ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right)","-\frac{2}{3} \tan ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right)",1,"(-2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/3","A",1
77,1,25,25,0.008953,"\int \frac{1+x}{(2-x) \sqrt{-1-x^3}} \, dx","Integrate[(1 + x)/((2 - x)*Sqrt[-1 - x^3]),x]","\frac{2}{3} \tan ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right)","\frac{2}{3} \tan ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right)",1,"(2*ArcTan[(1 + x)^2/(3*Sqrt[-1 - x^3])])/3","A",1
78,1,51,50,0.0239723,"\int \frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \left(\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1\right)^2}{3 \sqrt{a+b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}","\frac{2 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a+b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*ArcTanh[(Sqrt[a]*(1 + (b^(1/3)*x)/a^(1/3))^2)/(3*Sqrt[a + b*x^3])])/(3*a^(1/6)*b^(1/3))","A",1
79,1,53,52,0.028316,"\int \frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}\right)^2}{3 \sqrt{a-b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}","-\frac{2 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a-b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}",1,"(-2*ArcTanh[(Sqrt[a]*(1 - (b^(1/3)*x)/a^(1/3))^2)/(3*Sqrt[a - b*x^3])])/(3*a^(1/6)*b^(1/3))","A",1
80,1,54,53,0.0267172,"\int \frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \left(1-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}\right)^2}{3 \sqrt{b x^3-a}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}","-\frac{2 \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{b x^3-a}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}",1,"(-2*ArcTan[(Sqrt[a]*(1 - (b^(1/3)*x)/a^(1/3))^2)/(3*Sqrt[-a + b*x^3])])/(3*a^(1/6)*b^(1/3))","A",1
81,1,54,53,0.0233233,"\int \frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \left(\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1\right)^2}{3 \sqrt{-a-b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}","\frac{2 \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{-a-b x^3}}\right)}{3 \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*ArcTan[(Sqrt[a]*(1 + (b^(1/3)*x)/a^(1/3))^2)/(3*Sqrt[-a - b*x^3])])/(3*a^(1/6)*b^(1/3))","A",1
82,1,46,46,0.0376398,"\int \frac{c-2 d x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx","Integrate[(c - 2*d*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]","-\frac{2 \tanh ^{-1}\left(\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right)}{3 \sqrt{c} d}","-\frac{2 \tanh ^{-1}\left(\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right)}{3 \sqrt{c} d}",1,"(-2*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(3*Sqrt[c]*d)","A",1
83,1,273,139,0.3160209,"\int \frac{e+f x}{(2-x) \sqrt{1+x^3}} \, dx","Integrate[(e + f*x)/((2 - x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left(2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} (e+2 f) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-3 i f \sqrt{-2 i x+\sqrt{3}+i} \left(\left(\sqrt{3}-i\right) x-\sqrt{3}-i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(\sqrt{3}+3 i\right) \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^3+1}}","\frac{2}{9} (e+2 f) \tanh ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right)+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (e-f) F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*Sqrt[2/3]*Sqrt[((-I)*(1 + x))/(-3*I + Sqrt[3])]*((-3*I)*f*Sqrt[I + Sqrt[3] - (2*I)*x]*(-I - Sqrt[3] + (-I + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 2*Sqrt[3]*(e + 2*f)*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x^3])","C",0
84,1,271,153,0.2872017,"\int \frac{e+f x}{(2+x) \sqrt{1-x^3}} \, dx","Integrate[(e + f*x)/((2 + x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(3 f \sqrt{2 i x+\sqrt{3}+i} \left(i \sqrt{3} x+x+i \sqrt{3}-1\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} (e-2 f) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(\sqrt{3}+3 i\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}}","-\frac{2}{9} (e-2 f) \tanh ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right)-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} (e+f) F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(2*Sqrt[2/3]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(3*f*Sqrt[I + Sqrt[3] + (2*I)*x]*(-1 + I*Sqrt[3] + x + I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] - 2*Sqrt[3]*(e - 2*f)*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x^3])","C",0
85,1,269,156,0.1899959,"\int \frac{e+f x}{(2+x) \sqrt{-1+x^3}} \, dx","Integrate[(e + f*x)/((2 + x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(3 f \sqrt{2 i x+\sqrt{3}+i} \left(i \sqrt{3} x+x+i \sqrt{3}-1\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} (e-2 f) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(\sqrt{3}+3 i\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}}","-\frac{2}{9} (e-2 f) \tan ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right)-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} (e+f) F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*Sqrt[2/3]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(3*f*Sqrt[I + Sqrt[3] + (2*I)*x]*(-1 + I*Sqrt[3] + x + I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] - 2*Sqrt[3]*(e - 2*f)*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[-1 + x^3])","C",0
86,1,275,150,0.1967226,"\int \frac{e+f x}{(2-x) \sqrt{-1-x^3}} \, dx","Integrate[(e + f*x)/((2 - x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x+1)}{\sqrt{3}-3 i}} \left(2 \sqrt{3} \sqrt{2 i x+\sqrt{3}-i} \sqrt{x^2-x+1} (e+2 f) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-3 i f \sqrt{-2 i x+\sqrt{3}+i} \left(\left(\sqrt{3}-i\right) x-\sqrt{3}-i\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(\sqrt{3}+3 i\right) \sqrt{2 i x+\sqrt{3}-i} \sqrt{-x^3-1}}","\frac{2}{9} (e+2 f) \tan ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right)+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} (e-f) F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*Sqrt[2/3]*Sqrt[((-I)*(1 + x))/(-3*I + Sqrt[3])]*((-3*I)*f*Sqrt[I + Sqrt[3] - (2*I)*x]*(-I - Sqrt[3] + (-I + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 2*Sqrt[3]*(e + 2*f)*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I + Sqrt[3])*Sqrt[-I + Sqrt[3] + (2*I)*x]*Sqrt[-1 - x^3])","C",0
87,1,419,297,1.3849884,"\int \frac{e+f x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(e + f*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2 \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{\sqrt[4]{3} f \left(\left(\sqrt{3}+i\right) \sqrt[3]{a}-\left(\sqrt{3}-i\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a+b x^3}}","\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}+\frac{2 \left(2 \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a+b x^3}}\right)}{9 \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-1/2*(3^(1/4)*f*((I + Sqrt[3])*a^(1/3) - (-I + Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(b^(1/3)*e + 2*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])","C",0
88,1,447,304,1.3360234,"\int \frac{e+f x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(-i \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\sqrt[3]{b} e-2 \sqrt[3]{a} f\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{1}{2} i f \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \left(\left(\sqrt{3}-3 i\right) \sqrt[3]{a}-\left(\sqrt{3}+3 i\right) \sqrt[3]{b} x\right) F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{a} f+\sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{2 \left(\sqrt[3]{b} e-2 \sqrt[3]{a} f\right) \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a-b x^3}}\right)}{9 \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-1/2*I)*f*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*((-3*I + Sqrt[3])*a^(1/3) - (3*I + Sqrt[3])*b^(1/3)*x)*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2] - I*(b^(1/3)*e - 2*a^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",0
89,1,448,313,0.9169883,"\int \frac{e+f x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(-i \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\sqrt[3]{b} e-2 \sqrt[3]{a} f\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{1}{2} i f \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \left(\left(\sqrt{3}-3 i\right) \sqrt[3]{a}-\left(\sqrt{3}+3 i\right) \sqrt[3]{b} x\right) F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{a} f+\sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{2 \left(\sqrt[3]{b} e-2 \sqrt[3]{a} f\right) \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{b x^3-a}}\right)}{9 \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-1/2*I)*f*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*((-3*I + Sqrt[3])*a^(1/3) - (3*I + Sqrt[3])*b^(1/3)*x)*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2] - I*(b^(1/3)*e - 2*a^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",0
90,1,422,310,0.3554262,"\int \frac{e+f x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(e + f*x)/((2*a^(1/3) - b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(2 \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{\sqrt[4]{3} f \left(\left(\sqrt{3}+i\right) \sqrt[3]{a}-\left(\sqrt{3}-i\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}+\frac{2 \left(2 \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{-a-b x^3}}\right)}{9 \sqrt{a} b^{2/3}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-1/2*(3^(1/4)*f*((I + Sqrt[3])*a^(1/3) - (-I + Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(b^(1/3)*e + 2*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
91,1,384,221,1.1766663,"\int \frac{e+f x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]","-\frac{i \sqrt{\frac{c-2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(4 \sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{-\sqrt{3} c+3 i c}} \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{c^2}} (d e-c f) \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{3 i c-\sqrt{3} c}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)+f \sqrt{\frac{\left(\sqrt{3}-i\right) c+2 \left(\sqrt{3}+i\right) d x}{\left(\sqrt{3}-3 i\right) c}} \left(\left(\sqrt{3}-3 i\right) c-2 \left(\sqrt{3}+3 i\right) d x\right) F\left(\sin ^{-1}\left(\sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{3 i c-\sqrt{3} c}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{2 \left(\sqrt[3]{-1}-2\right) d^2 \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{c^3-8 d^3 x^3}}","-\frac{2 (d e-c f) \tanh ^{-1}\left(\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right)}{9 c^{3/2} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left(\left(1+\sqrt{3}\right) c-2 d x\right)^2}} (c f+2 d e) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) c-2 d x}{\left(1+\sqrt{3}\right) c-2 d x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left(\left(1+\sqrt{3}\right) c-2 d x\right)^2}} \sqrt{c^3-8 d^3 x^3}}",1,"((-1/2*I)*Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*(f*Sqrt[((-I + Sqrt[3])*c + 2*(I + Sqrt[3])*d*x)/((-3*I + Sqrt[3])*c)]*((-3*I + Sqrt[3])*c - 2*(3*I + Sqrt[3])*d*x)*EllipticF[ArcSin[Sqrt[2]*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2] + 4*Sqrt[2]*(d*e - c*f)*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[3]*c)]*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[2]*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*d^2*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3*x^3])","C",0
92,1,193,129,0.2568371,"\int \frac{x}{(2-x) \sqrt{1+x^3}} \, dx","Integrate[x/((2 - x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{2 i \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-2}\right)}{\sqrt{x^3+1}}","\frac{4}{9} \tanh ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{x^3+1}}\right)-\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((2*I)*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3))))/Sqrt[1 + x^3]","C",0
93,1,195,145,0.2428437,"\int \frac{x}{(2+x) \sqrt{1-x^3}} \, dx","Integrate[x/((2 + x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{2 i \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-2}\right)}{\sqrt{1-x^3}}","\frac{4}{9} \tanh ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right)-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((2*I)*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3))))/Sqrt[1 - x^3]","C",0
94,1,193,148,0.1081102,"\int \frac{x}{(2+x) \sqrt{-1+x^3}} \, dx","Integrate[x/((2 + x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{2 i \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-2}\right)}{\sqrt{x^3-1}}","\frac{4}{9} \tan ^{-1}\left(\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right)-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((2*I)*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3))))/Sqrt[-1 + x^3]","C",0
95,1,195,140,0.1912482,"\int \frac{x}{(2-x) \sqrt{-1-x^3}} \, dx","Integrate[x/((2 - x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{2 i \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-2}\right)}{\sqrt{-x^3-1}}","\frac{4}{9} \tan ^{-1}\left(\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right)-\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((2*I)*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3))))/Sqrt[-1 - x^3]","C",0
96,1,407,260,1.4401773,"\int \frac{x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[x/((2*a^(1/3) - b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{\sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(8 i \sqrt[3]{a} \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\sqrt{2} \sqrt[4]{3} \left(\left(\sqrt{3}+i\right) \sqrt[3]{a}-\left(\sqrt{3}-i\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{2 \left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a+b x^3}}","\frac{4 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a+b x^3}}\right)}{9 \sqrt[6]{a} b^{2/3}}-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}",1,"(Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-(Sqrt[2]*3^(1/4)*((I + Sqrt[3])*a^(1/3) - (-I + Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]) + (8*I)*a^(1/3)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/(2*(-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])","C",0
97,1,371,268,0.710914,"\int \frac{x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[x/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\left(\sqrt[3]{-1}-2\right) \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)+\frac{2 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","\frac{4 \tanh ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{a-b x^3}}\right)}{9 \sqrt[6]{a} b^{2/3}}-\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2 + (-1)^(1/3))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + (2*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",1
98,1,372,277,0.2324984,"\int \frac{x}{\left(2 \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[x/((2*a^(1/3) + b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\left(\sqrt[3]{-1}-2\right) \left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)+\frac{2 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","\frac{4 \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}-\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{b x^3-a}}\right)}{9 \sqrt[6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((-2 + (-1)^(1/3))*((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + (2*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",1
99,1,410,273,0.2995463,"\int \frac{x}{\left(2 \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[x/((2*a^(1/3) - b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{\sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(8 i \sqrt[3]{a} \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\sqrt{2} \sqrt[4]{3} \left(\left(\sqrt{3}+i\right) \sqrt[3]{a}-\left(\sqrt{3}-i\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{2 \left(\sqrt[3]{-1}-2\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{4 \tan ^{-1}\left(\frac{\left(\sqrt[3]{a}+\sqrt[3]{b} x\right)^2}{3 \sqrt[6]{a} \sqrt{-a-b x^3}}\right)}{9 \sqrt[6]{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3 \sqrt[4]{3} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}",1,"(Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-(Sqrt[2]*3^(1/4)*((I + Sqrt[3])*a^(1/3) - (-I + Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]) + (8*I)*a^(1/3)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/(2*(-2 + (-1)^(1/3))*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
100,1,295,202,0.6746825,"\int \frac{x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx","Integrate[x/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]","\frac{\sqrt{\frac{c-2 d x}{\left(1+\sqrt[3]{-1}\right) c}} \left(\left(\sqrt[3]{-1}-2\right) \left(\sqrt[3]{-1} c+2 d x\right) \sqrt{\frac{\sqrt[3]{-1} \left(c+2 \sqrt[3]{-1} d x\right)}{\left(1+\sqrt[3]{-1}\right) c}} F\left(\sin ^{-1}\left(\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}\right)|\sqrt[3]{-1}\right)+\frac{2 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) c \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{c^2}} \Pi \left(\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}}\right)|\sqrt[3]{-1}\right)}{\sqrt{3}}\right)}{\left(\sqrt[3]{-1}-2\right) d^2 \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left(1+\sqrt[3]{-1}\right) c}} \sqrt{c^3-8 d^3 x^3}}","\frac{2 \tanh ^{-1}\left(\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right)}{9 \sqrt{c} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left(\left(1+\sqrt{3}\right) c-2 d x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) c-2 d x}{\left(1+\sqrt{3}\right) c-2 d x}\right)|-7-4 \sqrt{3}\right)}{3 \sqrt[4]{3} d^2 \sqrt{\frac{c (c-2 d x)}{\left(\left(1+\sqrt{3}\right) c-2 d x\right)^2}} \sqrt{c^3-8 d^3 x^3}}",1,"(Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*((-2 + (-1)^(1/3))*((-1)^(1/3)*c + 2*d*x)*Sqrt[((-1)^(1/3)*(c + 2*(-1)^(1/3)*d*x))/((1 + (-1)^(1/3))*c)]*EllipticF[ArcSin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)] + (2*(-1)^(1/3)*(1 + (-1)^(1/3))*c*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]], (-1)^(1/3)])/Sqrt[3]))/((-2 + (-1)^(1/3))*d^2*Sqrt[(c - 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3*x^3])","C",0
101,1,267,42,0.4464352,"\int \frac{1+\sqrt{3}+x}{\left(1-\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]","-\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(4 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left(\sqrt{3}+(-2-i)\right) x-i \sqrt{3}+(1+2 i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(-3+(2+i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{2 \sqrt{3}-3}}",1,"(-2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((1 + 2*I) - I*Sqrt[3] + ((-2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (4*I)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(-3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
102,1,269,46,0.4891869,"\int \frac{1+\sqrt{3}-x}{\left(1-\sqrt{3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/((1 - Sqrt[3] - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(4 \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}+i} \left(\left((1+2 i)-i \sqrt{3}\right) x-\sqrt{3}+(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{1-x^3}}\right)}{\sqrt{2 \sqrt{3}-3}}",1,"(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((2 + I) - Sqrt[3] + ((1 + 2*I) - I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 4*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((-3*I + (1 + 2*I)*Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x^3])","C",0
103,1,267,44,0.3792338,"\int \frac{1+\sqrt{3}-x}{\left(1-\sqrt{3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/((1 - Sqrt[3] - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(4 \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}+i} \left(\left((1+2 i)-i \sqrt{3}\right) x-\sqrt{3}+(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{x^3-1}}\right)}{\sqrt{2 \sqrt{3}-3}}",1,"(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((2 + I) - Sqrt[3] + ((1 + 2*I) - I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 4*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((-3*I + (1 + 2*I)*Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[-1 + x^3])","C",0
104,1,269,44,0.3771394,"\int \frac{1+\sqrt{3}+x}{\left(1-\sqrt{3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[-1 - x^3]),x]","-\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(4 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left(\sqrt{3}+(-2-i)\right) x-i \sqrt{3}+(1+2 i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(-3+(2+i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{-x^3-1}}\right)}{\sqrt{2 \sqrt{3}-3}}",1,"(-2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((1 + 2*I) - I*Sqrt[3] + ((-2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (4*I)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(-3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
105,1,322,69,0.630908,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{4 (-1)^{5/6} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b}}-\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{a+b x^3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3^(1/4)*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))])) + (4*(-1)^(5/6)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3))))/Sqrt[a + b*x^3]","C",0
106,1,446,71,1.437469,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(4 \sqrt{3} \sqrt[3]{a} \sqrt{-\frac{2 i \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)+\sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \left(\left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b} x\right) F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*((-3 + (2 + I)*Sqrt[3])*a^(1/3) + (-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3)*x)*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2] + 4*Sqrt[3]*a^(1/3)*Sqrt[-(((2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3)))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",1
107,1,447,72,0.4318345,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(4 \sqrt{3} \sqrt[3]{a} \sqrt{-\frac{2 i \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)+\sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \left(\left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b} x\right) F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*((-3 + (2 + I)*Sqrt[3])*a^(1/3) + (-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3)*x)*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2] + 4*Sqrt[3]*a^(1/3)*Sqrt[-(((2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3)))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",1
108,1,325,72,0.5295462,"\int \frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{4 (-1)^{5/6} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt[3]{b}}-\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{-a-b x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3^(1/4)*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))])) + (4*(-1)^(5/6)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((-3*I + (1 + 2*I)*Sqrt[3])*b^(1/3))))/Sqrt[-a - b*x^3]","C",0
109,1,663,73,1.2758258,"\int \frac{1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\left(1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(1 + Sqrt[3] + (b/a)^(1/3)*x)/((1 - Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)-18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)+b x^3 \left(2 \left(3 \sqrt{3}-5\right) a-b x^3\right) \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)+8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)}{a \left(2 \left(3 \sqrt{3}-5\right) a-b x^3\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)+8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)}+12 \left(\sqrt{3}-3\right) x \sqrt[3]{\frac{b}{a}} \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{\frac{3 b x^3}{a}+3} F_1\left(1;\frac{1}{2},1;2;-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)\right)}{24 \left(3 \sqrt{3}-5\right) \sqrt{a+b x^3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left(x \sqrt[3]{\frac{b}{a}}+1\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(12*(-3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)]*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))]))))/(a*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[a + b*x^3])","C",0
110,1,648,75,1.1761406,"\int \frac{1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\left(1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-b x^3 \left(2 \left(3 \sqrt{3}-5\right) a+b x^3\right) \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right) \left(8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)\right)}{a \left(2 \left(3 \sqrt{3}-5\right) a+b x^3\right) \left(8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)}-12 \left(\sqrt{3}-3\right) x \sqrt[3]{\frac{b}{a}} \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{3-\frac{3 b x^3}{a}} F_1\left(1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)}{24 \left(3 \sqrt{3}-5\right) \sqrt{a-b x^3}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left(1-x \sqrt[3]{\frac{b}{a}}\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(-12*(-3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - b*x^3*(2*(-5 + 3*Sqrt[3])*a + b*x^3)*Sqrt[1 - (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)]*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)]))))/(a*(2*(-5 + 3*Sqrt[3])*a + b*x^3)*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[a - b*x^3])","C",0
111,1,649,76,0.5936085,"\int \frac{1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\left(1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-b x^3 \left(2 \left(3 \sqrt{3}-5\right) a+b x^3\right) \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right) \left(8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)\right)}{a \left(2 \left(3 \sqrt{3}-5\right) a+b x^3\right) \left(8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)}-12 \left(\sqrt{3}-3\right) x \sqrt[3]{\frac{b}{a}} \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{3-\frac{3 b x^3}{a}} F_1\left(1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)}{24 \left(3 \sqrt{3}-5\right) \sqrt{b x^3-a}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left(1-x \sqrt[3]{\frac{b}{a}}\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(-12*(-3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - b*x^3*(2*(-5 + 3*Sqrt[3])*a + b*x^3)*Sqrt[1 - (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)]*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)]))))/(a*(2*(-5 + 3*Sqrt[3])*a + b*x^3)*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a - 6*Sqrt[3]*a)])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[-a + b*x^3])","C",0
112,1,666,76,0.7813699,"\int \frac{1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\left(1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(1 + Sqrt[3] + (b/a)^(1/3)*x)/((1 - Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)-18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)+b x^3 \left(2 \left(3 \sqrt{3}-5\right) a-b x^3\right) \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)+8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)\right)}{a \left(2 \left(3 \sqrt{3}-5\right) a-b x^3\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)+\left(5-3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)+8 \left(3 \sqrt{3}-5\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{10 a-6 \sqrt{3} a}\right)\right)}+12 \left(\sqrt{3}-3\right) x \sqrt[3]{\frac{b}{a}} \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{\frac{3 b x^3}{a}+3} F_1\left(1;\frac{1}{2},1;2;-\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a-10 a}\right)\right)}{24 \left(3 \sqrt{3}-5\right) \sqrt{-a-b x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt{a} \left(x \sqrt[3]{\frac{b}{a}}+1\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{2 \sqrt{3}-3} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(12*(-3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] - (3*(-18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), (b*x^3)/(-10*a + 6*Sqrt[3]*a)]*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))]))))/(a*(2*(-5 + 3*Sqrt[3])*a - b*x^3)*(8*(-5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))] + (5 - 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a - 6*Sqrt[3]*a))])))))/(24*(-5 + 3*Sqrt[3])*Sqrt[-a - b*x^3])","C",0
113,1,269,42,0.4353521,"\int \frac{1-\sqrt{3}+x}{\left(1+\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(4 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left((1+2 i)+i \sqrt{3}\right) x-\sqrt{3}-(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{3+2 \sqrt{3}}}",1,"(2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 4*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
114,1,267,46,0.4874134,"\int \frac{1-\sqrt{3}-x}{\left(1+\sqrt{3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(\sqrt{2 i x+\sqrt{3}+i} \left(\left(\sqrt{3}+(2+i)\right) x+i \sqrt{3}+(1+2 i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-4 i \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right)}{\sqrt{3+2 \sqrt{3}}}",1,"(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((1 + 2*I) + I*Sqrt[3] + ((2 + I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] - (4*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3 + (2 + I)*Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x^3])","C",0
115,1,265,44,0.298008,"\int \frac{1-\sqrt{3}-x}{\left(1+\sqrt{3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left(\sqrt{2 i x+\sqrt{3}+i} \left(\left(\sqrt{3}+(2+i)\right) x+i \sqrt{3}+(1+2 i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)-4 i \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right)\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right)}{\sqrt{3+2 \sqrt{3}}}",1,"(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((1 + 2*I) + I*Sqrt[3] + ((2 + I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] - (4*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3 + (2 + I)*Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[-1 + x^3])","C",0
116,1,271,44,0.3486787,"\int \frac{1-\sqrt{3}+x}{\left(1+\sqrt{3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(4 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left((1+2 i)+i \sqrt{3}\right) x-\sqrt{3}-(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right)}{\sqrt{3+2 \sqrt{3}}}",1,"(2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 4*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
117,1,320,69,0.5942037,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{4 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt[3]{b}}-\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{a+b x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3^(1/4)*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))])) + (4*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((3 + (2 + I)*Sqrt[3])*b^(1/3))))/Sqrt[a + b*x^3]","C",0
118,1,329,71,0.8169908,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)/(((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}-\frac{4 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{3+(2+i) \sqrt{3}}\right)}{\sqrt[3]{b} \sqrt{a-b x^3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))] - (4*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3 + (2 + I)*Sqrt[3])))/(b^(1/3)*Sqrt[a - b*x^3])","C",0
119,1,330,72,0.3600274,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)/(((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{\sqrt[3]{-1} \left(\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b} x\right)}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}-\frac{4 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{3+(2+i) \sqrt{3}}\right)}{\sqrt[3]{b} \sqrt{b x^3-a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((((-1)^(1/3)*a^(1/3) + b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) + (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))] - (4*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3 + (2 + I)*Sqrt[3])))/(b^(1/3)*Sqrt[-a + b*x^3])","C",0
120,1,323,72,0.4887902,"\int \frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/(((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{2 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{4 \sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{a} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt[3]{b}}-\frac{\left(\sqrt[3]{-1} \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{b} x}{\sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{b} x+\sqrt[3]{a}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt[3]{b} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}}}\right)}{\sqrt{-a-b x^3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt[6]{a} \sqrt[3]{b}}",1,"(2*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(-((((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(3^(1/4)*b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))])) + (4*(-1)^(1/3)*(1 + (-1)^(1/3))*a^(1/3)*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/((3 + (2 + I)*Sqrt[3])*b^(1/3))))/Sqrt[-a - b*x^3]","C",0
121,1,667,73,1.2151964,"\int \frac{1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\left(1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/((1 + Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[a + b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-b x^3 \left(2 \left(5+3 \sqrt{3}\right) a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right) \left(8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)\right)}{a \left(2 \left(5+3 \sqrt{3}\right) a+b x^3\right) \left(8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)}+12 \left(3+\sqrt{3}\right) x \sqrt[3]{\frac{b}{a}} \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{\frac{3 b x^3}{a}+3} F_1\left(1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}{24 \left(5+3 \sqrt{3}\right) \sqrt{a+b x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left(x \sqrt[3]{\frac{b}{a}}+1\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 8*(b/a)^(2/3)*x^2*Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - b*x^3*(2*(5 + 3*Sqrt[3])*a + b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))]*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))]))))/(a*(2*(5 + 3*Sqrt[3])*a + b*x^3)*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])))))/(24*(5 + 3*Sqrt[3])*Sqrt[a + b*x^3])","C",0
122,1,649,75,1.1533515,"\int \frac{1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\left(1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(1 - Sqrt[3] - (b/a)^(1/3)*x)/((1 + Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[a - b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+b x^3 \left(2 \left(5+3 \sqrt{3}\right) a-b x^3\right) \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)+8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)}{a \left(2 \left(5+3 \sqrt{3}\right) a-b x^3\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)+8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}-12 \left(3+\sqrt{3}\right) x \sqrt[3]{\frac{b}{a}} \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{3-\frac{3 b x^3}{a}} F_1\left(1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}{24 \left(5+3 \sqrt{3}\right) \sqrt{a-b x^3}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left(1-x \sqrt[3]{\frac{b}{a}}\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(-12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(5 + 3*Sqrt[3])*a - b*x^3)*Sqrt[1 - (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]))))/(a*(2*(5 + 3*Sqrt[3])*a - b*x^3)*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])))))/(24*(5 + 3*Sqrt[3])*Sqrt[a - b*x^3])","C",0
123,1,650,76,0.6432815,"\int \frac{1-\sqrt{3}-\sqrt[3]{\frac{b}{a}} x}{\left(1+\sqrt{3}-\sqrt[3]{\frac{b}{a}} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(1 - Sqrt[3] - (b/a)^(1/3)*x)/((1 + Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[-a + b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+b x^3 \left(2 \left(5+3 \sqrt{3}\right) a-b x^3\right) \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)+8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)}{a \left(2 \left(5+3 \sqrt{3}\right) a-b x^3\right) \left(3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)+8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}-12 \left(3+\sqrt{3}\right) x \sqrt[3]{\frac{b}{a}} \sqrt{1-\frac{b x^3}{a}} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{3-\frac{3 b x^3}{a}} F_1\left(1;\frac{1}{2},1;2;\frac{b x^3}{a},\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}{24 \left(5+3 \sqrt{3}\right) \sqrt{b x^3-a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left(1-x \sqrt[3]{\frac{b}{a}}\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(-12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 - (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] - 8*(b/a)^(2/3)*x^2*Sqrt[3 - (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + b*x^3*(2*(5 + 3*Sqrt[3])*a - b*x^3)*Sqrt[1 - (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)]))))/(a*(2*(5 + 3*Sqrt[3])*a - b*x^3)*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, (b*x^3)/a, (b*x^3)/(10*a + 6*Sqrt[3]*a)])))))/(24*(5 + 3*Sqrt[3])*Sqrt[-a + b*x^3])","C",0
124,1,670,76,0.7164396,"\int \frac{1-\sqrt{3}+\sqrt[3]{\frac{b}{a}} x}{\left(1+\sqrt{3}+\sqrt[3]{\frac{b}{a}} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(1 - Sqrt[3] + (b/a)^(1/3)*x)/((1 + Sqrt[3] + (b/a)^(1/3)*x)*Sqrt[-a - b*x^3]),x]","\frac{x \left(-\frac{3 \left(10496 \sqrt{3} a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+18176 a^3 F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-b x^3 \left(2 \left(5+3 \sqrt{3}\right) a+b x^3\right) \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right) \left(8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)\right)}{a \left(2 \left(5+3 \sqrt{3}\right) a+b x^3\right) \left(8 \left(5+3 \sqrt{3}\right) a F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-3 b x^3 \left(F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)+\left(5+3 \sqrt{3}\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)\right)}+12 \left(3+\sqrt{3}\right) x \sqrt[3]{\frac{b}{a}} \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)-8 x^2 \left(\frac{b}{a}\right)^{2/3} \sqrt{\frac{3 b x^3}{a}+3} F_1\left(1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b x^3}{6 \sqrt{3} a+10 a}\right)\right)}{24 \left(5+3 \sqrt{3}\right) \sqrt{-a-b x^3}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} \sqrt{a} \left(x \sqrt[3]{\frac{b}{a}}+1\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{3+2 \sqrt{3}} \sqrt{a} \sqrt[3]{\frac{b}{a}}}",1,"(x*(12*(3 + Sqrt[3])*(b/a)^(1/3)*x*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 8*(b/a)^(2/3)*x^2*Sqrt[3 + (3*b*x^3)/a]*AppellF1[1, 1/2, 1, 2, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - (3*(18176*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + 10496*Sqrt[3]*a^3*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - b*x^3*(2*(5 + 3*Sqrt[3])*a + b*x^3)*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))]*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))]))))/(a*(2*(5 + 3*Sqrt[3])*a + b*x^3)*(8*(5 + 3*Sqrt[3])*a*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] - 3*b*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))] + (5 + 3*Sqrt[3])*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*x^3)/(10*a + 6*Sqrt[3]*a))])))))/(24*(5 + 3*Sqrt[3])*Sqrt[-a - b*x^3])","C",0
125,1,269,145,0.4159824,"\int \frac{1+x}{\left(1+\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(1 + x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left((1+2 i)+i \sqrt{3}\right) x-\sqrt{3}-(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{3+2 \sqrt{3}}}",1,"(2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
126,1,267,145,0.4162599,"\int \frac{1+x}{\left(1-\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(1 + x)/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]","-\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(2 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left(\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+\sqrt{2 i x+\sqrt{3}-i} \left(\left(\sqrt{3}+(-2-i)\right) x-i \sqrt{3}+(1+2 i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(-3+(2+i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{2 \sqrt{3}-3}}",1,"(-2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((1 + 2*I) - I*Sqrt[3] + ((-2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (2*I)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(-3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
127,1,291,173,0.6291528,"\int \frac{e+f x}{\left(1+\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left(\left(3+\sqrt{3}\right) f-\sqrt{3} e\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+3 f \sqrt{2 i x+\sqrt{3}-i} \left(\left((1+2 i)+i \sqrt{3}\right) x-\sqrt{3}-(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}}","\frac{\left(e-\sqrt{3} f-f\right) \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}+\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(e-\left(1-\sqrt{3}\right) f\right) F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*Sqrt[2/3]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*f*Sqrt[-I + Sqrt[3] + (2*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*(-(Sqrt[3]*e) + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x^3])","C",0
128,1,291,187,0.5792086,"\int \frac{e+f x}{\left(1+\sqrt{3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[(e + f*x)/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(2 \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left(\sqrt{3} e+\left(3+\sqrt{3}\right) f\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-3 i f \sqrt{-2 i x+\sqrt{3}-i} \left(\left(\sqrt{3}+(2-i)\right) x-i \left(\sqrt{3}+(2+i)\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}}","-\frac{\left(e+\sqrt{3} f+f\right) \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}-\frac{\sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(e+\left(1-\sqrt{3}\right) f\right) F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(2*Sqrt[2/3]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*((-I)*((2 + I) + Sqrt[3]) + ((2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*(Sqrt[3]*e + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])","C",0
129,1,289,190,0.4739645,"\int \frac{e+f x}{\left(1+\sqrt{3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[(e + f*x)/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left(2 \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left(\sqrt{3} e+\left(3+\sqrt{3}\right) f\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)-3 i f \sqrt{-2 i x+\sqrt{3}-i} \left(\left(\sqrt{3}+(2-i)\right) x-i \left(\sqrt{3}+(2+i)\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}}","-\frac{\left(e+\sqrt{3} f+f\right) \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(e+\left(1-\sqrt{3}\right) f\right) F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*Sqrt[2/3]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-3*I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*((-I)*((2 + I) + Sqrt[3]) + ((2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*(Sqrt[3]*e + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[-1 + x^3])","C",0
130,1,293,183,0.4948019,"\int \frac{e+f x}{\left(1+\sqrt{3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left(2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left(\left(3+\sqrt{3}\right) f-\sqrt{3} e\right) \Pi \left(\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)+3 f \sqrt{2 i x+\sqrt{3}-i} \left(\left((1+2 i)+i \sqrt{3}\right) x-\sqrt{3}-(2+i)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right)\right)}{\left(3 i+(1+2 i) \sqrt{3}\right) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}}","\frac{\left(e-\left(1+\sqrt{3}\right) f\right) \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right)}{\sqrt{3 \left(3+2 \sqrt{3}\right)}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(e-\left(1-\sqrt{3}\right) f\right) F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*Sqrt[2/3]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(3*f*Sqrt[-I + Sqrt[3] + (2*I)*x]*((-2 - I) - Sqrt[3] + ((1 + 2*I) + I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 2*(-(Sqrt[3]*e) + (3 + Sqrt[3])*f)*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])","C",0
131,1,438,332,1.8293959,"\int \frac{e+f x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\left(\sqrt{3}-1\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{i \sqrt[4]{3} f \left(\left(\sqrt{3}+(-2-i)\right) \sqrt[3]{a}+\left((1+2 i)-i \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a+b x^3}}","-\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} e-\left(1+\sqrt{3}\right) \sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\left(\sqrt[3]{b} e-\left(1-\sqrt{3}\right) \sqrt[3]{a} f\right) \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{\sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1/2*I)*3^(1/4)*f*(((-2 - I) + Sqrt[3])*a^(1/3) + ((1 + 2*I) - I*Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(b^(1/3)*e + (-1 + Sqrt[3])*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])","C",0
132,1,466,336,1.6705829,"\int \frac{e+f x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{1}{2} f \left(i \left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left(3-(2-i) \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-i \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\sqrt[3]{b} e-\left(\sqrt{3}-1\right) \sqrt[3]{a} f\right) \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","\frac{\sqrt{2+\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt[3]{a} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}+\frac{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{\sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*(I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 - I*(b^(1/3)*e - (-1 + Sqrt[3])*a^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",0
133,1,467,345,0.5765838,"\int \frac{e+f x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(\frac{1}{2} f \left(i \left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left(3-(2-i) \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-i \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\sqrt[3]{b} e-\left(\sqrt{3}-1\right) \sqrt[3]{a} f\right) \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \left(\left(1+\sqrt{3}\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}+\frac{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{\sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*((f*(I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 - I*(b^(1/3)*e - (-1 + Sqrt[3])*a^(1/3)*f)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",0
134,1,441,345,0.9284037,"\int \frac{e+f x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[(e + f*x)/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \left(\left(\sqrt{3}-1\right) \sqrt[3]{a} f+\sqrt[3]{b} e\right) \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{i \sqrt[4]{3} f \left(\left(\sqrt{3}+(-2-i)\right) \sqrt[3]{a}+\left((1+2 i)-i \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","-\frac{\sqrt{2-\sqrt{3}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \left(\sqrt[3]{b} e-\left(1+\sqrt{3}\right) \sqrt[3]{a} f\right) F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} \sqrt[3]{a} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{\left(\sqrt[3]{b} e-\left(1-\sqrt{3}\right) \sqrt[3]{a} f\right) \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{\sqrt{3 \left(2 \sqrt{3}-3\right)} \sqrt{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1/2*I)*3^(1/4)*f*(((-2 - I) + Sqrt[3])*a^(1/3) + ((1 + 2*I) - I*Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(b^(1/3)*e + (-1 + Sqrt[3])*a^(1/3)*f)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
135,1,209,136,0.4952839,"\int \frac{x}{\left(1+\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[x/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{2 i \left(1+\sqrt{3}\right) \sqrt{x^2-x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{3+(2+i) \sqrt{3}}\right)}{\sqrt{x^3+1}}","\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right)}{3^{3/4}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((2*I)*(1 + Sqrt[3])*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (2 + I)*Sqrt[3])))/Sqrt[1 + x^3]","C",0
136,1,232,152,0.6583474,"\int \frac{x}{\left(1+\sqrt{3}-x\right) \sqrt{1-x^3}} \, dx","Integrate[x/((1 + Sqrt[3] - x)*Sqrt[1 - x^3]),x]","\frac{2 i \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(2 \left(1+\sqrt{3}\right) \sqrt{x^2+x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\frac{i \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} \left(\left(3+(2+i) \sqrt{3}\right) x+(1+2 i) \sqrt{3}+3 i\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt{1-x^3}}","\frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{3^{3/4} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right)}{3^{3/4}}",1,"((2*I)*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((I*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*(3*I + (1 + 2*I)*Sqrt[3] + (3 + (2 + I)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + 2*(1 + Sqrt[3])*Sqrt[1 + x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((3 + (2 + I)*Sqrt[3])*Sqrt[1 - x^3])","C",0
137,1,230,164,0.2690387,"\int \frac{x}{\left(1+\sqrt{3}-x\right) \sqrt{-1+x^3}} \, dx","Integrate[x/((1 + Sqrt[3] - x)*Sqrt[-1 + x^3]),x]","\frac{2 i \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(2 \left(1+\sqrt{3}\right) \sqrt{x^2+x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\frac{i \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} \left(\left(3+(2+i) \sqrt{3}\right) x+(1+2 i) \sqrt{3}+3 i\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}\right)}{\left(3+(2+i) \sqrt{3}\right) \sqrt{x^3-1}}","\frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{x^3-1}}\right)}{3^{3/4}}",1,"((2*I)*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((I*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*(3*I + (1 + 2*I)*Sqrt[3] + (3 + (2 + I)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + 2*(1 + Sqrt[3])*Sqrt[1 + x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((3 + (2 + I)*Sqrt[3])*Sqrt[-1 + x^3])","C",0
138,1,211,156,0.1827176,"\int \frac{x}{\left(1+\sqrt{3}+x\right) \sqrt{-1-x^3}} \, dx","Integrate[x/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{2 i \left(1+\sqrt{3}\right) \sqrt{x^2-x+1} \Pi \left(\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{3+(2+i) \sqrt{3}}\right)}{\sqrt{-x^3-1}}","\frac{2 \sqrt{\frac{7}{6}-\frac{2}{\sqrt{3}}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right)}{3^{3/4}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((2*I)*(1 + Sqrt[3])*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (2 + I)*Sqrt[3])))/Sqrt[-1 - x^3]","C",0
139,1,225,147,0.5736912,"\int \frac{x}{\left(1-\sqrt{3}+x\right) \sqrt{1+x^3}} \, dx","Integrate[x/((1 - Sqrt[3] + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(\frac{\sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} \left(\left((1+2 i) \sqrt{3}-3 i\right) x-(2+i) \sqrt{3}+3\right) F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}-2 \left(\sqrt{3}-1\right) \sqrt{x^2-x+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)\right)}{\left((1+2 i) \sqrt{3}-3 i\right) \sqrt{x^3+1}}","\frac{2 \sqrt{\frac{7}{6}+\frac{2}{\sqrt{3}}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{x^3+1}}\right)}{3^{3/4}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*(3 - (2 + I)*Sqrt[3] + (-3*I + (1 + 2*I)*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] - 2*(-1 + Sqrt[3])*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((-3*I + (1 + 2*I)*Sqrt[3])*Sqrt[1 + x^3])","C",0
140,1,427,278,1.2525042,"\int \frac{x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{a+b x^3}} \, dx","Integrate[x/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a + b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \left(\sqrt{3}-1\right) \sqrt[3]{a} \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{i \sqrt[4]{3} \left(\left(\sqrt{3}+(-2-i)\right) \sqrt[3]{a}+\left((1+2 i)-i \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a+b x^3}}","\frac{2 \sqrt{\frac{7}{6}+\frac{2}{\sqrt{3}}} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{a+b x^3}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{a+b x^3}}\right)}{3^{3/4} \sqrt[6]{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1/2*I)*3^(1/4)*(((-2 - I) + Sqrt[3])*a^(1/3) + ((1 + 2*I) - I*Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])","C",0
141,1,454,286,1.3485224,"\int \frac{x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{a-b x^3}} \, dx","Integrate[x/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[a - b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \left(\sqrt{3}-1\right) \sqrt[3]{a} \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)+\frac{1}{2} \left(i \left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left(3-(2-i) \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{a-b x^3}}","\frac{2 \sqrt{\frac{7}{6}+\frac{2}{\sqrt{3}}} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^{2/3} \sqrt{\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{a-b x^3}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{a-b x^3}}\right)}{3^{3/4} \sqrt[6]{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a - b*x^3])","C",1
142,1,455,282,0.4307627,"\int \frac{x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{-a+b x^3}} \, dx","Integrate[x/(((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)*Sqrt[-a + b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}-\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \left(\sqrt{3}-1\right) \sqrt[3]{a} \sqrt{-\frac{i \left(2 \sqrt[3]{a}+\left(1-i \sqrt{3}\right) \sqrt[3]{b} x\right)}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)+\frac{1}{2} \left(i \left(-3+(2+i) \sqrt{3}\right) \sqrt[3]{a}+\left(3-(2-i) \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{\frac{\left(\sqrt{3}-i\right) \sqrt[3]{a}+\left(\sqrt{3}+i\right) \sqrt[3]{b} x}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} F\left(\sin ^{-1}\left(\sqrt{-\frac{i \left(\left(1-i \sqrt{3}\right) \sqrt[3]{b} x+2 \sqrt[3]{a}\right)}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{b x^3-a}}","\frac{\sqrt{2} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}+\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1+\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}{\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}-\sqrt[3]{b} x\right)^2}} \sqrt{b x^3-a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}-\sqrt[3]{b} x\right)}{\sqrt{b x^3-a}}\right)}{3^{3/4} \sqrt[6]{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) - b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((I*(-3 + (2 + I)*Sqrt[3])*a^(1/3) + (3 - (2 - I)*Sqrt[3])*b^(1/3)*x)*Sqrt[((-I + Sqrt[3])*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*EllipticF[ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/2 + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 + (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-I)*(2*a^(1/3) + (1 - I*Sqrt[3])*b^(1/3)*x))/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a + b*x^3])","C",1
143,1,430,278,0.9826164,"\int \frac{x}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{-a-b x^3}} \, dx","Integrate[x/(((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]","-\frac{4 \sqrt{\frac{\sqrt[3]{a}+\sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \left(i \left(\sqrt{3}-1\right) \sqrt[3]{a} \sqrt{\frac{\left(\sqrt{3}+i\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(\sqrt{3}-3 i\right) \sqrt[3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}+1} \Pi \left(\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)-\frac{i \sqrt[4]{3} \left(\left(\sqrt{3}+(-2-i)\right) \sqrt[3]{a}+\left((1+2 i)-i \sqrt{3}\right) \sqrt[3]{b} x\right) \sqrt{-\frac{2 i \sqrt[3]{b} x}{\sqrt[3]{a}}+\sqrt{3}+i} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(i+\sqrt{3}\right) \sqrt[3]{b} x-2 i \sqrt[3]{a}}{\left(-3 i+\sqrt{3}\right) \sqrt[3]{a}}}\right)|\frac{1}{2} \left(1+i \sqrt{3}\right)\right)}{2 \sqrt{2}}\right)}{\left(3-(2-i) \sqrt{3}\right) b^{2/3} \sqrt{\frac{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} x}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{a}}} \sqrt{-a-b x^3}}","\frac{\sqrt{2} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right) \sqrt{\frac{a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} x+b^{2/3} x^2}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{b} x+\left(1+\sqrt{3}\right) \sqrt[3]{a}}{\sqrt[3]{b} x+\left(1-\sqrt{3}\right) \sqrt[3]{a}}\right)|-7+4 \sqrt{3}\right)}{3^{3/4} b^{2/3} \sqrt{-\frac{\sqrt[3]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\left(\left(1-\sqrt{3}\right) \sqrt[3]{a}+\sqrt[3]{b} x\right)^2}} \sqrt{-a-b x^3}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2 \sqrt{3}-3} \sqrt[6]{a} \left(\sqrt[3]{a}+\sqrt[3]{b} x\right)}{\sqrt{-a-b x^3}}\right)}{3^{3/4} \sqrt[6]{a} b^{2/3}}",1,"(-4*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*(((-1/2*I)*3^(1/4)*(((-2 - I) + Sqrt[3])*a^(1/3) + ((1 + 2*I) - I*Sqrt[3])*b^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3)]*EllipticF[ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2])/Sqrt[2] + I*(-1 + Sqrt[3])*a^(1/3)*Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]*Sqrt[1 - (b^(1/3)*x)/a^(1/3) + (b^(2/3)*x^2)/a^(2/3)]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((-2*I)*a^(1/3) + (I + Sqrt[3])*b^(1/3)*x)/((-3*I + Sqrt[3])*a^(1/3))]], (1 + I*Sqrt[3])/2]))/((3 - (2 - I)*Sqrt[3])*b^(2/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[-a - b*x^3])","C",0
144,1,214,317,0.640919,"\int \frac{1+\sqrt{3}+x}{(c+d x) \sqrt{1+x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/((c + d*x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left(c-\left(1+\sqrt{3}\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{d \sqrt{x^3+1}}","\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\left(1+\sqrt{3}\right) d\right)^2}{\left(c-\left(1-\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \left(c-\left(1-\sqrt{3}\right) d\right)}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(c-\left(1+\sqrt{3}\right) d\right) \tan ^{-1}\left(\frac{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*(c - (1 + Sqrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(d*Sqrt[1 + x^3])","C",0
145,1,235,329,0.7637142,"\int \frac{1+\sqrt{3}-x}{(c+d x) \sqrt{1-x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/((c + d*x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} \left(\sqrt{3} c+\left(3+\sqrt{3}\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c-\sqrt[3]{-1} d}\right)}{3 d \sqrt{1-x^3}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(c+\sqrt{3} d+d\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c+\sqrt{3} d+d\right)^2}{\left(c-\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \left(c-\sqrt{3} d+d\right)}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (3 + Sqrt[3])*d)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c - (-1)^(1/3)*d)))/(3*d*Sqrt[1 - x^3])","C",0
146,1,233,325,0.2566139,"\int \frac{1+\sqrt{3}-x}{(c+d x) \sqrt{-1+x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} \left(\sqrt{3} c+\left(3+\sqrt{3}\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c-\sqrt[3]{-1} d}\right)}{3 d \sqrt{x^3-1}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(c+\sqrt{3} d+d\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c+\sqrt{3} d+d\right)^2}{\left(c-\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \left(c-\sqrt{3} d+d\right)}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (3 + Sqrt[3])*d)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c - (-1)^(1/3)*d)))/(3*d*Sqrt[-1 + x^3])","C",0
147,1,233,321,0.7436008,"\int \frac{1+\sqrt{3}+x}{(c+d x) \sqrt{-1-x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/((c + d*x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2-x+1} \left(\sqrt{3} c-\left(3+\sqrt{3}\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{3 d \sqrt{-x^3-1}}","\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\left(1+\sqrt{3}\right) d\right)^2}{\left(c-\left(1-\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \left(c-\left(1-\sqrt{3}\right) d\right)}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(c-\left(1+\sqrt{3}\right) d\right) \tan ^{-1}\left(\frac{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c - (3 + Sqrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(3*d*Sqrt[-1 - x^3])","C",0
148,1,213,358,0.5297847,"\int \frac{1-\sqrt{3}+x}{(c+d x) \sqrt{1+x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/((c + d*x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left(c+\left(\sqrt{3}-1\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{d \sqrt{x^3+1}}","-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(c-\left(1-\sqrt{3}\right) d\right) \tanh ^{-1}\left(\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left(x+\sqrt{3}+1\right)^2}{\left(x-\sqrt{3}+1\right)^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\left(1-\sqrt{3}\right) d\right)^2}{\left(c-\left(1+\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \left(c-\sqrt{3} d-d\right)}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*(c + (-1 + Sqrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(d*Sqrt[1 + x^3])","C",0
149,1,235,346,0.6973306,"\int \frac{1-\sqrt{3}-x}{(c+d x) \sqrt{1-x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} \left(\sqrt{3} c+\left(\sqrt{3}-3\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c-\sqrt[3]{-1} d}\right)}{3 d \sqrt{1-x^3}}","\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\sqrt{3} d+d\right)^2}{\left(c+\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \left(c+\sqrt{3} d+d\right)}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(c-\sqrt{3} d+d\right) \tan ^{-1}\left(\frac{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (-3 + Sqrt[3])*d)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c - (-1)^(1/3)*d)))/(3*d*Sqrt[1 - x^3])","C",0
150,1,233,342,0.1945105,"\int \frac{1-\sqrt{3}-x}{(c+d x) \sqrt{-1+x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/((c + d*x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} \left(\sqrt{3} c+\left(\sqrt{3}-3\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c-\sqrt[3]{-1} d}\right)}{3 d \sqrt{x^3-1}}","\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\sqrt{3} d+d\right)^2}{\left(c+\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \left(c+\sqrt{3} d+d\right)}-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(c-\sqrt{3} d+d\right) \tan ^{-1}\left(\frac{\sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c + (-3 + Sqrt[3])*d)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c - (-1)^(1/3)*d)))/(3*d*Sqrt[-1 + x^3])","C",0
151,1,233,362,0.609282,"\int \frac{1-\sqrt{3}+x}{(c+d x) \sqrt{-1-x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/((c + d*x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{3 \left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2-x+1} \left(\sqrt{3} c-\left(\sqrt{3}-3\right) d\right) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{3 d \sqrt{-x^3-1}}","-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(c-\left(1-\sqrt{3}\right) d\right) \tanh ^{-1}\left(\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left(x+\sqrt{3}+1\right)^2}{\left(x-\sqrt{3}+1\right)^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \Pi \left(\frac{\left(c-\left(1-\sqrt{3}\right) d\right)^2}{\left(c-\left(1+\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \left(c-\sqrt{3} d-d\right)}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((-3*((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*(1 + (-1)^(1/3))*(Sqrt[3]*c - (-3 + Sqrt[3])*d)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(3*d*Sqrt[-1 - x^3])","C",0
152,1,39,125,0.0276122,"\int \frac{1+\sqrt{3}+x}{x \sqrt{1+x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/(x*Sqrt[1 + x^3]),x]","x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)-\frac{2}{3} \left(1+\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{x^3+1}\right)","\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2}{3} \left(1+\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{x^3+1}\right)",1,"(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 + x^3]])/3 + x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3]","C",1
153,1,40,139,0.0277067,"\int \frac{1+\sqrt{3}-x}{x \sqrt{1-x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/(x*Sqrt[1 - x^3]),x]","-x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)-\frac{2}{3} \left(1+\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{1-x^3}\right)","\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2}{3} \left(1+\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{1-x^3}\right)",1,"(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 - x^3]])/3 - x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3]","C",1
154,1,58,142,0.0368285,"\int \frac{1+\sqrt{3}-x}{x \sqrt{-1+x^3}} \, dx","Integrate[(1 + Sqrt[3] - x)/(x*Sqrt[-1 + x^3]),x]","\frac{2}{3} \left(1+\sqrt{3}\right) \tan ^{-1}\left(\sqrt{x^3-1}\right)-\frac{x \sqrt{1-x^3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)}{\sqrt{x^3-1}}","\frac{2}{3} \left(1+\sqrt{3}\right) \tan ^{-1}\left(\sqrt{x^3-1}\right)+\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*(1 + Sqrt[3])*ArcTan[Sqrt[-1 + x^3]])/3 - (x*Sqrt[1 - x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, x^3])/Sqrt[-1 + x^3]","C",1
155,1,61,136,0.0374069,"\int \frac{1+\sqrt{3}+x}{x \sqrt{-1-x^3}} \, dx","Integrate[(1 + Sqrt[3] + x)/(x*Sqrt[-1 - x^3]),x]","\frac{x \sqrt{x^3+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)}{\sqrt{-x^3-1}}+\frac{2}{3} \left(1+\sqrt{3}\right) \tan ^{-1}\left(\sqrt{-x^3-1}\right)","\frac{2}{3} \left(1+\sqrt{3}\right) \tan ^{-1}\left(\sqrt{-x^3-1}\right)+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*(1 + Sqrt[3])*ArcTan[Sqrt[-1 - x^3]])/3 + (x*Sqrt[1 + x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3])/Sqrt[-1 - x^3]","C",1
156,1,41,127,0.0278995,"\int \frac{1-\sqrt{3}+x}{x \sqrt{1+x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/(x*Sqrt[1 + x^3]),x]","x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)-\frac{2}{3} \left(1-\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{x^3+1}\right)","\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2}{3} \left(1-\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{x^3+1}\right)",1,"(-2*(1 - Sqrt[3])*ArcTanh[Sqrt[1 + x^3]])/3 + x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3]","C",1
157,1,42,141,0.0245533,"\int \frac{1-\sqrt{3}-x}{x \sqrt{1-x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/(x*Sqrt[1 - x^3]),x]","-x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)-\frac{2}{3} \left(1-\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{1-x^3}\right)","\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2}{3} \left(1-\sqrt{3}\right) \tanh ^{-1}\left(\sqrt{1-x^3}\right)",1,"(-2*(1 - Sqrt[3])*ArcTanh[Sqrt[1 - x^3]])/3 - x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3]","C",1
158,1,60,144,0.0446751,"\int \frac{1-\sqrt{3}-x}{x \sqrt{-1+x^3}} \, dx","Integrate[(1 - Sqrt[3] - x)/(x*Sqrt[-1 + x^3]),x]","\frac{2}{3} \left(1-\sqrt{3}\right) \tan ^{-1}\left(\sqrt{x^3-1}\right)-\frac{x \sqrt{1-x^3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)}{\sqrt{x^3-1}}","\frac{2}{3} \left(1-\sqrt{3}\right) \tan ^{-1}\left(\sqrt{x^3-1}\right)+\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*(1 - Sqrt[3])*ArcTan[Sqrt[-1 + x^3]])/3 - (x*Sqrt[1 - x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, x^3])/Sqrt[-1 + x^3]","C",1
159,1,63,138,0.0339969,"\int \frac{1-\sqrt{3}+x}{x \sqrt{-1-x^3}} \, dx","Integrate[(1 - Sqrt[3] + x)/(x*Sqrt[-1 - x^3]),x]","\frac{x \sqrt{x^3+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)}{\sqrt{-x^3-1}}+\frac{2}{3} \left(1-\sqrt{3}\right) \tan ^{-1}\left(\sqrt{-x^3-1}\right)","\frac{2}{3} \left(1-\sqrt{3}\right) \tan ^{-1}\left(\sqrt{-x^3-1}\right)+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*(1 - Sqrt[3])*ArcTan[Sqrt[-1 - x^3]])/3 + (x*Sqrt[1 + x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3])/Sqrt[-1 - x^3]","C",1
160,1,194,332,0.2220732,"\int \frac{x}{(3+x) \sqrt{1+x^3}} \, dx","Integrate[x/((3 + x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{3 i \sqrt{x^2-x+1} \Pi \left(\frac{i \sqrt{3}}{3+\sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{3+\sqrt[3]{-1}}\right)}{\sqrt{x^3+1}}","-\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \tan ^{-1}\left(\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}}}{\sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}}}\right)}{\sqrt{26} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2 \sqrt{2 \left(97+56 \sqrt{3}\right)} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}+\frac{12 \sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(97-56 \sqrt{3};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((3*I)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/(3 + (-1)^(1/3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (-1)^(1/3))))/Sqrt[1 + x^3]","C",0
161,1,195,377,0.2260726,"\int \frac{x}{(3+x) \sqrt{1-x^3}} \, dx","Integrate[x/((3 + x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{3 i \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-3}\right)}{\sqrt{1-x^3}}","\frac{3 (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \tanh ^{-1}\left(\frac{\sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}}}{2 \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}}}\right)}{2 \sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}-\frac{2 \sqrt{2 \left(37+20 \sqrt{3}\right)} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}+\frac{12 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{1}{169} \left(553+304 \sqrt{3}\right);\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((3*I)*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-3 + (-1)^(1/3))))/Sqrt[1 - x^3]","C",0
162,1,193,373,0.1157574,"\int \frac{x}{(3+x) \sqrt{-1+x^3}} \, dx","Integrate[x/((3 + x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{\left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{3 i \sqrt{x^2+x+1} \Pi \left(\frac{2 \sqrt{3}}{5 i+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1}-3}\right)}{\sqrt{x^3-1}}","\frac{3 (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \tanh ^{-1}\left(\frac{\sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}}}{2 \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}}}\right)}{2 \sqrt{7} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}-\frac{2 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \left(4+\sqrt{3}\right) \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}+\frac{12 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \Pi \left(\frac{1}{169} \left(553+304 \sqrt{3}\right);\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{13 \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((3*I)*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(5*I + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-3 + (-1)^(1/3))))/Sqrt[-1 + x^3]","C",0
163,1,196,341,0.2078723,"\int \frac{x}{(3+x) \sqrt{-1-x^3}} \, dx","Integrate[x/((3 + x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{\left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{3 i \sqrt{x^2-x+1} \Pi \left(\frac{i \sqrt{3}}{3+\sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{3+\sqrt[3]{-1}}\right)}{\sqrt{-x^3-1}}","-\frac{3 (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \tan ^{-1}\left(\frac{\sqrt{\frac{13}{2}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}}}{\sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}}}\right)}{\sqrt{26} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}-\frac{2 \sqrt{14+8 \sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}+\frac{12 \sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \Pi \left(97-56 \sqrt{3};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{2-\sqrt{3}} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + ((3*I)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3])/(3 + (-1)^(1/3)), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 + (-1)^(1/3))))/Sqrt[-1 - x^3]","C",0
164,1,211,450,0.5628488,"\int \frac{e+f x}{(c+d x) \sqrt{1+x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{f \left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i \sqrt{x^2-x+1} (c f-d e) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{d \sqrt{x^3+1}}","\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (d e-c f) \tan ^{-1}\left(\frac{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (d e-c f) \Pi \left(\frac{\left(c-\left(1+\sqrt{3}\right) d\right)^2}{\left(c-\left(1-\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \left(c^2-2 c d-2 d^2\right)}+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \left(e-\sqrt{3} f-f\right) F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1} \left(c-\sqrt{3} d-d\right)}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((f*((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*(-(d*e) + c*f)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(d*Sqrt[1 + x^3])","C",0
165,1,233,474,0.7015169,"\int \frac{e+f x}{(c+d x) \sqrt{1-x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{3 f \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \sqrt{3} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} (c f-d e) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1} d-c}\right)}{3 d \sqrt{1-x^3}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} (d e-c f) \tanh ^{-1}\left(\frac{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} (d e-c f) \Pi \left(\frac{\left(c+\sqrt{3} d+d\right)^2}{\left(c-\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \left(c^2+2 c d-2 d^2\right)}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \left(e+\sqrt{3} f+f\right) F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3} \left(c+\sqrt{3} d+d\right)}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-c + (-1)^(1/3)*d)))/(3*d*Sqrt[1 - x^3])","C",0
166,1,231,475,0.2362901,"\int \frac{e+f x}{(c+d x) \sqrt{-1+x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \left(\frac{3 f \left(x+\sqrt[3]{-1}\right) \sqrt{\frac{(-1)^{2/3} x+\sqrt[3]{-1}}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}}+\frac{\sqrt[3]{-1} \sqrt{3} \left(1+\sqrt[3]{-1}\right) \sqrt{x^2+x+1} (c f-d e) \Pi \left(\frac{i \sqrt{3} d}{\sqrt[3]{-1} d-c};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{-1} d-c}\right)}{3 d \sqrt{x^3-1}}","-\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} (d e-c f) \tanh ^{-1}\left(\frac{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{c+d}}\right)}{\sqrt{d} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} (d e-c f) \Pi \left(\frac{\left(c+\sqrt{3} d+d\right)^2}{\left(c-\sqrt{3} d+d\right)^2};\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \left(c^2+2 c d-2 d^2\right)}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} \left(e+\sqrt{3} f+f\right) F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1} \left(c+\sqrt{3} d+d\right)}",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-c + (-1)^(1/3)*d)))/(3*d*Sqrt[-1 + x^3])","C",0
167,1,213,463,0.353075,"\int \frac{e+f x}{(c+d x) \sqrt{-1-x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \left(-\frac{f \left(\sqrt[3]{-1}-x\right) \sqrt{\frac{\sqrt[3]{-1}-(-1)^{2/3} x}{1+\sqrt[3]{-1}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}}+\frac{i \sqrt{x^2-x+1} (c f-d e) \Pi \left(\frac{i \sqrt{3} d}{c+\sqrt[3]{-1} d};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{c+\sqrt[3]{-1} d}\right)}{d \sqrt{-x^3-1}}","\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (d e-c f) \tan ^{-1}\left(\frac{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{c-d}}\right)}{\sqrt{d} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}-\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} (d e-c f) \Pi \left(\frac{\left(c-\left(1+\sqrt{3}\right) d\right)^2}{\left(c-\left(1-\sqrt{3}\right) d\right)^2};\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \left(c^2-2 c d-2 d^2\right)}+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} \left(e-\sqrt{3} f-f\right) F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1} \left(c-\sqrt{3} d-d\right)}",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*(-((f*((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*(-(d*e) + c*f)*Sqrt[1 - x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(c + (-1)^(1/3)*d), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(c + (-1)^(1/3)*d)))/(d*Sqrt[-1 - x^3])","C",0
168,1,34,120,0.0180664,"\int \frac{e+f x}{x \sqrt{1+x^3}} \, dx","Integrate[(e + f*x)/(x*Sqrt[1 + x^3]),x]","f x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)-\frac{2}{3} e \tanh ^{-1}\left(\sqrt{x^3+1}\right)","\frac{2 \sqrt{2+\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{2}{3} e \tanh ^{-1}\left(\sqrt{x^3+1}\right)",1,"(-2*e*ArcTanh[Sqrt[1 + x^3]])/3 + f*x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3]","C",1
169,1,34,134,0.0227834,"\int \frac{e+f x}{x \sqrt{1-x^3}} \, dx","Integrate[(e + f*x)/(x*Sqrt[1 - x^3]),x]","f x \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)-\frac{2}{3} e \tanh ^{-1}\left(\sqrt{1-x^3}\right)","-\frac{2}{3} e \tanh ^{-1}\left(\sqrt{1-x^3}\right)-\frac{2 \sqrt{2+\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left(-x+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{1-x}{\left(-x+\sqrt{3}+1\right)^2}} \sqrt{1-x^3}}",1,"(-2*e*ArcTanh[Sqrt[1 - x^3]])/3 + f*x*Hypergeometric2F1[1/3, 1/2, 4/3, x^3]","C",1
170,1,52,137,0.0272016,"\int \frac{e+f x}{x \sqrt{-1+x^3}} \, dx","Integrate[(e + f*x)/(x*Sqrt[-1 + x^3]),x]","\frac{2}{3} e \tan ^{-1}\left(\sqrt{x^3-1}\right)+\frac{f x \sqrt{1-x^3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right)}{\sqrt{x^3-1}}","\frac{2}{3} e \tan ^{-1}\left(\sqrt{x^3-1}\right)-\frac{2 \sqrt{2-\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left(-x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{1-x}{\left(-x-\sqrt{3}+1\right)^2}} \sqrt{x^3-1}}",1,"(2*e*ArcTan[Sqrt[-1 + x^3]])/3 + (f*x*Sqrt[1 - x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, x^3])/Sqrt[-1 + x^3]","C",1
171,1,56,131,0.02939,"\int \frac{e+f x}{x \sqrt{-1-x^3}} \, dx","Integrate[(e + f*x)/(x*Sqrt[-1 - x^3]),x]","\frac{2}{3} e \tan ^{-1}\left(\sqrt{-x^3-1}\right)+\frac{f x \sqrt{x^3+1} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right)}{\sqrt{-x^3-1}}","\frac{2}{3} e \tan ^{-1}\left(\sqrt{-x^3-1}\right)+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left(x-\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right)|-7+4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{-\frac{x+1}{\left(x-\sqrt{3}+1\right)^2}} \sqrt{-x^3-1}}",1,"(2*e*ArcTan[Sqrt[-1 - x^3]])/3 + (f*x*Sqrt[1 + x^3]*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3])/Sqrt[-1 - x^3]","C",1
172,0,0,95,0.1431779,"\int \frac{c-d x}{(c+d x) \sqrt[3]{2 c^3+d^3 x^3}} \, dx","Integrate[(c - d*x)/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)),x]","\int \frac{c-d x}{(c+d x) \sqrt[3]{2 c^3+d^3 x^3}} \, dx","\frac{3 \log \left(d (2 c+d x)-d \sqrt[3]{2 c^3+d^3 x^3}\right)}{2 d}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 (2 c+d x)}{\sqrt[3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right)}{d}-\frac{\log (c+d x)}{d}",1,"Integrate[(c - d*x)/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x]","F",-1
173,0,0,234,0.1844311,"\int \frac{e+f x}{(c+d x) \sqrt[3]{-c^3+d^3 x^3}} \, dx","Integrate[(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)),x]","\int \frac{e+f x}{(c+d x) \sqrt[3]{-c^3+d^3 x^3}} \, dx","-\frac{3 (d e-c f) \log \left(2^{2/3} d \sqrt[3]{d^3 x^3-c^3}+d (c-d x)\right)}{4 \sqrt[3]{2} c d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left(\frac{1-\frac{\sqrt[3]{2} (c-d x)}{\sqrt[3]{d^3 x^3-c^3}}}{\sqrt{3}}\right)}{2 \sqrt[3]{2} c d^2}-\frac{f \log \left(\sqrt[3]{d^3 x^3-c^3}-d x\right)}{2 d^2}+\frac{f \tan ^{-1}\left(\frac{\frac{2 d x}{\sqrt[3]{d^3 x^3-c^3}}+1}{\sqrt{3}}\right)}{\sqrt{3} d^2}+\frac{(d e-c f) \log \left((c-d x) (c+d x)^2\right)}{4 \sqrt[3]{2} c d^2}",1,"Integrate[(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)), x]","F",-1
174,1,133,160,0.123881,"\int x^2 (a+b x)^n \left(c+d x^3\right) \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^3),x]","\frac{(a+b x)^{n+1} \left(\frac{(a+b x)^2 \left(b^3 c-10 a^3 d\right)}{n+3}+\frac{a (a+b x) \left(5 a^3 d-2 b^3 c\right)}{n+2}+\frac{10 a^2 d (a+b x)^3}{n+4}+\frac{a^2 b^3 c-a^5 d}{n+1}+\frac{d (a+b x)^5}{n+6}-\frac{5 a d (a+b x)^4}{n+5}\right)}{b^6}","-\frac{a \left(2 b^3 c-5 a^3 d\right) (a+b x)^{n+2}}{b^6 (n+2)}+\frac{\left(b^3 c-10 a^3 d\right) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{10 a^2 d (a+b x)^{n+4}}{b^6 (n+4)}+\frac{a^2 \left(b^3 c-a^3 d\right) (a+b x)^{n+1}}{b^6 (n+1)}-\frac{5 a d (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d (a+b x)^{n+6}}{b^6 (n+6)}",1,"((a + b*x)^(1 + n)*((a^2*b^3*c - a^5*d)/(1 + n) + (a*(-2*b^3*c + 5*a^3*d)*(a + b*x))/(2 + n) + ((b^3*c - 10*a^3*d)*(a + b*x)^2)/(3 + n) + (10*a^2*d*(a + b*x)^3)/(4 + n) - (5*a*d*(a + b*x)^4)/(5 + n) + (d*(a + b*x)^5)/(6 + n)))/b^6","A",1
175,1,104,126,0.0797183,"\int x (a+b x)^n \left(c+d x^3\right) \, dx","Integrate[x*(a + b*x)^n*(c + d*x^3),x]","\frac{(a+b x)^{n+1} \left(\frac{(a+b x) \left(b^3 c-4 a^3 d\right)}{n+2}+\frac{a \left(a^3 d-b^3 c\right)}{n+1}+\frac{6 a^2 d (a+b x)^2}{n+3}+\frac{d (a+b x)^4}{n+5}-\frac{4 a d (a+b x)^3}{n+4}\right)}{b^5}","-\frac{a \left(b^3 c-a^3 d\right) (a+b x)^{n+1}}{b^5 (n+1)}+\frac{\left(b^3 c-4 a^3 d\right) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)}",1,"((a + b*x)^(1 + n)*((a*(-(b^3*c) + a^3*d))/(1 + n) + ((b^3*c - 4*a^3*d)*(a + b*x))/(2 + n) + (6*a^2*d*(a + b*x)^2)/(3 + n) - (4*a*d*(a + b*x)^3)/(4 + n) + (d*(a + b*x)^4)/(5 + n)))/b^5","A",1
176,1,94,94,0.071227,"\int (a+b x)^n \left(c+d x^3\right) \, dx","Integrate[(a + b*x)^n*(c + d*x^3),x]","\frac{\left(b^3 c-a^3 d\right) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)}","\frac{\left(b^3 c-a^3 d\right) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)}",1,"((b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*a^2*d*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d*(a + b*x)^(4 + n))/(b^4*(4 + n))","A",1
177,1,94,99,0.0572182,"\int \frac{(a+b x)^n \left(c+d x^3\right)}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^3))/x,x]","\frac{(a+b x)^{n+1} \left(a d \left(2 a^2-2 a b (n+1) x+b^2 \left(n^2+3 n+2\right) x^2\right)-b^3 c \left(n^2+5 n+6\right) \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)\right)}{a b^3 (n+1) (n+2) (n+3)}","\frac{a^2 d (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac{d (a+b x)^{n+3}}{b^3 (n+3)}-\frac{c (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"((a + b*x)^(1 + n)*(a*d*(2*a^2 - 2*a*b*(1 + n)*x + b^2*(2 + 3*n + n^2)*x^2) - b^3*c*(6 + 5*n + n^2)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a*b^3*(1 + n)*(2 + n)*(3 + n))","A",1
178,1,252,294,0.2699058,"\int x^2 (a+b x)^n \left(c+d x^3\right)^2 \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^3)^2,x]","\frac{(a+b x)^{n+1} \left(\frac{\left(a b^3 c-a^4 d\right)^2}{n+1}+\frac{2 d (a+b x)^5 \left(b^3 c-28 a^3 d\right)}{n+6}+\frac{10 a d (a+b x)^4 \left(7 a^3 d-b^3 c\right)}{n+5}-\frac{2 a (a+b x) \left(b^3 c-4 a^3 d\right) \left(b^3 c-a^3 d\right)}{n+2}+\frac{28 a^2 d^2 (a+b x)^6}{n+7}+\frac{(a+b x)^2 \left(28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right)}{n+3}+\frac{4 a^2 d (a+b x)^3 \left(5 b^3 c-14 a^3 d\right)}{n+4}+\frac{d^2 (a+b x)^8}{n+9}-\frac{8 a d^2 (a+b x)^7}{n+8}\right)}{b^9}","-\frac{2 a \left(b^3 c-4 a^3 d\right) \left(b^3 c-a^3 d\right) (a+b x)^{n+2}}{b^9 (n+2)}-\frac{10 a d \left(b^3 c-7 a^3 d\right) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{2 d \left(b^3 c-28 a^3 d\right) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}+\frac{\left(28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{a^2 \left(b^3 c-a^3 d\right)^2 (a+b x)^{n+1}}{b^9 (n+1)}+\frac{4 a^2 d \left(5 b^3 c-14 a^3 d\right) (a+b x)^{n+4}}{b^9 (n+4)}-\frac{8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^2 (a+b x)^{n+9}}{b^9 (n+9)}",1,"((a + b*x)^(1 + n)*((a*b^3*c - a^4*d)^2/(1 + n) - (2*a*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x))/(2 + n) + ((b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^2)/(3 + n) + (4*a^2*d*(5*b^3*c - 14*a^3*d)*(a + b*x)^3)/(4 + n) + (10*a*d*(-(b^3*c) + 7*a^3*d)*(a + b*x)^4)/(5 + n) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^5)/(6 + n) + (28*a^2*d^2*(a + b*x)^6)/(7 + n) - (8*a*d^2*(a + b*x)^7)/(8 + n) + (d^2*(a + b*x)^8)/(9 + n)))/b^9","A",1
179,1,211,248,0.205118,"\int x (a+b x)^n \left(c+d x^3\right)^2 \, dx","Integrate[x*(a + b*x)^n*(c + d*x^3)^2,x]","\frac{(a+b x)^{n+1} \left(\frac{d (a+b x)^4 \left(2 b^3 c-35 a^3 d\right)}{n+5}+\frac{a d (a+b x)^3 \left(35 a^3 d-8 b^3 c\right)}{n+4}+\frac{(a+b x) \left(b^3 c-7 a^3 d\right) \left(b^3 c-a^3 d\right)}{n+2}-\frac{a \left(b^3 c-a^3 d\right)^2}{n+1}+\frac{21 a^2 d^2 (a+b x)^5}{n+6}+\frac{3 a^2 d (a+b x)^2 \left(4 b^3 c-7 a^3 d\right)}{n+3}+\frac{d^2 (a+b x)^7}{n+8}-\frac{7 a d^2 (a+b x)^6}{n+7}\right)}{b^8}","-\frac{a \left(b^3 c-a^3 d\right)^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left(b^3 c-7 a^3 d\right) \left(b^3 c-a^3 d\right) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{a d \left(8 b^3 c-35 a^3 d\right) (a+b x)^{n+4}}{b^8 (n+4)}+\frac{d \left(2 b^3 c-35 a^3 d\right) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac{3 a^2 d \left(4 b^3 c-7 a^3 d\right) (a+b x)^{n+3}}{b^8 (n+3)}-\frac{7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^2 (a+b x)^{n+8}}{b^8 (n+8)}",1,"((a + b*x)^(1 + n)*(-((a*(b^3*c - a^3*d)^2)/(1 + n)) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x))/(2 + n) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^2)/(3 + n) + (a*d*(-8*b^3*c + 35*a^3*d)*(a + b*x)^3)/(4 + n) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^4)/(5 + n) + (21*a^2*d^2*(a + b*x)^5)/(6 + n) - (7*a*d^2*(a + b*x)^6)/(7 + n) + (d^2*(a + b*x)^7)/(8 + n)))/b^8","A",1
180,1,172,203,0.1700968,"\int (a+b x)^n \left(c+d x^3\right)^2 \, dx","Integrate[(a + b*x)^n*(c + d*x^3)^2,x]","\frac{(a+b x)^{n+1} \left(\frac{2 d (a+b x)^3 \left(b^3 c-10 a^3 d\right)}{n+4}+\frac{3 a d (a+b x)^2 \left(5 a^3 d-2 b^3 c\right)}{n+3}+\frac{\left(b^3 c-a^3 d\right)^2}{n+1}+\frac{15 a^2 d^2 (a+b x)^4}{n+5}+\frac{6 a^2 d (a+b x) \left(b^3 c-a^3 d\right)}{n+2}+\frac{d^2 (a+b x)^6}{n+7}-\frac{6 a d^2 (a+b x)^5}{n+6}\right)}{b^7}","\frac{\left(b^3 c-a^3 d\right)^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac{3 a d \left(2 b^3 c-5 a^3 d\right) (a+b x)^{n+3}}{b^7 (n+3)}+\frac{2 d \left(b^3 c-10 a^3 d\right) (a+b x)^{n+4}}{b^7 (n+4)}+\frac{15 a^2 d^2 (a+b x)^{n+5}}{b^7 (n+5)}+\frac{6 a^2 d \left(b^3 c-a^3 d\right) (a+b x)^{n+2}}{b^7 (n+2)}-\frac{6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac{d^2 (a+b x)^{n+7}}{b^7 (n+7)}",1,"((a + b*x)^(1 + n)*((b^3*c - a^3*d)^2/(1 + n) + (6*a^2*d*(b^3*c - a^3*d)*(a + b*x))/(2 + n) + (3*a*d*(-2*b^3*c + 5*a^3*d)*(a + b*x)^2)/(3 + n) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^3)/(4 + n) + (15*a^2*d^2*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))/b^7","A",1
181,1,188,209,0.1666321,"\int \frac{(a+b x)^n \left(c+d x^3\right)^2}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^3)^2)/x,x]","(a+b x)^{n+1} \left(\frac{2 d (a+b x)^2 \left(b^3 c-5 a^3 d\right)}{b^6 (n+3)}+\frac{a d (a+b x) \left(5 a^3 d-4 b^3 c\right)}{b^6 (n+2)}+\frac{10 a^2 d^2 (a+b x)^3}{b^6 (n+4)}+\frac{a^2 d \left(2 b^3 c-a^3 d\right)}{b^6 (n+1)}+\frac{d^2 (a+b x)^5}{b^6 (n+6)}-\frac{5 a d^2 (a+b x)^4}{b^6 (n+5)}-\frac{c^2 \, _2F_1\left(1,n+1;n+2;\frac{a+b x}{a}\right)}{a n+a}\right)","-\frac{a d \left(4 b^3 c-5 a^3 d\right) (a+b x)^{n+2}}{b^6 (n+2)}+\frac{2 d \left(b^3 c-5 a^3 d\right) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{10 a^2 d^2 (a+b x)^{n+4}}{b^6 (n+4)}+\frac{a^2 d \left(2 b^3 c-a^3 d\right) (a+b x)^{n+1}}{b^6 (n+1)}-\frac{5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d^2 (a+b x)^{n+6}}{b^6 (n+6)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"(a + b*x)^(1 + n)*((a^2*d*(2*b^3*c - a^3*d))/(b^6*(1 + n)) + (a*d*(-4*b^3*c + 5*a^3*d)*(a + b*x))/(b^6*(2 + n)) + (2*d*(b^3*c - 5*a^3*d)*(a + b*x)^2)/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^3)/(b^6*(4 + n)) - (5*a*d^2*(a + b*x)^4)/(b^6*(5 + n)) + (d^2*(a + b*x)^5)/(b^6*(6 + n)) - (c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))","A",1
182,1,402,459,0.4713129,"\int x^2 (a+b x)^n \left(c+d x^3\right)^3 \, dx","Integrate[x^2*(a + b*x)^n*(c + d*x^3)^3,x]","\frac{(a+b x)^{n+1} \left(\frac{3 d^2 (a+b x)^8 \left(b^3 c-55 a^3 d\right)}{n+9}+\frac{6 a d^2 (a+b x)^7 \left(55 a^3 d-4 b^3 c\right)}{n+8}+\frac{a (a+b x) \left(b^3 c-a^3 d\right)^2 \left(11 a^3 d-2 b^3 c\right)}{n+2}+\frac{55 a^2 d^3 (a+b x)^9}{n+10}+\frac{3 d (a+b x)^5 \left(154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right)}{n+6}-\frac{15 a d (a+b x)^4 \left(22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right)}{n+5}+\frac{(a+b x)^2 \left(b^3 c-a^3 d\right) \left(55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right)}{n+3}+\frac{42 a^2 d^2 (a+b x)^6 \left(2 b^3 c-11 a^3 d\right)}{n+7}+\frac{a^2 \left(b^3 c-a^3 d\right)^3}{n+1}+\frac{3 a^2 d (a+b x)^3 \left(55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right)}{n+4}+\frac{d^3 (a+b x)^{11}}{n+12}-\frac{11 a d^3 (a+b x)^{10}}{n+11}\right)}{b^{12}}","-\frac{6 a d^2 \left(4 b^3 c-55 a^3 d\right) (a+b x)^{n+8}}{b^{12} (n+8)}+\frac{3 d^2 \left(b^3 c-55 a^3 d\right) (a+b x)^{n+9}}{b^{12} (n+9)}-\frac{a \left(2 b^3 c-11 a^3 d\right) \left(b^3 c-a^3 d\right)^2 (a+b x)^{n+2}}{b^{12} (n+2)}+\frac{55 a^2 d^3 (a+b x)^{n+10}}{b^{12} (n+10)}+\frac{\left(b^3 c-a^3 d\right) \left(55 a^6 d^2-29 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+3}}{b^{12} (n+3)}-\frac{15 a d \left(22 a^6 d^2-14 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+5}}{b^{12} (n+5)}+\frac{3 d \left(154 a^6 d^2-56 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+6}}{b^{12} (n+6)}+\frac{42 a^2 d^2 \left(2 b^3 c-11 a^3 d\right) (a+b x)^{n+7}}{b^{12} (n+7)}+\frac{a^2 \left(b^3 c-a^3 d\right)^3 (a+b x)^{n+1}}{b^{12} (n+1)}+\frac{3 a^2 d \left(55 a^6 d^2-56 a^3 b^3 c d+10 b^6 c^2\right) (a+b x)^{n+4}}{b^{12} (n+4)}-\frac{11 a d^3 (a+b x)^{n+11}}{b^{12} (n+11)}+\frac{d^3 (a+b x)^{n+12}}{b^{12} (n+12)}",1,"((a + b*x)^(1 + n)*((a^2*(b^3*c - a^3*d)^3)/(1 + n) + (a*(b^3*c - a^3*d)^2*(-2*b^3*c + 11*a^3*d)*(a + b*x))/(2 + n) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^2)/(3 + n) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^3)/(4 + n) - (15*a*d*(b^6*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^4)/(5 + n) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a^6*d^2)*(a + b*x)^5)/(6 + n) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*(a + b*x)^6)/(7 + n) + (6*a*d^2*(-4*b^3*c + 55*a^3*d)*(a + b*x)^7)/(8 + n) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^8)/(9 + n) + (55*a^2*d^3*(a + b*x)^9)/(10 + n) - (11*a*d^3*(a + b*x)^10)/(11 + n) + (d^3*(a + b*x)^11)/(12 + n)))/b^12","A",1
183,1,345,396,0.3814589,"\int x (a+b x)^n \left(c+d x^3\right)^3 \, dx","Integrate[x*(a + b*x)^n*(c + d*x^3)^3,x]","\frac{(a+b x)^{n+1} \left(\frac{3 d^2 (a+b x)^7 \left(b^3 c-40 a^3 d\right)}{n+8}+\frac{21 a d^2 (a+b x)^6 \left(10 a^3 d-b^3 c\right)}{n+7}+\frac{(a+b x) \left(b^3 c-10 a^3 d\right) \left(b^3 c-a^3 d\right)^2}{n+2}+\frac{a \left(a^3 d-b^3 c\right)^3}{n+1}+\frac{45 a^2 d^3 (a+b x)^8}{n+9}+\frac{3 d (a+b x)^4 \left(70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right)}{n+5}-\frac{3 a d (a+b x)^3 \left(40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right)}{n+4}+\frac{63 a^2 d^2 (a+b x)^5 \left(b^3 c-4 a^3 d\right)}{n+6}+\frac{9 a^2 d (a+b x)^2 \left(a^3 d-b^3 c\right) \left(5 a^3 d-2 b^3 c\right)}{n+3}+\frac{d^3 (a+b x)^{10}}{n+11}-\frac{10 a d^3 (a+b x)^9}{n+10}\right)}{b^{11}}","-\frac{21 a d^2 \left(b^3 c-10 a^3 d\right) (a+b x)^{n+7}}{b^{11} (n+7)}+\frac{3 d^2 \left(b^3 c-40 a^3 d\right) (a+b x)^{n+8}}{b^{11} (n+8)}-\frac{a \left(b^3 c-a^3 d\right)^3 (a+b x)^{n+1}}{b^{11} (n+1)}+\frac{\left(b^3 c-10 a^3 d\right) \left(b^3 c-a^3 d\right)^2 (a+b x)^{n+2}}{b^{11} (n+2)}+\frac{45 a^2 d^3 (a+b x)^{n+9}}{b^{11} (n+9)}-\frac{3 a d \left(40 a^6 d^2-35 a^3 b^3 c d+4 b^6 c^2\right) (a+b x)^{n+4}}{b^{11} (n+4)}+\frac{3 d \left(70 a^6 d^2-35 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+5}}{b^{11} (n+5)}+\frac{63 a^2 d^2 \left(b^3 c-4 a^3 d\right) (a+b x)^{n+6}}{b^{11} (n+6)}+\frac{9 a^2 d \left(2 b^3 c-5 a^3 d\right) \left(b^3 c-a^3 d\right) (a+b x)^{n+3}}{b^{11} (n+3)}-\frac{10 a d^3 (a+b x)^{n+10}}{b^{11} (n+10)}+\frac{d^3 (a+b x)^{n+11}}{b^{11} (n+11)}",1,"((a + b*x)^(1 + n)*((a*(-(b^3*c) + a^3*d)^3)/(1 + n) + ((b^3*c - 10*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x))/(2 + n) + (9*a^2*d*(-(b^3*c) + a^3*d)*(-2*b^3*c + 5*a^3*d)*(a + b*x)^2)/(3 + n) - (3*a*d*(4*b^6*c^2 - 35*a^3*b^3*c*d + 40*a^6*d^2)*(a + b*x)^3)/(4 + n) + (3*d*(b^6*c^2 - 35*a^3*b^3*c*d + 70*a^6*d^2)*(a + b*x)^4)/(5 + n) + (63*a^2*d^2*(b^3*c - 4*a^3*d)*(a + b*x)^5)/(6 + n) + (21*a*d^2*(-(b^3*c) + 10*a^3*d)*(a + b*x)^6)/(7 + n) + (3*d^2*(b^3*c - 40*a^3*d)*(a + b*x)^7)/(8 + n) + (45*a^2*d^3*(a + b*x)^8)/(9 + n) - (10*a*d^3*(a + b*x)^9)/(10 + n) + (d^3*(a + b*x)^10)/(11 + n)))/b^11","A",1
184,1,290,337,0.3593726,"\int (a+b x)^n \left(c+d x^3\right)^3 \, dx","Integrate[(a + b*x)^n*(c + d*x^3)^3,x]","\frac{(a+b x)^{n+1} \left(\frac{9 d (a+b x) \left(a b^3 c-a^4 d\right)^2}{n+2}+\frac{3 d^2 (a+b x)^6 \left(b^3 c-28 a^3 d\right)}{n+7}+\frac{18 a d^2 (a+b x)^5 \left(7 a^3 d-b^3 c\right)}{n+6}-\frac{9 a d (a+b x)^2 \left(b^3 c-4 a^3 d\right) \left(b^3 c-a^3 d\right)}{n+3}+\frac{\left(b^3 c-a^3 d\right)^3}{n+1}+\frac{36 a^2 d^3 (a+b x)^7}{n+8}+\frac{3 d (a+b x)^3 \left(28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right)}{n+4}+\frac{9 a^2 d^2 (a+b x)^4 \left(5 b^3 c-14 a^3 d\right)}{n+5}+\frac{d^3 (a+b x)^9}{n+10}-\frac{9 a d^3 (a+b x)^8}{n+9}\right)}{b^{10}}","-\frac{18 a d^2 \left(b^3 c-7 a^3 d\right) (a+b x)^{n+6}}{b^{10} (n+6)}+\frac{3 d^2 \left(b^3 c-28 a^3 d\right) (a+b x)^{n+7}}{b^{10} (n+7)}+\frac{\left(b^3 c-a^3 d\right)^3 (a+b x)^{n+1}}{b^{10} (n+1)}-\frac{9 a d \left(b^3 c-4 a^3 d\right) \left(b^3 c-a^3 d\right) (a+b x)^{n+3}}{b^{10} (n+3)}+\frac{36 a^2 d^3 (a+b x)^{n+8}}{b^{10} (n+8)}+\frac{3 d \left(28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right) (a+b x)^{n+4}}{b^{10} (n+4)}+\frac{9 a^2 d^2 \left(5 b^3 c-14 a^3 d\right) (a+b x)^{n+5}}{b^{10} (n+5)}+\frac{9 a^2 d \left(b^3 c-a^3 d\right)^2 (a+b x)^{n+2}}{b^{10} (n+2)}-\frac{9 a d^3 (a+b x)^{n+9}}{b^{10} (n+9)}+\frac{d^3 (a+b x)^{n+10}}{b^{10} (n+10)}",1,"((a + b*x)^(1 + n)*((b^3*c - a^3*d)^3/(1 + n) + (9*d*(a*b^3*c - a^4*d)^2*(a + b*x))/(2 + n) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^2)/(3 + n) + (3*d*(b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^3)/(4 + n) + (9*a^2*d^2*(5*b^3*c - 14*a^3*d)*(a + b*x)^4)/(5 + n) + (18*a*d^2*(-(b^3*c) + 7*a^3*d)*(a + b*x)^5)/(6 + n) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x)^6)/(7 + n) + (36*a^2*d^3*(a + b*x)^7)/(8 + n) - (9*a*d^3*(a + b*x)^8)/(9 + n) + (d^3*(a + b*x)^9)/(10 + n)))/b^10","A",1
185,1,332,358,0.3467615,"\int \frac{(a+b x)^n \left(c+d x^3\right)^3}{x} \, dx","Integrate[((a + b*x)^n*(c + d*x^3)^3)/x,x]","(a+b x)^{n+1} \left(\frac{d^2 (a+b x)^5 \left(3 b^3 c-56 a^3 d\right)}{b^9 (n+6)}+\frac{5 a d^2 (a+b x)^4 \left(14 a^3 d-3 b^3 c\right)}{b^9 (n+5)}+\frac{28 a^2 d^3 (a+b x)^6}{b^9 (n+7)}+\frac{d (a+b x)^2 \left(28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right)}{b^9 (n+3)}-\frac{a d (a+b x) \left(8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right)}{b^9 (n+2)}+\frac{2 a^2 d^2 (a+b x)^3 \left(15 b^3 c-28 a^3 d\right)}{b^9 (n+4)}+\frac{a^2 d \left(a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right)}{b^9 (n+1)}+\frac{d^3 (a+b x)^8}{b^9 (n+9)}-\frac{8 a d^3 (a+b x)^7}{b^9 (n+8)}-\frac{c^3 \, _2F_1\left(1,n+1;n+2;\frac{a+b x}{a}\right)}{a n+a}\right)","-\frac{5 a d^2 \left(3 b^3 c-14 a^3 d\right) (a+b x)^{n+5}}{b^9 (n+5)}+\frac{d^2 \left(3 b^3 c-56 a^3 d\right) (a+b x)^{n+6}}{b^9 (n+6)}+\frac{28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac{a d \left(8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right) (a+b x)^{n+2}}{b^9 (n+2)}+\frac{d \left(28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right) (a+b x)^{n+3}}{b^9 (n+3)}+\frac{2 a^2 d^2 \left(15 b^3 c-28 a^3 d\right) (a+b x)^{n+4}}{b^9 (n+4)}+\frac{a^2 d \left(a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right) (a+b x)^{n+1}}{b^9 (n+1)}-\frac{8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac{d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac{c^3 (a+b x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b x}{a}+1\right)}{a (n+1)}",1,"(a + b*x)^(1 + n)*((a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2))/(b^9*(1 + n)) - (a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x))/(b^9*(2 + n)) + (d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^2)/(b^9*(3 + n)) + (2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^3)/(b^9*(4 + n)) + (5*a*d^2*(-3*b^3*c + 14*a^3*d)*(a + b*x)^4)/(b^9*(5 + n)) + (d^2*(3*b^3*c - 56*a^3*d)*(a + b*x)^5)/(b^9*(6 + n)) + (28*a^2*d^3*(a + b*x)^6)/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^7)/(b^9*(8 + n)) + (d^3*(a + b*x)^8)/(b^9*(9 + n)) - (c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))","A",1
186,1,284,324,0.6137223,"\int \frac{x^5 (e+f x)^n}{a+b x^3} \, dx","Integrate[(x^5*(e + f*x)^n)/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(\frac{a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{(n+1) \left(\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e\right)}+\frac{a \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{(n+1) \left(\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f\right)}+\frac{3 b^{2/3} e^2}{f^3 (n+1)}-\frac{6 b^{2/3} e (e+f x)}{f^3 (n+2)}+\frac{3 b^{2/3} (e+f x)^2}{f^3 (n+3)}\right)}{3 b^{5/3}}","\frac{a (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b^{5/3} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{a (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{3 b^{5/3} (n+1) \left(\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e\right)}+\frac{a (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{3 b^{5/3} (n+1) \left(\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f\right)}+\frac{e^2 (e+f x)^{n+1}}{b f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b f^3 (n+3)}",1,"((e + f*x)^(1 + n)*((3*b^(2/3)*e^2)/(f^3*(1 + n)) - (6*b^(2/3)*e*(e + f*x))/(f^3*(2 + n)) + (3*b^(2/3)*(e + f*x)^2)/(f^3*(3 + n)) + (a*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)])/((b^(1/3)*e - a^(1/3)*f)*(1 + n)) + (a*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)])/((b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) + (a*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)])/((b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n))))/(3*b^(5/3))","A",1
187,1,292,332,0.6827473,"\int \frac{x^4 (e+f x)^n}{a+b x^3} \, dx","Integrate[(x^4*(e + f*x)^n)/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(-\frac{a^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{\sqrt[3]{-1} a^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{(-1)^{2/3} a^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{(n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}+\frac{3 \sqrt[3]{b} (e+f x)}{f^2 (n+2)}-\frac{3 \sqrt[3]{b} e}{f^2 (n+1)}\right)}{3 b^{4/3}}","-\frac{a^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b^{4/3} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{\sqrt[3]{-1} a^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b^{4/3} (n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{(-1)^{2/3} a^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{3 b^{4/3} (n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}-\frac{e (e+f x)^{n+1}}{b f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b f^2 (n+2)}",1,"((e + f*x)^(1 + n)*((-3*b^(1/3)*e)/(f^2*(1 + n)) + (3*b^(1/3)*(e + f*x))/(f^2*(2 + n)) - (a^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)])/((b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(1/3)*a^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/(((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(2/3)*a^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)])/(((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))))/(3*b^(4/3))","A",1
188,1,239,293,0.2400585,"\int \frac{x^3 (e+f x)^n}{a+b x^3} \, dx","Integrate[(x^3*(e + f*x)^n)/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(\frac{\sqrt[3]{a} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}+\frac{\sqrt[3]{a} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}-\frac{\sqrt[3]{a} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e}+\frac{3}{f}\right)}{3 b (n+1)}","\frac{\sqrt[3]{a} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{\sqrt[3]{a} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b (n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}-\frac{\sqrt[3]{a} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{3 b (n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}+\frac{(e+f x)^{n+1}}{b f (n+1)}",1,"((e + f*x)^(1 + n)*(3/f + (a^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)])/(b^(1/3)*e - a^(1/3)*f) + (a^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f) - (a^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)])/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*b*(1 + n))","A",1
189,1,213,253,0.1559805,"\int \frac{x^2 (e+f x)^n}{a+b x^3} \, dx","Integrate[(x^2*(e + f*x)^n)/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{3 b^{2/3} (n+1)}","-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 b^{2/3} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{3 b^{2/3} (n+1) \left(\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e\right)}-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{3 b^{2/3} (n+1) \left(\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f\right)}",1,"((e + f*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)]/(b^(1/3)*e - a^(1/3)*f)) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)]/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)]/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)))/(3*b^(2/3)*(1 + n))","A",1
190,1,237,288,0.186837,"\int \frac{x (e+f x)^n}{a+b x^3} \, dx","Integrate[(x*(e + f*x)^n)/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}-\frac{\sqrt[3]{-1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}-\frac{(-1)^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} (n+1)}","\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}-\frac{\sqrt[3]{-1} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} (n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}-\frac{(-1)^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} (n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}",1,"((e + f*x)^(1 + n)*(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)]/(b^(1/3)*e - a^(1/3)*f) - ((-1)^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f) - ((-1)^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)])/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(1/3)*b^(1/3)*(1 + n))","A",1
191,1,222,263,0.1212446,"\int \frac{(e+f x)^n}{a+b x^3} \, dx","Integrate[(e + f*x)^n/(a + b*x^3),x]","\frac{(e+f x)^{n+1} \left(-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}+\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e}\right)}{3 a^{2/3} (n+1)}","-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 a^{2/3} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 a^{2/3} (n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{3 a^{2/3} (n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}",1,"((e + f*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)]/(b^(1/3)*e - a^(1/3)*f)) - Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)]/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f) + Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)]/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(2/3)*(1 + n))","A",1
192,1,244,300,0.2437219,"\int \frac{(e+f x)^n}{x \left(a+b x^3\right)} \, dx","Integrate[(e + f*x)^n/(x*(a + b*x^3)),x]","\frac{(e+f x)^{n+1} \left(\frac{\sqrt[3]{b} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}+\frac{\sqrt[3]{b} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e}+\frac{\sqrt[3]{b} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}-\frac{3 \, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{e}\right)}{3 a (n+1)}","\frac{\sqrt[3]{b} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 a (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{\sqrt[3]{b} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e+\sqrt[3]{-1} \sqrt[3]{a} f}\right)}{3 a (n+1) \left(\sqrt[3]{-1} \sqrt[3]{a} f+\sqrt[3]{b} e\right)}+\frac{\sqrt[3]{b} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f}\right)}{3 a (n+1) \left(\sqrt[3]{b} e-(-1)^{2/3} \sqrt[3]{a} f\right)}-\frac{(e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{f x}{e}+1\right)}{a e (n+1)}",1,"((e + f*x)^(1 + n)*((b^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)])/(b^(1/3)*e - a^(1/3)*f) + (b^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)])/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f) + (b^(1/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)])/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f) - (3*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e])/e))/(3*a*(1 + n))","A",1
193,1,273,326,0.2694059,"\int \frac{(e+f x)^n}{x^2 \left(a+b x^3\right)} \, dx","Integrate[(e + f*x)^n/(x^2*(a + b*x^3)),x]","\frac{(e+f x)^{n+1} \left(-\frac{b^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{\sqrt[3]{b} e-\sqrt[3]{a} f}+\frac{\sqrt[3]{-1} b^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}+\frac{(-1)^{2/3} b^{2/3} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e}+\frac{3 \sqrt[3]{a} f \, _2F_1\left(2,n+1;n+2;\frac{f x}{e}+1\right)}{e^2}\right)}{3 a^{4/3} (n+1)}","-\frac{b^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{b} (e+f x)}{\sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 a^{4/3} (n+1) \left(\sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{\sqrt[3]{-1} b^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{(-1)^{2/3} \sqrt[3]{b} (e+f x)}{(-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f}\right)}{3 a^{4/3} (n+1) \left((-1)^{2/3} \sqrt[3]{b} e-\sqrt[3]{a} f\right)}+\frac{(-1)^{2/3} b^{2/3} (e+f x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[3]{-1} \sqrt[3]{b} (e+f x)}{\sqrt[3]{-1} \sqrt[3]{b} e+\sqrt[3]{a} f}\right)}{3 a^{4/3} (n+1) \left(\sqrt[3]{a} f+\sqrt[3]{-1} \sqrt[3]{b} e\right)}+\frac{f (e+f x)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{f x}{e}+1\right)}{a e^2 (n+1)}",1,"((e + f*x)^(1 + n)*(-((b^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)])/(b^(1/3)*e - a^(1/3)*f)) + ((-1)^(1/3)*b^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f) + ((-1)^(2/3)*b^(2/3)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)])/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f) + (3*a^(1/3)*f*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (f*x)/e])/e^2))/(3*a^(4/3)*(1 + n))","A",1
194,1,213,253,0.3852416,"\int \frac{x^2 (c+d x)^{1+n}}{a+b x^3} \, dx","Integrate[(x^2*(c + d*x)^(1 + n))/(a + b*x^3),x]","\frac{(c+d x)^{n+2} \left(-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c-\sqrt[3]{a} d}\right)}{\sqrt[3]{b} c-\sqrt[3]{a} d}-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c+\sqrt[3]{-1} \sqrt[3]{a} d}\right)}{\sqrt[3]{-1} \sqrt[3]{a} d+\sqrt[3]{b} c}-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c-(-1)^{2/3} \sqrt[3]{a} d}\right)}{\sqrt[3]{b} c-(-1)^{2/3} \sqrt[3]{a} d}\right)}{3 b^{2/3} (n+2)}","-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c-\sqrt[3]{a} d}\right)}{3 b^{2/3} (n+2) \left(\sqrt[3]{b} c-\sqrt[3]{a} d\right)}-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c+\sqrt[3]{-1} \sqrt[3]{a} d}\right)}{3 b^{2/3} (n+2) \left(\sqrt[3]{-1} \sqrt[3]{a} d+\sqrt[3]{b} c\right)}-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[3]{b} (c+d x)}{\sqrt[3]{b} c-(-1)^{2/3} \sqrt[3]{a} d}\right)}{3 b^{2/3} (n+2) \left(\sqrt[3]{b} c-(-1)^{2/3} \sqrt[3]{a} d\right)}",1,"((c + d*x)^(2 + n)*(-(Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(b^(1/3)*c - a^(1/3)*d)) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)]/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)))/(3*b^(2/3)*(2 + n))","A",1
195,0,0,211,0.1715921,"\int \frac{x^m (e+f x)^n}{a+b x^3} \, dx","Integrate[(x^m*(e + f*x)^n)/(a + b*x^3),x]","\int \frac{x^m (e+f x)^n}{a+b x^3} \, dx","\frac{x^{m+1} (e+f x)^n \left(\frac{f x}{e}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{f x}{e},-\frac{\sqrt[3]{b} x}{\sqrt[3]{a}}\right)}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left(\frac{f x}{e}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{f x}{e},\frac{\sqrt[3]{-1} \sqrt[3]{b} x}{\sqrt[3]{a}}\right)}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left(\frac{f x}{e}+1\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{f x}{e},-\frac{(-1)^{2/3} \sqrt[3]{b} x}{\sqrt[3]{a}}\right)}{3 a (m+1)}",1,"Integrate[(x^m*(e + f*x)^n)/(a + b*x^3), x]","F",-1
196,1,820,1480,2.1488048,"\int \frac{\sqrt{c+d x^3}}{a+b x} \, dx","Integrate[Sqrt[c + d*x^3]/(a + b*x),x]","\frac{2 \left(\frac{\sqrt[3]{-1} \sqrt{3} \left(1+\sqrt[3]{-1}\right) \sqrt[3]{c} d \sqrt{\frac{\sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}} \sqrt{\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt[3]{d} x}{\sqrt[3]{c}}+1} \Pi \left(\frac{i \sqrt{3} b \sqrt[3]{c}}{\sqrt[3]{d} a+\sqrt[3]{-1} b \sqrt[3]{c}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}}\right)|\sqrt[3]{-1}\right) a^3}{b^2 \left(\sqrt[3]{d} a+\sqrt[3]{-1} b \sqrt[3]{c}\right)}-\frac{3^{3/4} d^{2/3} \left(\sqrt[3]{-1} \sqrt[3]{c}-\sqrt[3]{d} x\right) \sqrt{\frac{\sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}} \sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{d} x}{\sqrt[3]{c}}} F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}}\right)|\sqrt[3]{-1}\right) a^2}{b^2 \sqrt{\frac{(-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}}}+\frac{3^{3/4} \sqrt[3]{c} \sqrt[3]{d} \left(\sqrt[3]{-1} \sqrt[3]{c}-\sqrt[3]{d} x\right) \sqrt{-\frac{2 i \sqrt[3]{d} x}{\sqrt[3]{c}}+\sqrt{3}+i} \sqrt{\frac{i \left(\frac{\sqrt[3]{d} x}{\sqrt[3]{c}}+1\right)}{3 i+\sqrt{3}}} \left(\left(-1+(-1)^{2/3}\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{d} x}{\sqrt[3]{c}}}}{\sqrt[4]{3}}\right)|\frac{\sqrt[3]{-1}}{-1+\sqrt[3]{-1}}\right)+F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt[6]{-1}-\frac{i \sqrt[3]{d} x}{\sqrt[3]{c}}}}{\sqrt[4]{3}}\right)|\frac{\sqrt[3]{-1}}{-1+\sqrt[3]{-1}}\right)\right) a}{b \sqrt{\frac{(-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}}}+d x^3+c-\frac{3 i b c^{4/3} \sqrt{\frac{\sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}} \sqrt{\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt[3]{d} x}{\sqrt[3]{c}}+1} \Pi \left(\frac{i \sqrt{3} b \sqrt[3]{c}}{\sqrt[3]{d} a+\sqrt[3]{-1} b \sqrt[3]{c}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} \sqrt[3]{d} x+\sqrt[3]{c}}{\left(1+\sqrt[3]{-1}\right) \sqrt[3]{c}}}\right)|\sqrt[3]{-1}\right)}{\sqrt[3]{d} a+\sqrt[3]{-1} b \sqrt[3]{c}}\right)}{3 b \sqrt{d x^3+c}}","\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} \sqrt[3]{c} \sqrt[3]{d} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right) \sqrt{\frac{d^{2/3} x^2-\sqrt[3]{c} \sqrt[3]{d} x+c^{2/3}}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right) a}{b^2 \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \sqrt{d x^3+c}}+\frac{2 \sqrt{2+\sqrt{3}} \left(\sqrt[3]{d} a+\left(1-\sqrt{3}\right) b \sqrt[3]{c}\right) \sqrt[3]{d} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right) \sqrt{\frac{d^{2/3} x^2-\sqrt[3]{c} \sqrt[3]{d} x+c^{2/3}}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right) a}{\sqrt[4]{3} b^3 \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \sqrt{d x^3+c}}-\frac{2 \sqrt[3]{d} \sqrt{d x^3+c} a}{b^2 \left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)}-\frac{\sqrt[6]{c} \sqrt{b \sqrt[3]{c}-a \sqrt[3]{d}} \sqrt{d^{2/3} a^2+b \sqrt[3]{c} \sqrt[3]{d} a+b^2 c^{2/3}} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right) \sqrt{\frac{c^{2/3} \left(\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt[3]{d} x}{\sqrt[3]{c}}+1\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \tanh ^{-1}\left(\frac{\sqrt{2-\sqrt{3}} \sqrt{d^{2/3} a^2+b \sqrt[3]{c} \sqrt[3]{d} a+b^2 c^{2/3}} \sqrt{1-\frac{\left(\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}\right)^2}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}}}{\sqrt[4]{3} \sqrt{b} \sqrt[6]{c} \sqrt{b \sqrt[3]{c}-a \sqrt[3]{d}} \sqrt{\frac{\left(\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}\right)^2}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}-4 \sqrt{3}+7}}\right)}{b^{5/2} \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \sqrt{d x^3+c}}-\frac{2 \sqrt{2+\sqrt{3}} \left(b^3 c-a^3 d\right) \left(\sqrt[3]{d} x+\sqrt[3]{c}\right) \sqrt{\frac{d^{2/3} x^2-\sqrt[3]{c} \sqrt[3]{d} x+c^{2/3}}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} b^3 \left(\left(1+\sqrt{3}\right) b \sqrt[3]{c}-a \sqrt[3]{d}\right) \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \sqrt{d x^3+c}}+\frac{4 \sqrt[4]{3} \sqrt{2+\sqrt{3}} \sqrt[3]{c} \left(b^3 c-a^3 d\right) \left(\sqrt[3]{d} x+\sqrt[3]{c}\right) \sqrt{\frac{c^{2/3} \left(\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt[3]{d} x}{\sqrt[3]{c}}+1\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \Pi \left(\frac{\left(\left(1+\sqrt{3}\right) b \sqrt[3]{c}-a \sqrt[3]{d}\right)^2}{\left(\left(1-\sqrt{3}\right) b \sqrt[3]{c}-a \sqrt[3]{d}\right)^2};\sin ^{-1}\left(\frac{\sqrt[3]{d} x+\left(1-\sqrt{3}\right) \sqrt[3]{c}}{\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}}\right)|-7-4 \sqrt{3}\right)}{b^2 \left(-d^{2/3} a^2+2 b \sqrt[3]{c} \sqrt[3]{d} a+2 b^2 c^{2/3}\right) \sqrt{\frac{\sqrt[3]{c} \left(\sqrt[3]{d} x+\sqrt[3]{c}\right)}{\left(\sqrt[3]{d} x+\left(1+\sqrt{3}\right) \sqrt[3]{c}\right)^2}} \sqrt{d x^3+c}}+\frac{2 \sqrt{d x^3+c}}{3 b}",1,"(2*(c + d*x^3 - (3^(3/4)*a^2*d^(2/3)*((-1)^(1/3)*c^(1/3) - d^(1/3)*x)*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]*Sqrt[(-1)^(1/6) - (I*d^(1/3)*x)/c^(1/3)]*EllipticF[ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/(b^2*Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]) + (3^(3/4)*a*c^(1/3)*d^(1/3)*((-1)^(1/3)*c^(1/3) - d^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*d^(1/3)*x)/c^(1/3)]*Sqrt[(I*(1 + (d^(1/3)*x)/c^(1/3)))/(3*I + Sqrt[3])]*((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*d^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))] + EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*d^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]))/(b*Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]) - ((3*I)*b*c^(4/3)*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]*Sqrt[1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)]*EllipticPi[(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)), ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)) + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*a^3*c^(1/3)*d*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]*Sqrt[1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)]*EllipticPi[(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)), ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/(b^2*((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)))))/(3*b*Sqrt[c + d*x^3])","C",0
197,0,0,135,0.0475693,"\int \frac{\left(d^3+e^3 x^3\right)^p}{d+e x} \, dx","Integrate[(d^3 + e^3*x^3)^p/(d + e*x),x]","\int \frac{\left(d^3+e^3 x^3\right)^p}{d+e x} \, dx","\frac{\left(d^3+e^3 x^3\right)^p \left(1+\frac{2 (d+e x)}{\left(-3+i \sqrt{3}\right) d}\right)^{-p} \left(1-\frac{2 (d+e x)}{\left(3+i \sqrt{3}\right) d}\right)^{-p} F_1\left(p;-p,-p;p+1;-\frac{2 (d+e x)}{\left(-3+i \sqrt{3}\right) d},\frac{2 (d+e x)}{\left(3+i \sqrt{3}\right) d}\right)}{e p}",1,"Integrate[(d^3 + e^3*x^3)^p/(d + e*x), x]","F",-1
198,1,296,16,0.868615,"\int \frac{2-2 x-x^2}{\left(2+x^2\right) \sqrt{1+x^3}} \, dx","Integrate[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[1 + x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \sqrt{x^2-x+1} \left(\frac{\sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1}-x\right) F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x+1}-\frac{3 i \left(\sqrt{2}-i\right) \Pi \left(\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{5/6}+\sqrt{2}}+\frac{3 \left(5+i \sqrt{2}+i \sqrt{3}+\sqrt{6}\right) \Pi \left(\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{5 i+2 \sqrt{2}+\sqrt{3}+2 i \sqrt{6}}\right)}{3 \sqrt{x^3+1}}","2 \tan ^{-1}\left(\frac{x+1}{\sqrt{x^3+1}}\right)",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) - x)*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(1 + (-1)^(2/3)*x) - ((3*I)*(-I + Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I - 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(5/6) + Sqrt[2]) + (3*(5 + I*Sqrt[2] + I*Sqrt[3] + Sqrt[6])*EllipticPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(5*I + 2*Sqrt[2] + Sqrt[3] + (2*I)*Sqrt[6])))/(3*Sqrt[1 + x^3])","C",1
199,1,280,20,0.7267589,"\int \frac{2+2 x-x^2}{\left(2+x^2\right) \sqrt{1-x^3}} \, dx","Integrate[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \sqrt{x^2+x+1} \left(\frac{\sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(x+\sqrt[3]{-1}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x-1}+\frac{6 \left(1+i \sqrt{2}\right) \Pi \left(\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{i+2 \sqrt{2}-\sqrt{3}}+\frac{3 \left(1-i \sqrt{2}\right) \Pi \left(\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{5/6}-\sqrt{2}}\right)}{3 \sqrt{1-x^3}}","-2 \tan ^{-1}\left(\frac{1-x}{\sqrt{1-x^3}}\right)",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x) + (6*(1 + I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I - 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(I + 2*Sqrt[2] - Sqrt[3]) + (3*(1 - I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(5/6) - Sqrt[2])))/(3*Sqrt[1 - x^3])","C",1
200,1,278,18,0.2341686,"\int \frac{2+2 x-x^2}{\left(2+x^2\right) \sqrt{-1+x^3}} \, dx","Integrate[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[-1 + x^3]),x]","\frac{2 \sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \sqrt{x^2+x+1} \left(\frac{\sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(x+\sqrt[3]{-1}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x-1}+\frac{6 \left(1+i \sqrt{2}\right) \Pi \left(\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{i+2 \sqrt{2}-\sqrt{3}}+\frac{3 \left(1-i \sqrt{2}\right) \Pi \left(\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{5/6}-\sqrt{2}}\right)}{3 \sqrt{x^3-1}}","-2 \tanh ^{-1}\left(\frac{1-x}{\sqrt{x^3-1}}\right)",1,"(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x) + (6*(1 + I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I - 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(I + 2*Sqrt[2] - Sqrt[3]) + (3*(1 - I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(5/6) - Sqrt[2])))/(3*Sqrt[-1 + x^3])","C",1
201,1,298,18,0.5586436,"\int \frac{2-2 x-x^2}{\left(2+x^2\right) \sqrt{-1-x^3}} \, dx","Integrate[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[-1 - x^3]),x]","\frac{2 \sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \sqrt{x^2-x+1} \left(\frac{\sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1}-x\right) F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x+1}-\frac{3 i \left(\sqrt{2}-i\right) \Pi \left(\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{5/6}+\sqrt{2}}+\frac{3 \left(5+i \sqrt{2}+i \sqrt{3}+\sqrt{6}\right) \Pi \left(\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{5 i+2 \sqrt{2}+\sqrt{3}+2 i \sqrt{6}}\right)}{3 \sqrt{-x^3-1}}","2 \tanh ^{-1}\left(\frac{x+1}{\sqrt{-x^3-1}}\right)",1,"(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) - x)*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(1 + (-1)^(2/3)*x) - ((3*I)*(-I + Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I - 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(5/6) + Sqrt[2]) + (3*(5 + I*Sqrt[2] + I*Sqrt[3] + Sqrt[6])*EllipticPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(5*I + 2*Sqrt[2] + Sqrt[3] + (2*I)*Sqrt[6])))/(3*Sqrt[-1 - x^3])","C",1
202,1,424,30,1.3558659,"\int \frac{2-2 x-x^2}{\left(2+d+d x+x^2\right) \sqrt{1+x^3}} \, dx","Integrate[(2 - 2*x - x^2)/((2 + d + d*x + x^2)*Sqrt[1 + x^3]),x]","\frac{\sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \sqrt{x^2-x+1} \left(\frac{2 \sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1}-x\right) F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x+1}-\frac{3 i \left(\left(-\left(\left(1+\sqrt[3]{-1}\right) d^2\right)+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2-4 d-8}+4\right) d-2 \sqrt[3]{-1} \sqrt{d^2-4 d-8}+4 \sqrt{d^2-4 d-8}+8 \sqrt[3]{-1}+8\right) \Pi \left(\frac{2 i \sqrt{3}}{d-\sqrt{d^2-4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\left(\left(1+\sqrt[3]{-1}\right) d^2+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2-4 d-8}-4\right) d-2 \left(\sqrt[3]{-1} \sqrt{d^2-4 d-8}-2 \sqrt{d^2-4 d-8}+4 \sqrt[3]{-1}+4\right)\right) \Pi \left(\frac{2 i \sqrt{3}}{d+\sqrt{d^2-4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)\right)}{\left(\sqrt[3]{-1} d+d+(-1)^{2/3}+2\right) \sqrt{d^2-4 d-8}}\right)}{3 \sqrt{x^3+1}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{d+1} (x+1)}{\sqrt{x^3+1}}\right)}{\sqrt{d+1}}",1,"(Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*((2*Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) - x)*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(1 + (-1)^(2/3)*x) - ((3*I)*((8 + 8*(-1)^(1/3) - (1 + (-1)^(1/3))*d^2 + 4*Sqrt[-8 - 4*d + d^2] - 2*(-1)^(1/3)*Sqrt[-8 - 4*d + d^2] + (1 + (-1)^(1/3))*d*(4 + Sqrt[-8 - 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d - Sqrt[-8 - 4*d + d^2]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)] + ((1 + (-1)^(1/3))*d^2 + (1 + (-1)^(1/3))*d*(-4 + Sqrt[-8 - 4*d + d^2]) - 2*(4 + 4*(-1)^(1/3) - 2*Sqrt[-8 - 4*d + d^2] + (-1)^(1/3)*Sqrt[-8 - 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d + Sqrt[-8 - 4*d + d^2]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((2 + (-1)^(2/3) + d + (-1)^(1/3)*d)*Sqrt[-8 - 4*d + d^2])))/(3*Sqrt[1 + x^3])","C",0
203,1,427,38,1.4851979,"\int \frac{2+2 x-x^2}{\left(2-d+d x+x^2\right) \sqrt{1-x^3}} \, dx","Integrate[(2 + 2*x - x^2)/((2 - d + d*x + x^2)*Sqrt[1 - x^3]),x]","\frac{\sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \sqrt{x^2+x+1} \left(\frac{2 \sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(x+\sqrt[3]{-1}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x-1}+\frac{3 i \left(\left(-\left(\left(1+\sqrt[3]{-1}\right) d^2\right)+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2+4 d-8}-4\right) d+2 \sqrt[3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}+8 \sqrt[3]{-1}+8\right) \Pi \left(\frac{2 i \sqrt{3}}{-d+\sqrt{d^2+4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\left(\left(1+\sqrt[3]{-1}\right) d^2+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2+4 d-8}+4\right) d+2 \sqrt[3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}-8 \sqrt[3]{-1}-8\right) \Pi \left(-\frac{2 i \sqrt{3}}{d+\sqrt{d^2+4 d-8}-2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)\right)}{\left(\sqrt[3]{-1} d+d-(-1)^{2/3}-2\right) \sqrt{d^2+4 d-8}}\right)}{3 \sqrt{1-x^3}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{1-d} (1-x)}{\sqrt{1-x^3}}\right)}{\sqrt{1-d}}",1,"(Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((2*Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x) + ((3*I)*((8 + 8*(-1)^(1/3) - (1 + (-1)^(1/3))*d^2 - 4*Sqrt[-8 + 4*d + d^2] + 2*(-1)^(1/3)*Sqrt[-8 + 4*d + d^2] + (1 + (-1)^(1/3))*d*(-4 + Sqrt[-8 + 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) - d + Sqrt[-8 + 4*d + d^2]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)] + (-8 - 8*(-1)^(1/3) + (1 + (-1)^(1/3))*d^2 - 4*Sqrt[-8 + 4*d + d^2] + 2*(-1)^(1/3)*Sqrt[-8 + 4*d + d^2] + (1 + (-1)^(1/3))*d*(4 + Sqrt[-8 + 4*d + d^2]))*EllipticPi[((-2*I)*Sqrt[3])/(-2*(-1)^(1/3) + d + Sqrt[-8 + 4*d + d^2]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((-2 - (-1)^(2/3) + d + (-1)^(1/3)*d)*Sqrt[-8 + 4*d + d^2])))/(3*Sqrt[1 - x^3])","C",0
204,1,425,36,0.4573014,"\int \frac{2+2 x-x^2}{\left(2-d+d x+x^2\right) \sqrt{-1+x^3}} \, dx","Integrate[(2 + 2*x - x^2)/((2 - d + d*x + x^2)*Sqrt[-1 + x^3]),x]","\frac{\sqrt{\frac{1-x}{1+\sqrt[3]{-1}}} \sqrt{x^2+x+1} \left(\frac{2 \sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(x+\sqrt[3]{-1}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x-1}+\frac{3 i \left(\left(-\left(\left(1+\sqrt[3]{-1}\right) d^2\right)+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2+4 d-8}-4\right) d+2 \sqrt[3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}+8 \sqrt[3]{-1}+8\right) \Pi \left(\frac{2 i \sqrt{3}}{-d+\sqrt{d^2+4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\left(\left(1+\sqrt[3]{-1}\right) d^2+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2+4 d-8}+4\right) d+2 \sqrt[3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}-8 \sqrt[3]{-1}-8\right) \Pi \left(-\frac{2 i \sqrt{3}}{d+\sqrt{d^2+4 d-8}-2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)\right)}{\left(\sqrt[3]{-1} d+d-(-1)^{2/3}-2\right) \sqrt{d^2+4 d-8}}\right)}{3 \sqrt{x^3-1}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{1-d} (1-x)}{\sqrt{x^3-1}}\right)}{\sqrt{1-d}}",1,"(Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((2*Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x) + ((3*I)*((8 + 8*(-1)^(1/3) - (1 + (-1)^(1/3))*d^2 - 4*Sqrt[-8 + 4*d + d^2] + 2*(-1)^(1/3)*Sqrt[-8 + 4*d + d^2] + (1 + (-1)^(1/3))*d*(-4 + Sqrt[-8 + 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) - d + Sqrt[-8 + 4*d + d^2]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)] + (-8 - 8*(-1)^(1/3) + (1 + (-1)^(1/3))*d^2 - 4*Sqrt[-8 + 4*d + d^2] + 2*(-1)^(1/3)*Sqrt[-8 + 4*d + d^2] + (1 + (-1)^(1/3))*d*(4 + Sqrt[-8 + 4*d + d^2]))*EllipticPi[((-2*I)*Sqrt[3])/(-2*(-1)^(1/3) + d + Sqrt[-8 + 4*d + d^2]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((-2 - (-1)^(2/3) + d + (-1)^(1/3)*d)*Sqrt[-8 + 4*d + d^2])))/(3*Sqrt[-1 + x^3])","C",0
205,1,426,32,0.545743,"\int \frac{2-2 x-x^2}{\left(2+d+d x+x^2\right) \sqrt{-1-x^3}} \, dx","Integrate[(2 - 2*x - x^2)/((2 + d + d*x + x^2)*Sqrt[-1 - x^3]),x]","\frac{\sqrt{\frac{x+1}{1+\sqrt[3]{-1}}} \sqrt{x^2-x+1} \left(\frac{2 \sqrt{3} \left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1}-x\right) F\left(\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)}{(-1)^{2/3} x+1}-\frac{3 i \left(\left(-\left(\left(1+\sqrt[3]{-1}\right) d^2\right)+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2-4 d-8}+4\right) d-2 \sqrt[3]{-1} \sqrt{d^2-4 d-8}+4 \sqrt{d^2-4 d-8}+8 \sqrt[3]{-1}+8\right) \Pi \left(\frac{2 i \sqrt{3}}{d-\sqrt{d^2-4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)+\left(\left(1+\sqrt[3]{-1}\right) d^2+\left(1+\sqrt[3]{-1}\right) \left(\sqrt{d^2-4 d-8}-4\right) d-2 \left(\sqrt[3]{-1} \sqrt{d^2-4 d-8}-2 \sqrt{d^2-4 d-8}+4 \sqrt[3]{-1}+4\right)\right) \Pi \left(\frac{2 i \sqrt{3}}{d+\sqrt{d^2-4 d-8}+2 \sqrt[3]{-1}};\sin ^{-1}\left(\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt[3]{-1}}}\right)|\sqrt[3]{-1}\right)\right)}{\left(\sqrt[3]{-1} d+d+(-1)^{2/3}+2\right) \sqrt{d^2-4 d-8}}\right)}{3 \sqrt{-x^3-1}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d+1} (x+1)}{\sqrt{-x^3-1}}\right)}{\sqrt{d+1}}",1,"(Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*((2*Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) - x)*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(1 + (-1)^(2/3)*x) - ((3*I)*((8 + 8*(-1)^(1/3) - (1 + (-1)^(1/3))*d^2 + 4*Sqrt[-8 - 4*d + d^2] - 2*(-1)^(1/3)*Sqrt[-8 - 4*d + d^2] + (1 + (-1)^(1/3))*d*(4 + Sqrt[-8 - 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d - Sqrt[-8 - 4*d + d^2]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)] + ((1 + (-1)^(1/3))*d^2 + (1 + (-1)^(1/3))*d*(-4 + Sqrt[-8 - 4*d + d^2]) - 2*(4 + 4*(-1)^(1/3) - 2*Sqrt[-8 - 4*d + d^2] + (-1)^(1/3)*Sqrt[-8 - 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d + Sqrt[-8 - 4*d + d^2]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((2 + (-1)^(2/3) + d + (-1)^(1/3)*d)*Sqrt[-8 - 4*d + d^2])))/(3*Sqrt[-1 - x^3])","C",0
206,1,186,355,0.1381182,"\int (d+e x)^3 \sqrt{a+c x^4} \, dx","Integrate[(d + e*x)^3*Sqrt[a + c*x^4],x]","\frac{\sqrt{a+c x^4} \left(12 c d^3 x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^4}{a}\right)+9 \sqrt{a} \sqrt{c} d^2 e \sinh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)+9 c d^2 e x^2 \sqrt{\frac{c x^4}{a}+1}+12 c d e^2 x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)+2 c e^3 x^4 \sqrt{\frac{c x^4}{a}+1}+2 a e^3 \sqrt{\frac{c x^4}{a}+1}\right)}{12 c \sqrt{\frac{c x^4}{a}+1}}","\frac{a^{3/4} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(9 \sqrt{a} e^2+5 \sqrt{c} d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{6 a^{5/4} d e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{15} d x \sqrt{a+c x^4} \left(5 d^2+9 e^2 x^2\right)+\frac{3}{4} d^2 e x^2 \sqrt{a+c x^4}+\frac{3 a d^2 e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{4 \sqrt{c}}+\frac{6 a d e^2 x \sqrt{a+c x^4}}{5 \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{e^3 \left(a+c x^4\right)^{3/2}}{6 c}",1,"(Sqrt[a + c*x^4]*(2*a*e^3*Sqrt[1 + (c*x^4)/a] + 9*c*d^2*e*x^2*Sqrt[1 + (c*x^4)/a] + 2*c*e^3*x^4*Sqrt[1 + (c*x^4)/a] + 9*Sqrt[a]*Sqrt[c]*d^2*e*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]] + 12*c*d^3*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^4)/a)] + 12*c*d*e^2*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^4)/a)]))/(12*c*Sqrt[1 + (c*x^4)/a])","C",1
207,1,146,326,0.1385032,"\int (d+e x)^2 \sqrt{a+c x^4} \, dx","Integrate[(d + e*x)^2*Sqrt[a + c*x^4],x]","\frac{\sqrt{a+c x^4} \left(6 \sqrt{c} d^2 x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^4}{a}\right)+e \left(3 d \left(\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)+\sqrt{c} x^2 \sqrt{\frac{c x^4}{a}+1}\right)+2 \sqrt{c} e x^3 \, _2F_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)\right)\right)}{6 \sqrt{c} \sqrt{\frac{c x^4}{a}+1}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(3 \sqrt{a} e^2+5 \sqrt{c} d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{2 a^{5/4} e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{15} x \sqrt{a+c x^4} \left(5 d^2+3 e^2 x^2\right)+\frac{1}{2} d e x^2 \sqrt{a+c x^4}+\frac{a d e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}+\frac{2 a e^2 x \sqrt{a+c x^4}}{5 \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(Sqrt[a + c*x^4]*(6*Sqrt[c]*d^2*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^4)/a)] + e*(3*d*(Sqrt[c]*x^2*Sqrt[1 + (c*x^4)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]]) + 2*Sqrt[c]*e*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^4)/a)])))/(6*Sqrt[c]*Sqrt[1 + (c*x^4)/a])","C",1
208,1,109,158,0.0657713,"\int (d+e x) \sqrt{a+c x^4} \, dx","Integrate[(d + e*x)*Sqrt[a + c*x^4],x]","\frac{\sqrt{a+c x^4} \left(4 \sqrt{c} d x \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^4}{a}\right)+\sqrt{a} e \sinh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a}}\right)+\sqrt{c} e x^2 \sqrt{\frac{c x^4}{a}+1}\right)}{4 \sqrt{c} \sqrt{\frac{c x^4}{a}+1}}","\frac{a^{3/4} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{1}{3} d x \sqrt{a+c x^4}+\frac{1}{4} e x^2 \sqrt{a+c x^4}+\frac{a e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{4 \sqrt{c}}",1,"(Sqrt[a + c*x^4]*(Sqrt[c]*e*x^2*Sqrt[1 + (c*x^4)/a] + Sqrt[a]*e*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]] + 4*Sqrt[c]*d*x*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^4)/a)]))/(4*Sqrt[c]*Sqrt[1 + (c*x^4)/a])","C",1
209,1,89,105,0.1068752,"\int \sqrt{a+c x^4} \, dx","Integrate[Sqrt[a + c*x^4],x]","\frac{x \left(a+c x^4\right)-\frac{2 i a \sqrt{\frac{c x^4}{a}+1} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{3 \sqrt{a+c x^4}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{1}{3} x \sqrt{a+c x^4}",1,"(x*(a + c*x^4) - ((2*I)*a*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(3*Sqrt[a + c*x^4])","C",1
210,1,405,730,0.8359929,"\int \frac{\sqrt{a+c x^4}}{d+e x} \, dx","Integrate[Sqrt[a + c*x^4]/(d + e*x),x]","\frac{2 c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \left(\sqrt{a} e^2+i \sqrt{c} d^2\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-2 \sqrt{a} c^{3/4} d^2 e^2 \sqrt{\frac{c x^4}{a}+1} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(\sqrt[4]{c} d e \left(\sqrt{c} d^2 \sqrt{a+c x^4} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)-\sqrt{a+c x^4} \sqrt{a e^4+c d^4} \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)+e^2 \left(a+c x^4\right)\right)-2 \sqrt[4]{-1} \sqrt[4]{a} \sqrt{\frac{c x^4}{a}+1} \left(a e^4+c d^4\right) \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)\right)}{2 \sqrt[4]{c} d e^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}","-\frac{\sqrt{-a e^4-c d^4} \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{2 e^3}+\frac{\sqrt{c} d^2 \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 e^3}-\frac{\sqrt[4]{a} \sqrt[4]{c} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 e^4 \sqrt{a+c x^4}}+\frac{\sqrt[4]{c} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(a e^4+c d^4\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} e^4 \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(a e^4+c d^4\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} d e^4 \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\sqrt{a e^4+c d^4} \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{2 e^3}-\frac{\sqrt{c} d x \sqrt{a+c x^4}}{e^2 \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{\sqrt[4]{a} \sqrt[4]{c} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{e^2 \sqrt{a+c x^4}}+\frac{\sqrt{a+c x^4}}{2 e}",1,"(-2*Sqrt[a]*c^(3/4)*d^2*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + 2*c^(3/4)*d^2*(I*Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c^(1/4)*d*e*(e^2*(a + c*x^4) + Sqrt[c]*d^2*Sqrt[a + c*x^4]*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]] - Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])]) - 2*(-1)^(1/4)*a^(1/4)*(c*d^4 + a*e^4)*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(1/4)*d*e^4*Sqrt[a + c*x^4])","C",1
211,1,382,1221,1.9314421,"\int \frac{\sqrt{a+c x^4}}{(d+e x)^2} \, dx","Integrate[Sqrt[a + c*x^4]/(d + e*x)^2,x]","\frac{2 \sqrt[4]{-1} \sqrt[4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)-\frac{2 \sqrt{c} \sqrt{\frac{c x^4}{a}+1} \left(\sqrt{a} e^2+i \sqrt{c} d^2\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}-\frac{c d^3 e \sqrt{a+c x^4} \tanh ^{-1}\left(\frac{-a e^2-c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{\sqrt{a e^4+c d^4}}-\frac{e^3 \left(a+c x^4\right)}{d+e x}-\sqrt{c} d e \sqrt{a+c x^4} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)-2 i a e^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{\frac{c x^4}{a}+1} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{e^4 \sqrt{a+c x^4}}","\frac{c \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{c x^4+a}}\right) d^3}{e^3 \sqrt{c d^4+a e^4}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{c x^4+a}}\right) d}{e^3}-\frac{\sqrt{c x^4+a} d}{e \left(d^2-e^2 x^2\right)}-\frac{2 \sqrt[4]{a} \sqrt[4]{c} \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{e^2 \sqrt{c x^4+a}}+\frac{3 \sqrt[4]{a} \sqrt[4]{c} \left(\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 e^4 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} e^4 \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} e^4 \sqrt{c x^4+a}}-\frac{\sqrt[4]{c} \left(c d^4+a e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} e^4 \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \sqrt{c x^4+a}}+\frac{2 \sqrt{c} x \sqrt{c x^4+a}}{e^2 \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{x \sqrt{c x^4+a}}{d^2-e^2 x^2}-\frac{\left(c d^4-a e^4\right) \tan ^{-1}\left(\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{c x^4+a}}\right)}{2 e^3 \sqrt{-c d^4-a e^4} d}+\frac{\sqrt{-c d^4-a e^4} \tan ^{-1}\left(\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{c x^4+a}}\right)}{2 e^3 d}+\frac{\left(\sqrt{c} d^2-\sqrt{a} e^2\right)^2 \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e^4 \sqrt{c x^4+a} d^2}+\frac{\left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(c d^4+a e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} e^4 \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \sqrt{c x^4+a} d^2}",1,"(-((e^3*(a + c*x^4))/(d + e*x)) - Sqrt[c]*d*e*Sqrt[a + c*x^4]*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]] - (c*d^3*e*Sqrt[a + c*x^4]*ArcTanh[(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/Sqrt[c*d^4 + a*e^4] - (2*I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (2*Sqrt[c]*(I*Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]] + 2*(-1)^(1/4)*a^(1/4)*c^(3/4)*d^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1])/(e^4*Sqrt[a + c*x^4])","C",1
212,1,157,295,0.1359101,"\int \frac{(d+e x)^3}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x)^3/Sqrt[a + c*x^4],x]","\frac{d^3 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)}{\sqrt{a+c x^4}}+\frac{3 d^2 e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}+\frac{d e^2 x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)}{\sqrt{a+c x^4}}+\frac{e^3 \sqrt{a+c x^4}}{2 c}","\frac{d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(3 \sqrt{a} e^2+\sqrt{c} d^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} c^{3/4} \sqrt{a+c x^4}}-\frac{3 \sqrt[4]{a} d e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{3 d^2 e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}+\frac{3 d e^2 x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}+\frac{e^3 \sqrt{a+c x^4}}{2 c}",1,"(e^3*Sqrt[a + c*x^4])/(2*c) + (3*d^2*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(2*Sqrt[c]) + (d^3*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)])/Sqrt[a + c*x^4] + (d*e^2*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/Sqrt[a + c*x^4]","C",1
213,1,133,263,0.1471607,"\int \frac{(d+e x)^2}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x)^2/Sqrt[a + c*x^4],x]","\frac{d^2 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)}{\sqrt{a+c x^4}}+\frac{d e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{\sqrt{c}}+\frac{e^2 x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^4}{a}\right)}{3 \sqrt{a+c x^4}}","\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 c^{3/4} \sqrt{a+c x^4}}-\frac{\sqrt[4]{a} e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{c^{3/4} \sqrt{a+c x^4}}+\frac{d e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{\sqrt{c}}+\frac{e^2 x \sqrt{a+c x^4}}{\sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(d*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/Sqrt[c] + (d^2*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)])/Sqrt[a + c*x^4] + (e^2*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^4)/a)])/(3*Sqrt[a + c*x^4])","C",1
214,1,79,121,0.0410904,"\int \frac{d+e x}{\sqrt{a+c x^4}} \, dx","Integrate[(d + e*x)/Sqrt[a + c*x^4],x]","\frac{d x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)}{\sqrt{a+c x^4}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}","\frac{d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{e \tanh ^{-1}\left(\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right)}{2 \sqrt{c}}",1,"(e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(2*Sqrt[c]) + (d*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)])/Sqrt[a + c*x^4]","C",1
215,1,74,88,0.0319547,"\int \frac{1}{\sqrt{a+c x^4}} \, dx","Integrate[1/Sqrt[a + c*x^4],x]","-\frac{i \sqrt{\frac{c x^4}{a}+1} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{c} \sqrt{a+c x^4}}",1,"((-I)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*Sqrt[a + c*x^4])","C",1
216,1,200,405,0.2619098,"\int \frac{1}{(d+e x) \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x)*Sqrt[a + c*x^4]),x]","\frac{\sqrt{\frac{c x^4}{a}+1} \left(\sqrt[4]{c} d \log \left(\frac{e^2 x^2-d^2}{a e^2 \left(\sqrt{\frac{c x^4}{a}+1} \sqrt{\frac{c d^4}{a e^4}+1}+1\right)+c d^2 x^2}\right)-2 \sqrt[4]{-1} \sqrt[4]{a} e \sqrt{\frac{c d^4}{a e^4}+1} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)\right)}{2 \sqrt[4]{c} d e \sqrt{a+c x^4} \sqrt{\frac{c d^4}{a e^4}+1}}","\frac{e \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{2 \sqrt{-a e^4-c d^4}}+\frac{\sqrt[4]{c} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{c} d \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{e \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{2 \sqrt{a e^4+c d^4}}",1,"(Sqrt[1 + (c*x^4)/a]*(-2*(-1)^(1/4)*a^(1/4)*Sqrt[1 + (c*d^4)/(a*e^4)]*e*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1] + c^(1/4)*d*Log[(-d^2 + e^2*x^2)/(c*d^2*x^2 + a*e^2*(1 + Sqrt[1 + (c*d^4)/(a*e^4)]*Sqrt[1 + (c*x^4)/a]))]))/(2*c^(1/4)*d*Sqrt[1 + (c*d^4)/(a*e^4)]*e*Sqrt[a + c*x^4])","C",1
217,1,425,610,1.1061466,"\int \frac{1}{(d+e x)^2 \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x)^2*Sqrt[a + c*x^4]),x]","\frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(2 \sqrt[4]{-1} \sqrt[4]{a} c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)+e^3 \left(a+c x^4\right) \sqrt{a e^4+c d^4}+c d^3 e \sqrt{a+c x^4} (d+e x) \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)\right)+\sqrt{a} \sqrt{c} e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x) \sqrt{a e^4+c d^4} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+i \sqrt{c} \sqrt{\frac{c x^4}{a}+1} (d+e x) \left(\sqrt{c} d^2+i \sqrt{a} e^2\right) \sqrt{a e^4+c d^4} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x) \left(a e^4+c d^4\right)^{3/2}}","-\frac{c^{3/4} d^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right) \left(a e^4+c d^4\right)}-\frac{e^3 \sqrt{a+c x^4}}{(d+e x) \left(a e^4+c d^4\right)}+\frac{\sqrt{c} e^2 x \sqrt{a+c x^4}}{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^4+c d^4\right)}-\frac{\sqrt[4]{a} \sqrt[4]{c} e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+c x^4} \left(a e^4+c d^4\right)}+\frac{\sqrt[4]{c} \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(\sqrt{a} e^2+\sqrt{c} d^2\right)}-\frac{c d^3 e \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{\left(-a e^4-c d^4\right)^{3/2}}-\frac{c d^3 e \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{\left(a e^4+c d^4\right)^{3/2}}",1,"(Sqrt[a]*Sqrt[c]*e^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*Sqrt[c]*(Sqrt[c]*d^2 + I*Sqrt[a]*e^2)*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[(I*Sqrt[c])/Sqrt[a]]*(e^3*Sqrt[c*d^4 + a*e^4]*(a + c*x^4) + c*d^3*e*(d + e*x)*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])] + 2*(-1)^(1/4)*a^(1/4)*c^(3/4)*d^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^4 + a*e^4)^(3/2)*(d + e*x)*Sqrt[a + c*x^4])","C",1
218,1,513,659,2.5155193,"\int \frac{1}{(d+e x)^3 \sqrt{a+c x^4}} \, dx","Integrate[1/((d + e*x)^3*Sqrt[a + c*x^4]),x]","\frac{-\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(e^3 \sqrt{a e^4+c d^4} \left(a^2 e^4+a c \left(7 d^4+6 d^3 e x+e^4 x^4\right)+c^2 d^3 x^4 (7 d+6 e x)\right)+6 \sqrt[4]{-1} \sqrt[4]{a} c^{3/4} d \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \left(c d^4-a e^4\right) \sqrt{a e^4+c d^4} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)+3 c d^2 e \sqrt{a+c x^4} (d+e x)^2 \left(c d^4-a e^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)\right)+6 \sqrt{a} c^{3/2} d^3 e^2 \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \sqrt{a e^4+c d^4} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)+2 i c d \sqrt{\frac{c x^4}{a}+1} (d+e x)^2 \left(3 i \sqrt{a} \sqrt{c} d^2 e^2-a e^4+2 c d^4\right) \sqrt{a e^4+c d^4} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{2 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} (d+e x)^2 \left(a e^4+c d^4\right)^{5/2}}","\frac{c^{3/4} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+c x^4} \left(a e^4+c d^4\right)}+\frac{3 c^{3/2} d^3 e^2 x \sqrt{a+c x^4}}{\left(\sqrt{a}+\sqrt{c} x^2\right) \left(a e^4+c d^4\right)^2}-\frac{3 \sqrt[4]{a} c^{5/4} d^3 e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+c x^4} \left(a e^4+c d^4\right)^2}-\frac{3 c^{3/4} d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right)^2 \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt{a+c x^4} \left(a e^4+c d^4\right)^2}-\frac{e^3 \sqrt{a+c x^4}}{2 (d+e x)^2 \left(a e^4+c d^4\right)}-\frac{3 c d^3 e^3 \sqrt{a+c x^4}}{(d+e x) \left(a e^4+c d^4\right)^2}+\frac{3 c d^2 e \left(c d^4-a e^4\right) \tan ^{-1}\left(\frac{x \sqrt{-a e^4-c d^4}}{d e \sqrt{a+c x^4}}\right)}{2 \left(-a e^4-c d^4\right)^{5/2}}-\frac{3 c d^2 e \left(c d^4-a e^4\right) \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)}{2 \left(a e^4+c d^4\right)^{5/2}}",1,"(6*Sqrt[a]*c^(3/2)*d^3*e^2*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (2*I)*c*d*(2*c*d^4 + (3*I)*Sqrt[a]*Sqrt[c]*d^2*e^2 - a*e^4)*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[(I*Sqrt[c])/Sqrt[a]]*(e^3*Sqrt[c*d^4 + a*e^4]*(a^2*e^4 + c^2*d^3*x^4*(7*d + 6*e*x) + a*c*(7*d^4 + 6*d^3*e*x + e^4*x^4)) + 3*c*d^2*e*(c*d^4 - a*e^4)*(d + e*x)^2*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])] + 6*(-1)^(1/4)*a^(1/4)*c^(3/4)*d*(c*d^4 - a*e^4)*Sqrt[c*d^4 + a*e^4]*(d + e*x)^2*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^4 + a*e^4)^(5/2)*(d + e*x)^2*Sqrt[a + c*x^4])","C",1
219,1,126,298,0.066673,"\int \frac{(d+e x)^3}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x)^3/(a + c*x^4)^(3/2),x]","\frac{c d^3 x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+2 c d e^2 x^3 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right)-a e^3+c d^3 x+3 c d^2 e x^2}{2 a c \sqrt{a+c x^4}}","\frac{d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-3 \sqrt{a} e^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{3 d e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}-\frac{a e^3-c x \left(d^3+3 d^2 e x+3 d e^2 x^2\right)}{2 a c \sqrt{a+c x^4}}-\frac{3 d e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(-(a*e^3) + c*d^3*x + 3*c*d^2*e*x^2 + c*d^3*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*c*d*e^2*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(2*a*c*Sqrt[a + c*x^4])","C",1
220,1,108,270,0.063302,"\int \frac{(d+e x)^2}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x)^2/(a + c*x^4)^(3/2),x]","\frac{x \left(3 d^2 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+2 e^2 x^2 \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^4}{a}\right)+3 d (d+2 e x)\right)}{6 a \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} \left(\sqrt{c} d^2-\sqrt{a} e^2\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{e^2 \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}+\frac{x (d+e x)^2}{2 a \sqrt{a+c x^4}}-\frac{e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left(\sqrt{a}+\sqrt{c} x^2\right)}",1,"(x*(3*d*(d + 2*e*x) + 3*d^2*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*e^2*x^2*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)]))/(6*a*Sqrt[a + c*x^4])","C",1
221,1,59,114,0.0311803,"\int \frac{d+e x}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[(d + e*x)/(a + c*x^4)^(3/2),x]","\frac{x \left(d \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+d+e x\right)}{2 a \sqrt{a+c x^4}}","\frac{d \left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{x (d+e x)}{2 a \sqrt{a+c x^4}}",1,"(x*(d + e*x + d*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)]))/(2*a*Sqrt[a + c*x^4])","C",1
222,1,55,108,0.0106509,"\int \frac{1}{\left(a+c x^4\right)^{3/2}} \, dx","Integrate[(a + c*x^4)^(-3/2),x]","\frac{x \sqrt{\frac{c x^4}{a}+1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^4}{a}\right)+x}{2 a \sqrt{a+c x^4}}","\frac{\left(\sqrt{a}+\sqrt{c} x^2\right) \sqrt{\frac{a+c x^4}{\left(\sqrt{a}+\sqrt{c} x^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} \sqrt[4]{c} \sqrt{a+c x^4}}+\frac{x}{2 a \sqrt{a+c x^4}}",1,"(x + x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)])/(2*a*Sqrt[a + c*x^4])","C",1
223,1,434,818,0.8848403,"\int \frac{1}{(d+e x) \left(a+c x^4\right)^{3/2}} \, dx","Integrate[1/((d + e*x)*(a + c*x^4)^(3/2)),x]","\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left(\sqrt[4]{c} d \left(\sqrt{a e^4+c d^4} \left(a e^3+c d x \left(d^2-d e x+e^2 x^2\right)\right)-a e^5 \sqrt{a+c x^4} \tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right)\right)-2 \sqrt[4]{-1} a^{5/4} e^4 \sqrt{\frac{c x^4}{a}+1} \sqrt{a e^4+c d^4} \Pi \left(\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left.\sin ^{-1}\left(\frac{(-1)^{3/4} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)\right|-1\right)\right)+c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \left(\sqrt{a} e^2-i \sqrt{c} d^2\right) \sqrt{a e^4+c d^4} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)-\sqrt{a} c^{3/4} d^2 e^2 \sqrt{\frac{c x^4}{a}+1} \sqrt{a e^4+c d^4} E\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right)\right|-1\right)}{2 a \sqrt[4]{c} d \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left(a e^4+c d^4\right)^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-c d^4-a e^4} x}{d e \sqrt{c x^4+a}}\right) e^5}{2 \left(-c d^4-a e^4\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{a e^2+c d^2 x^2}{\sqrt{c d^4+a e^4} \sqrt{c x^4+a}}\right) e^5}{2 \left(c d^4+a e^4\right)^{3/2}}+\frac{\sqrt[4]{c} d \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^4}{2 \sqrt[4]{a} \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \left(c d^4+a e^4\right) \sqrt{c x^4+a}}-\frac{\left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} \Pi \left(\frac{\left(\sqrt{c} d^2+\sqrt{a} e^2\right)^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^4}{4 \sqrt[4]{a} \sqrt[4]{c} d \left(\sqrt{c} d^2+\sqrt{a} e^2\right) \left(c d^4+a e^4\right) \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} d \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e^2}{2 a^{3/4} \left(c d^4+a e^4\right) \sqrt{c x^4+a}}-\frac{\sqrt{c} d x \sqrt{c x^4+a} e^2}{2 a \left(c d^4+a e^4\right) \left(\sqrt{c} x^2+\sqrt{a}\right)}+\frac{\left(a e^2-c d^2 x^2\right) e}{2 a \left(c d^4+a e^4\right) \sqrt{c x^4+a}}+\frac{\sqrt[4]{c} d \left(\sqrt{c} d^2-\sqrt{a} e^2\right) \left(\sqrt{c} x^2+\sqrt{a}\right) \sqrt{\frac{c x^4+a}{\left(\sqrt{c} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{c} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 a^{5/4} \left(c d^4+a e^4\right) \sqrt{c x^4+a}}+\frac{c d x \left(d^2+e^2 x^2\right)}{2 a \left(c d^4+a e^4\right) \sqrt{c x^4+a}}",1,"(-(Sqrt[a]*c^(3/4)*d^2*e^2*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]) + c^(3/4)*d^2*((-I)*Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c^(1/4)*d*(Sqrt[c*d^4 + a*e^4]*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)) - a*e^5*Sqrt[a + c*x^4]*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])]) - 2*(-1)^(1/4)*a^(5/4)*e^4*Sqrt[c*d^4 + a*e^4]*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4)], -1]))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(1/4)*d*(c*d^4 + a*e^4)^(3/2)*Sqrt[a + c*x^4])","C",1
224,1,274,349,0.4274302,"\int \frac{x^3 (c+d x)^n}{a+b x^4} \, dx","Integrate[(x^3*(c + d*x)^n)/(a + b*x^4),x]","\frac{(c+d x)^{n+1} \left(-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-i \sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c-i \sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+i \sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c+i \sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt[4]{-a} d}\right)}{\sqrt[4]{-a} d+\sqrt[4]{b} c}\right)}{4 b^{3/4} (n+1)}","-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt{-\sqrt{-a}} d}\right)}{4 b^{3/4} (n+1) \left(\sqrt[4]{b} c-\sqrt{-\sqrt{-a}} d\right)}-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt{-\sqrt{-a}} d}\right)}{4 b^{3/4} (n+1) \left(\sqrt{-\sqrt{-a}} d+\sqrt[4]{b} c\right)}-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}\right)}{4 b^{3/4} (n+1) \left(\sqrt[4]{b} c-\sqrt[4]{-a} d\right)}-\frac{(c+d x)^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt[4]{-a} d}\right)}{4 b^{3/4} (n+1) \left(\sqrt[4]{-a} d+\sqrt[4]{b} c\right)}",1,"((c + d*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(b^(1/4)*c - (-a)^(1/4)*d)) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(b^(1/4)*c - I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]/(b^(1/4)*c + I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(1 + n))","C",1
225,1,274,349,0.2462219,"\int \frac{x^3 (c+d x)^{1+n}}{a+b x^4} \, dx","Integrate[(x^3*(c + d*x)^(1 + n))/(a + b*x^4),x]","\frac{(c+d x)^{n+2} \left(-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-i \sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c-i \sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+i \sqrt[4]{-a} d}\right)}{\sqrt[4]{b} c+i \sqrt[4]{-a} d}-\frac{\, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt[4]{-a} d}\right)}{\sqrt[4]{-a} d+\sqrt[4]{b} c}\right)}{4 b^{3/4} (n+2)}","-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt{-\sqrt{-a}} d}\right)}{4 b^{3/4} (n+2) \left(\sqrt[4]{b} c-\sqrt{-\sqrt{-a}} d\right)}-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt{-\sqrt{-a}} d}\right)}{4 b^{3/4} (n+2) \left(\sqrt{-\sqrt{-a}} d+\sqrt[4]{b} c\right)}-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c-\sqrt[4]{-a} d}\right)}{4 b^{3/4} (n+2) \left(\sqrt[4]{b} c-\sqrt[4]{-a} d\right)}-\frac{(c+d x)^{n+2} \, _2F_1\left(1,n+2;n+3;\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{b} c+\sqrt[4]{-a} d}\right)}{4 b^{3/4} (n+2) \left(\sqrt[4]{-a} d+\sqrt[4]{b} c\right)}",1,"((c + d*x)^(2 + n)*(-(Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(b^(1/4)*c - (-a)^(1/4)*d)) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(b^(1/4)*c - I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]/(b^(1/4)*c + I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(2 + n))","C",1
226,1,1416,1605,7.3739563,"\int \frac{1}{\left(c+d x+e x^2\right) \sqrt{a+b x^4}} \, dx","Integrate[1/((c + d*x + e*x^2)*Sqrt[a + b*x^4]),x]","-\frac{i \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left(-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left(-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right)};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right)\right|-1\right) d^2}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left(-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right) \left(\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}\right) \sqrt{b x^4+a}}-\frac{i \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left(-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left(-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right)};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right)\right|-1\right) d^2}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left(-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right) \left(\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}\right) \sqrt{b x^4+a}}-\frac{\sqrt{2} e^2 \left(\frac{\tanh ^{-1}\left(\frac{2 a e^2+b \left(d^2-\sqrt{d^2-4 c e} d-2 c e\right) x^2}{\sqrt{4 a e^4+b \left(2 d^4-2 \sqrt{d^2-4 c e} d^3-8 c e d^2+4 c e \sqrt{d^2-4 c e} d+4 c^2 e^2\right)} \sqrt{b x^4+a}}\right)}{2 \sqrt{2 a e^4+b \left(d^4-\sqrt{d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right)}}-\frac{\tanh ^{-1}\left(\frac{2 a e^2+b \left(d^2+\sqrt{d^2-4 c e} d-2 c e\right) x^2}{\sqrt{4 a e^4+2 b \left(d^4+\sqrt{d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right)} \sqrt{b x^4+a}}\right)}{2 \sqrt{2 a e^4+b \left(d^4+\sqrt{d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right)}}\right)}{\sqrt{d^2-4 c e}}-\frac{i \sqrt{d^4-4 c d^2 e} \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left(-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left(-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right)};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left(-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right) \left(\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}\right) \sqrt{b x^4+a}}+\frac{i \sqrt{d^4-4 c d^2 e} \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left(-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left(-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right)};\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right)\right|-1\right)}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left(-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right) \left(\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}\right) \sqrt{b x^4+a}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-b d^4+4 b c e d^2-b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d-2 a e^4-2 b c^2 e^2} x}{e \left(d+\sqrt{d^2-4 c e}\right) \sqrt{b x^4+a}}\right) e^2}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left(d^4+\sqrt{d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-b d^4+4 b c e d^2+b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d-2 a e^4-2 b c^2 e^2} x}{e \left(d-\sqrt{d^2-4 c e}\right) \sqrt{b x^4+a}}\right) e^2}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left(d^4-\sqrt{d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right)}}-\frac{\tanh ^{-1}\left(\frac{4 a e^2+b \left(d-\sqrt{d^2-4 c e}\right)^2 x^2}{2 \sqrt{2} \sqrt{b d^4-4 b c e d^2-b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d+2 a e^4+2 b c^2 e^2} \sqrt{b x^4+a}}\right) e^2}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{b d^4-4 b c e d^2-b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d+2 a e^4+2 b c^2 e^2}}+\frac{\tanh ^{-1}\left(\frac{4 a e^2+b \left(d+\sqrt{d^2-4 c e}\right)^2 x^2}{2 \sqrt{2} \sqrt{b d^4-4 b c e d^2+b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d+2 a e^4+2 b c^2 e^2} \sqrt{b x^4+a}}\right) e^2}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{b d^4-4 b c e d^2+b \sqrt{d^2-4 c e} \left(d^2-2 c e\right) d+2 a e^4+2 b c^2 e^2}}+\frac{\sqrt[4]{b} \left(d-\sqrt{d^2-4 c e}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \sqrt{d^2-4 c e} \left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2-\sqrt{d^2-4 c e} d-2 c e\right)\right) \sqrt{b x^4+a}}-\frac{\sqrt[4]{b} \left(d+\sqrt{d^2-4 c e}\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \sqrt{d^2-4 c e} \left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2+\sqrt{d^2-4 c e} d-2 c e\right)\right) \sqrt{b x^4+a}}+\frac{\left(2 \sqrt{a} e^2-\sqrt{b} \left(d^2-\sqrt{d^2-4 c e} d-2 c e\right)\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} \Pi \left(\frac{\left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2-\sqrt{d^2-4 c e} d-2 c e\right)\right)^2}{4 \sqrt{a} \sqrt{b} e^2 \left(d-\sqrt{d^2-4 c e}\right)^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{d^2-4 c e} \left(d-\sqrt{d^2-4 c e}\right) \left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2-\sqrt{d^2-4 c e} d-2 c e\right)\right) \sqrt{b x^4+a}}-\frac{\left(2 \sqrt{a} e^2-\sqrt{b} \left(d^2+\sqrt{d^2-4 c e} d-2 c e\right)\right) \left(\sqrt{b} x^2+\sqrt{a}\right) \sqrt{\frac{b x^4+a}{\left(\sqrt{b} x^2+\sqrt{a}\right)^2}} \Pi \left(\frac{\left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2+\sqrt{d^2-4 c e} d-2 c e\right)\right)^2}{4 \sqrt{a} \sqrt{b} e^2 \left(d+\sqrt{d^2-4 c e}\right)^2};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} x}{\sqrt[4]{a}}\right)|\frac{1}{2}\right) e}{2 \sqrt[4]{a} \sqrt[4]{b} \sqrt{d^2-4 c e} \left(d+\sqrt{d^2-4 c e}\right) \left(2 \sqrt{a} e^2+\sqrt{b} \left(d^2+\sqrt{d^2-4 c e} d-2 c e\right)\right) \sqrt{b x^4+a}}",1,"-((Sqrt[2]*e^2*(ArcTanh[(2*a*e^2 + b*(d^2 - 2*c*e - d*Sqrt[d^2 - 4*c*e])*x^2)/(Sqrt[4*a*e^4 + b*(2*d^4 - 8*c*d^2*e + 4*c^2*e^2 - 2*d^3*Sqrt[d^2 - 4*c*e] + 4*c*d*e*Sqrt[d^2 - 4*c*e])]*Sqrt[a + b*x^4])]/(2*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 - d^3*Sqrt[d^2 - 4*c*e] + 2*c*d*e*Sqrt[d^2 - 4*c*e])]) - ArcTanh[(2*a*e^2 + b*(d^2 - 2*c*e + d*Sqrt[d^2 - 4*c*e])*x^2)/(Sqrt[4*a*e^4 + 2*b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 + d^3*Sqrt[d^2 - 4*c*e] - 2*c*d*e*Sqrt[d^2 - 4*c*e])]*Sqrt[a + b*x^4])]/(2*Sqrt[2*a*e^4 + b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 + d^3*Sqrt[d^2 - 4*c*e] - 2*c*d*e*Sqrt[d^2 - 4*c*e])])))/Sqrt[d^2 - 4*c*e]) - (I*d^2*Sqrt[1 - (I*Sqrt[b]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[b]*x^2)/Sqrt[a]]*EllipticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e - Sqrt[d^4 - 4*c*d^2*e])), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*e*(-d^2 + 2*c*e - Sqrt[d^4 - 4*c*d^2*e])*(-1/2*(d^2 - 2*c*e - Sqrt[d^4 - 4*c*d^2*e])/e^2 + (d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])/(2*e^2))*Sqrt[a + b*x^4]) - (I*Sqrt[d^4 - 4*c*d^2*e]*Sqrt[1 - (I*Sqrt[b]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[b]*x^2)/Sqrt[a]]*EllipticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e - Sqrt[d^4 - 4*c*d^2*e])), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*e*(-d^2 + 2*c*e - Sqrt[d^4 - 4*c*d^2*e])*(-1/2*(d^2 - 2*c*e - Sqrt[d^4 - 4*c*d^2*e])/e^2 + (d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])/(2*e^2))*Sqrt[a + b*x^4]) - (I*d^2*Sqrt[1 - (I*Sqrt[b]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[b]*x^2)/Sqrt[a]]*EllipticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e])), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*e*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e])*((d^2 - 2*c*e - Sqrt[d^4 - 4*c*d^2*e])/(2*e^2) - (d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])/(2*e^2))*Sqrt[a + b*x^4]) + (I*Sqrt[d^4 - 4*c*d^2*e]*Sqrt[1 - (I*Sqrt[b]*x^2)/Sqrt[a]]*Sqrt[1 + (I*Sqrt[b]*x^2)/Sqrt[a]]*EllipticPi[((-2*I)*Sqrt[a]*e^2)/(Sqrt[b]*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e])), I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*e*(-d^2 + 2*c*e + Sqrt[d^4 - 4*c*d^2*e])*((d^2 - 2*c*e - Sqrt[d^4 - 4*c*d^2*e])/(2*e^2) - (d^2 - 2*c*e + Sqrt[d^4 - 4*c*d^2*e])/(2*e^2))*Sqrt[a + b*x^4])","C",0
227,1,132,161,0.1039915,"\int x^m \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x^m*(c*(a + b*x^2)^2)^(3/2),x]","\frac{x^{m+1} \left(c \left(a+b x^2\right)^2\right)^{3/2} \left(a^3 \left(m^3+15 m^2+71 m+105\right)+3 a^2 b \left(m^3+13 m^2+47 m+35\right) x^2+3 a b^2 \left(m^3+11 m^2+31 m+21\right) x^4+b^3 \left(m^3+9 m^2+23 m+15\right) x^6\right)}{(m+1) (m+3) (m+5) (m+7) \left(a+b x^2\right)^3}","\frac{a^3 c x^{m+1} \sqrt{c \left(a+b x^2\right)^2}}{(m+1) \left(a+b x^2\right)}+\frac{3 a^2 b c x^{m+3} \sqrt{c \left(a+b x^2\right)^2}}{(m+3) \left(a+b x^2\right)}+\frac{b^3 c x^{m+7} \sqrt{c \left(a+b x^2\right)^2}}{(m+7) \left(a+b x^2\right)}+\frac{3 a b^2 c x^{m+5} \sqrt{c \left(a+b x^2\right)^2}}{(m+5) \left(a+b x^2\right)}",1,"(x^(1 + m)*(c*(a + b*x^2)^2)^(3/2)*(a^3*(105 + 71*m + 15*m^2 + m^3) + 3*a^2*b*(35 + 47*m + 13*m^2 + m^3)*x^2 + 3*a*b^2*(21 + 31*m + 11*m^2 + m^3)*x^4 + b^3*(15 + 23*m + 9*m^2 + m^3)*x^6))/((1 + m)*(3 + m)*(5 + m)*(7 + m)*(a + b*x^2)^3)","A",1
228,1,63,143,0.0239786,"\int x^5 \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x^5*(c*(a + b*x^2)^2)^(3/2),x]","\frac{x^6 \left(20 a^3+45 a^2 b x^2+36 a b^2 x^4+10 b^3 x^6\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{120 \left(a+b x^2\right)^3}","\frac{a^3 c x^6 \sqrt{c \left(a+b x^2\right)^2}}{6 \left(a+b x^2\right)}+\frac{3 a^2 b c x^8 \sqrt{c \left(a+b x^2\right)^2}}{8 \left(a+b x^2\right)}+\frac{b^3 c x^{12} \sqrt{c \left(a+b x^2\right)^2}}{12 \left(a+b x^2\right)}+\frac{3 a b^2 c x^{10} \sqrt{c \left(a+b x^2\right)^2}}{10 \left(a+b x^2\right)}",1,"(x^6*(c*(a + b*x^2)^2)^(3/2)*(20*a^3 + 45*a^2*b*x^2 + 36*a*b^2*x^4 + 10*b^3*x^6))/(120*(a + b*x^2)^3)","A",1
229,1,63,143,0.023422,"\int x^4 \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x^4*(c*(a + b*x^2)^2)^(3/2),x]","\frac{x^5 \left(231 a^3+495 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{1155 \left(a+b x^2\right)^3}","\frac{a^3 c x^5 \sqrt{c \left(a+b x^2\right)^2}}{5 \left(a+b x^2\right)}+\frac{3 a^2 b c x^7 \sqrt{c \left(a+b x^2\right)^2}}{7 \left(a+b x^2\right)}+\frac{b^3 c x^{11} \sqrt{c \left(a+b x^2\right)^2}}{11 \left(a+b x^2\right)}+\frac{a b^2 c x^9 \sqrt{c \left(a+b x^2\right)^2}}{3 \left(a+b x^2\right)}",1,"(x^5*(c*(a + b*x^2)^2)^(3/2)*(231*a^3 + 495*a^2*b*x^2 + 385*a*b^2*x^4 + 105*b^3*x^6))/(1155*(a + b*x^2)^3)","A",1
230,1,63,66,0.0231703,"\int x^3 \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x^3*(c*(a + b*x^2)^2)^(3/2),x]","\frac{x^4 \left(10 a^3+20 a^2 b x^2+15 a b^2 x^4+4 b^3 x^6\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{40 \left(a+b x^2\right)^3}","\frac{c \left(a+b x^2\right)^4 \sqrt{c \left(a+b x^2\right)^2}}{10 b^2}-\frac{a c \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^2}}{8 b^2}",1,"(x^4*(c*(a + b*x^2)^2)^(3/2)*(10*a^3 + 20*a^2*b*x^2 + 15*a*b^2*x^4 + 4*b^3*x^6))/(40*(a + b*x^2)^3)","A",1
231,1,63,143,0.0195626,"\int x^2 \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x^2*(c*(a + b*x^2)^2)^(3/2),x]","\frac{\left(105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{315 \left(a+b x^2\right)^3}","\frac{a^3 c x^3 \sqrt{c \left(a+b x^2\right)^2}}{3 \left(a+b x^2\right)}+\frac{3 a^2 b c x^5 \sqrt{c \left(a+b x^2\right)^2}}{5 \left(a+b x^2\right)}+\frac{b^3 c x^9 \sqrt{c \left(a+b x^2\right)^2}}{9 \left(a+b x^2\right)}+\frac{3 a b^2 c x^7 \sqrt{c \left(a+b x^2\right)^2}}{7 \left(a+b x^2\right)}",1,"((c*(a + b*x^2)^2)^(3/2)*(105*a^3*x^3 + 189*a^2*b*x^5 + 135*a*b^2*x^7 + 35*b^3*x^9))/(315*(a + b*x^2)^3)","A",1
232,1,29,32,0.0127412,"\int x \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[x*(c*(a + b*x^2)^2)^(3/2),x]","\frac{\left(a+b x^2\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{8 b}","\frac{c \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^2}}{8 b}",1,"((a + b*x^2)*(c*(a + b*x^2)^2)^(3/2))/(8*b)","A",1
233,1,61,135,0.0158936,"\int \left(c \left(a+b x^2\right)^2\right)^{3/2} \, dx","Integrate[(c*(a + b*x^2)^2)^(3/2),x]","\frac{\left(35 a^3 x+35 a^2 b x^3+21 a b^2 x^5+5 b^3 x^7\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{35 \left(a+b x^2\right)^3}","\frac{a^3 c x \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}+\frac{a^2 b c x^3 \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}+\frac{b^3 c x^7 \sqrt{c \left(a+b x^2\right)^2}}{7 \left(a+b x^2\right)}+\frac{3 a b^2 c x^5 \sqrt{c \left(a+b x^2\right)^2}}{5 \left(a+b x^2\right)}",1,"((c*(a + b*x^2)^2)^(3/2)*(35*a^3*x + 35*a^2*b*x^3 + 21*a*b^2*x^5 + 5*b^3*x^7))/(35*(a + b*x^2)^3)","A",1
234,1,62,139,0.0239977,"\int \frac{\left(c \left(a+b x^2\right)^2\right)^{3/2}}{x} \, dx","Integrate[(c*(a + b*x^2)^2)^(3/2)/x,x]","\frac{\left(c \left(a+b x^2\right)^2\right)^{3/2} \left(12 a^3 \log (x)+b x^2 \left(18 a^2+9 a b x^2+2 b^2 x^4\right)\right)}{12 \left(a+b x^2\right)^3}","\frac{a^3 c \log (x) \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}+\frac{3 a^2 b c x^2 \sqrt{c \left(a+b x^2\right)^2}}{2 \left(a+b x^2\right)}+\frac{b^3 c x^6 \sqrt{c \left(a+b x^2\right)^2}}{6 \left(a+b x^2\right)}+\frac{3 a b^2 c x^4 \sqrt{c \left(a+b x^2\right)^2}}{4 \left(a+b x^2\right)}",1,"((c*(a + b*x^2)^2)^(3/2)*(b*x^2*(18*a^2 + 9*a*b*x^2 + 2*b^2*x^4) + 12*a^3*Log[x]))/(12*(a + b*x^2)^3)","A",1
235,1,62,134,0.0260423,"\int \frac{\left(c \left(a+b x^2\right)^2\right)^{3/2}}{x^2} \, dx","Integrate[(c*(a + b*x^2)^2)^(3/2)/x^2,x]","\frac{\left(-5 a^3+15 a^2 b x^2+5 a b^2 x^4+b^3 x^6\right) \left(c \left(a+b x^2\right)^2\right)^{3/2}}{5 x \left(a+b x^2\right)^3}","-\frac{a^3 c \sqrt{c \left(a+b x^2\right)^2}}{x \left(a+b x^2\right)}+\frac{3 a^2 b c x \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}+\frac{b^3 c x^5 \sqrt{c \left(a+b x^2\right)^2}}{5 \left(a+b x^2\right)}+\frac{a b^2 c x^3 \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}",1,"((c*(a + b*x^2)^2)^(3/2)*(-5*a^3 + 15*a^2*b*x^2 + 5*a*b^2*x^4 + b^3*x^6))/(5*x*(a + b*x^2)^3)","A",1
236,1,65,140,0.0266503,"\int \frac{\left(c \left(a+b x^2\right)^2\right)^{3/2}}{x^3} \, dx","Integrate[(c*(a + b*x^2)^2)^(3/2)/x^3,x]","-\frac{\left(c \left(a+b x^2\right)^2\right)^{3/2} \left(2 a^3-12 a^2 b x^2 \log (x)-6 a b^2 x^4-b^3 x^6\right)}{4 x^2 \left(a+b x^2\right)^3}","-\frac{a^3 c \sqrt{c \left(a+b x^2\right)^2}}{2 x^2 \left(a+b x^2\right)}+\frac{3 a^2 b c \log (x) \sqrt{c \left(a+b x^2\right)^2}}{a+b x^2}+\frac{b^3 c x^4 \sqrt{c \left(a+b x^2\right)^2}}{4 \left(a+b x^2\right)}+\frac{3 a b^2 c x^2 \sqrt{c \left(a+b x^2\right)^2}}{2 \left(a+b x^2\right)}",1,"-1/4*((c*(a + b*x^2)^2)^(3/2)*(2*a^3 - 6*a*b^2*x^4 - b^3*x^6 - 12*a^2*b*x^2*Log[x]))/(x^2*(a + b*x^2)^3)","A",1
237,1,143,253,0.1530772,"\int x^2 \left(c \left(a+b x^2\right)^3\right)^{3/2} \, dx","Integrate[x^2*(c*(a + b*x^2)^3)^(3/2),x]","\frac{\left(c \left(a+b x^2\right)^3\right)^{3/2} \left(\sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \left(315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right)-315 a^{11/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right)}{15360 b^{3/2} \left(a+b x^2\right)^4 \sqrt{\frac{b x^2}{a}+1}}","-\frac{21 a^{9/2} c \sqrt{c \left(a+b x^2\right)^3} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{1024 b^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/2}}+\frac{21 a^5 c x \sqrt{c \left(a+b x^2\right)^3}}{1024 b \left(a+b x^2\right)}+\frac{21 a^4 c x^3 \sqrt{c \left(a+b x^2\right)^3}}{512 \left(a+b x^2\right)}+\frac{7}{128} a^3 c x^3 \sqrt{c \left(a+b x^2\right)^3}+\frac{21}{320} a^2 c x^3 \left(a+b x^2\right) \sqrt{c \left(a+b x^2\right)^3}+\frac{3}{40} a c x^3 \left(a+b x^2\right)^2 \sqrt{c \left(a+b x^2\right)^3}+\frac{1}{12} c x^3 \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^3}",1,"((c*(a + b*x^2)^3)^(3/2)*(Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*(315*a^5 + 4910*a^4*b*x^2 + 11432*a^3*b^2*x^4 + 12144*a^2*b^3*x^6 + 6272*a*b^4*x^8 + 1280*b^5*x^10) - 315*a^(11/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(15360*b^(3/2)*(a + b*x^2)^4*Sqrt[1 + (b*x^2)/a])","A",1
238,1,29,32,0.0169606,"\int x \left(c \left(a+b x^2\right)^3\right)^{3/2} \, dx","Integrate[x*(c*(a + b*x^2)^3)^(3/2),x]","\frac{\left(a+b x^2\right) \left(c \left(a+b x^2\right)^3\right)^{3/2}}{11 b}","\frac{c \left(a+b x^2\right)^4 \sqrt{c \left(a+b x^2\right)^3}}{11 b}",1,"((a + b*x^2)*(c*(a + b*x^2)^3)^(3/2))/(11*b)","A",1
239,1,132,207,0.1133023,"\int \left(c \left(a+b x^2\right)^3\right)^{3/2} \, dx","Integrate[(c*(a + b*x^2)^3)^(3/2),x]","\frac{\left(c \left(a+b x^2\right)^3\right)^{3/2} \left(315 a^{9/2} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)+\sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \left(965 a^4+1490 a^3 b x^2+1368 a^2 b^2 x^4+656 a b^3 x^6+128 b^4 x^8\right)\right)}{1280 \sqrt{b} \left(a+b x^2\right)^4 \sqrt{\frac{b x^2}{a}+1}}","\frac{63 a^{7/2} c \sqrt{c \left(a+b x^2\right)^3} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{256 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/2}}+\frac{63 a^4 c x \sqrt{c \left(a+b x^2\right)^3}}{256 \left(a+b x^2\right)}+\frac{21}{128} a^3 c x \sqrt{c \left(a+b x^2\right)^3}+\frac{21}{160} a^2 c x \left(a+b x^2\right) \sqrt{c \left(a+b x^2\right)^3}+\frac{9}{80} a c x \left(a+b x^2\right)^2 \sqrt{c \left(a+b x^2\right)^3}+\frac{1}{10} c x \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^3}",1,"((c*(a + b*x^2)^3)^(3/2)*(Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*(965*a^4 + 1490*a^3*b*x^2 + 1368*a^2*b^2*x^4 + 656*a*b^3*x^6 + 128*b^4*x^8) + 315*a^(9/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(1280*Sqrt[b]*(a + b*x^2)^4*Sqrt[1 + (b*x^2)/a])","A",1
240,1,111,192,0.0759327,"\int \frac{\left(c \left(a+b x^2\right)^3\right)^{3/2}}{x} \, dx","Integrate[(c*(a + b*x^2)^3)^(3/2)/x,x]","\frac{\left(c \left(a+b x^2\right)^3\right)^{3/2} \left(\sqrt{a+b x^2} \left(563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right)-315 a^{9/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)\right)}{315 \left(a+b x^2\right)^{9/2}}","\frac{a^4 c \sqrt{c \left(a+b x^2\right)^3}}{a+b x^2}+\frac{1}{3} a^3 c \sqrt{c \left(a+b x^2\right)^3}-\frac{a^3 c \sqrt{c \left(a+b x^2\right)^3} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/2}}+\frac{1}{5} a^2 c \left(a+b x^2\right) \sqrt{c \left(a+b x^2\right)^3}+\frac{1}{7} a c \left(a+b x^2\right)^2 \sqrt{c \left(a+b x^2\right)^3}+\frac{1}{9} c \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^3}",1,"((c*(a + b*x^2)^3)^(3/2)*(Sqrt[a + b*x^2]*(563*a^4 + 506*a^3*b*x^2 + 408*a^2*b^2*x^4 + 185*a*b^3*x^6 + 35*b^4*x^8) - 315*a^(9/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/(315*(a + b*x^2)^(9/2))","A",1
241,1,65,208,0.017043,"\int \frac{\left(c \left(a+b x^2\right)^3\right)^{3/2}}{x^2} \, dx","Integrate[(c*(a + b*x^2)^3)^(3/2)/x^2,x]","-\frac{a^4 \left(c \left(a+b x^2\right)^3\right)^{3/2} \, _2F_1\left(-\frac{9}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(a+b x^2\right)^4 \sqrt{\frac{b x^2}{a}+1}}","\frac{315 a^{5/2} \sqrt{b} c \sqrt{c \left(a+b x^2\right)^3} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{128 \left(\frac{b x^2}{a}+1\right)^{3/2}}+\frac{315 a^3 b c x \sqrt{c \left(a+b x^2\right)^3}}{128 \left(a+b x^2\right)}+\frac{105}{64} a^2 b c x \sqrt{c \left(a+b x^2\right)^3}+\frac{21}{16} a b c x \left(a+b x^2\right) \sqrt{c \left(a+b x^2\right)^3}-\frac{c \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^3}}{x}+\frac{9}{8} b c x \left(a+b x^2\right)^2 \sqrt{c \left(a+b x^2\right)^3}",1,"-((a^4*(c*(a + b*x^2)^3)^(3/2)*Hypergeometric2F1[-9/2, -1/2, 1/2, -((b*x^2)/a)])/(x*(a + b*x^2)^4*Sqrt[1 + (b*x^2)/a]))","C",1
242,1,48,202,0.0192599,"\int \frac{\left(c \left(a+b x^2\right)^3\right)^{3/2}}{x^3} \, dx","Integrate[(c*(a + b*x^2)^3)^(3/2)/x^3,x]","\frac{b \left(a+b x^2\right) \left(c \left(a+b x^2\right)^3\right)^{3/2} \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right)}{11 a^2}","\frac{9 a^3 b c \sqrt{c \left(a+b x^2\right)^3}}{2 \left(a+b x^2\right)}+\frac{3}{2} a^2 b c \sqrt{c \left(a+b x^2\right)^3}-\frac{9 a^2 b c \sqrt{c \left(a+b x^2\right)^3} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{2 \left(\frac{b x^2}{a}+1\right)^{3/2}}+\frac{9}{10} a b c \left(a+b x^2\right) \sqrt{c \left(a+b x^2\right)^3}-\frac{c \left(a+b x^2\right)^3 \sqrt{c \left(a+b x^2\right)^3}}{2 x^2}+\frac{9}{14} b c \left(a+b x^2\right)^2 \sqrt{c \left(a+b x^2\right)^3}",1,"(b*(a + b*x^2)*(c*(a + b*x^2)^3)^(3/2)*Hypergeometric2F1[2, 11/2, 13/2, 1 + (b*x^2)/a])/(11*a^2)","C",1
243,1,89,77,0.0672052,"\int x^2 \left(\frac{c}{a+b x^2}\right)^{3/2} \, dx","Integrate[x^2*(c/(a + b*x^2))^(3/2),x]","\frac{\sqrt{a} \sqrt{\frac{b x^2}{a}+1} \left(\frac{c}{a+b x^2}\right)^{3/2} \left(\left(a+b x^2\right) \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)-\sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1}\right)}{b^{3/2}}","\frac{\sqrt{a} c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{c}{a+b x^2}} \sinh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)}{b^{3/2}}-\frac{c x \sqrt{\frac{c}{a+b x^2}}}{b}",1,"(Sqrt[a]*(c/(a + b*x^2))^(3/2)*Sqrt[1 + (b*x^2)/a]*(-(Sqrt[a]*Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]) + (a + b*x^2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/b^(3/2)","A",1
244,1,21,21,0.0049174,"\int x \left(\frac{c}{a+b x^2}\right)^{3/2} \, dx","Integrate[x*(c/(a + b*x^2))^(3/2),x]","-\frac{c \sqrt{\frac{c}{a+b x^2}}}{b}","-\frac{c \sqrt{\frac{c}{a+b x^2}}}{b}",1,"-((c*Sqrt[c/(a + b*x^2)])/b)","A",1
245,1,21,21,0.0079614,"\int \left(\frac{c}{a+b x^2}\right)^{3/2} \, dx","Integrate[(c/(a + b*x^2))^(3/2),x]","\frac{c x \sqrt{\frac{c}{a+b x^2}}}{a}","\frac{c x \sqrt{\frac{c}{a+b x^2}}}{a}",1,"(c*x*Sqrt[c/(a + b*x^2)])/a","A",1
246,1,38,71,0.0085174,"\int \frac{\left(\frac{c}{a+b x^2}\right)^{3/2}}{x} \, dx","Integrate[(c/(a + b*x^2))^(3/2)/x,x]","\frac{c \sqrt{\frac{c}{a+b x^2}} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right)}{a}","\frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{a}",1,"(c*Sqrt[c/(a + b*x^2)]*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^2)/a])/a","C",1
247,1,32,48,0.0094695,"\int \frac{\left(\frac{c}{a+b x^2}\right)^{3/2}}{x^2} \, dx","Integrate[(c/(a + b*x^2))^(3/2)/x^2,x]","-\frac{c \left(a+2 b x^2\right) \sqrt{\frac{c}{a+b x^2}}}{a^2 x}","-\frac{2 b c x \sqrt{\frac{c}{a+b x^2}}}{a^2}-\frac{c \sqrt{\frac{c}{a+b x^2}}}{a x}",1,"-((c*Sqrt[c/(a + b*x^2)]*(a + 2*b*x^2))/(a^2*x))","A",1
248,1,40,104,0.0108598,"\int \frac{\left(\frac{c}{a+b x^2}\right)^{3/2}}{x^3} \, dx","Integrate[(c/(a + b*x^2))^(3/2)/x^3,x]","-\frac{b c \sqrt{\frac{c}{a+b x^2}} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{b x^2}{a}+1\right)}{a^2}","-\frac{3 b c \sqrt{\frac{c}{a+b x^2}}}{2 a^2}+\frac{3 b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)}{2 a^2}-\frac{c \sqrt{\frac{c}{a+b x^2}}}{2 a x^2}",1,"-((b*c*Sqrt[c/(a + b*x^2)]*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (b*x^2)/a])/a^2)","C",1
249,1,63,138,0.0375009,"\int x^7 \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[x^7*(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{2 \left(a+b x^2\right) \left(-128 a^3+224 a^2 b x^2-308 a b^2 x^4+385 b^3 x^6\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{7315 b^4}","-\frac{2 a^3 \left(a+b x^2\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{7 b^4}+\frac{6 a^2 \left(a+b x^2\right)^2 \left(c \sqrt{a+b x^2}\right)^{3/2}}{11 b^4}+\frac{2 \left(a+b x^2\right)^4 \left(c \sqrt{a+b x^2}\right)^{3/2}}{19 b^4}-\frac{2 a \left(a+b x^2\right)^3 \left(c \sqrt{a+b x^2}\right)^{3/2}}{5 b^4}",1,"(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*(-128*a^3 + 224*a^2*b*x^2 - 308*a*b^2*x^4 + 385*b^3*x^6))/(7315*b^4)","A",1
250,1,52,102,0.0261983,"\int x^5 \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[x^5*(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{2 \left(a+b x^2\right) \left(32 a^2-56 a b x^2+77 b^2 x^4\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{1155 b^3}","\frac{2 a^2 \left(a+b x^2\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{7 b^3}+\frac{2 \left(a+b x^2\right)^3 \left(c \sqrt{a+b x^2}\right)^{3/2}}{15 b^3}-\frac{4 a \left(a+b x^2\right)^2 \left(c \sqrt{a+b x^2}\right)^{3/2}}{11 b^3}",1,"(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*(32*a^2 - 56*a*b*x^2 + 77*b^2*x^4))/(1155*b^3)","A",1
251,1,41,66,0.0205353,"\int x^3 \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[x^3*(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{2 \left(a+b x^2\right) \left(7 b x^2-4 a\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{77 b^2}","\frac{2 \left(a+b x^2\right)^2 \left(c \sqrt{a+b x^2}\right)^{3/2}}{11 b^2}-\frac{2 a \left(a+b x^2\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{7 b^2}",1,"(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*(-4*a + 7*b*x^2))/(77*b^2)","A",1
252,1,31,36,0.0070925,"\int x \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[x*(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{2 \left(a+b x^2\right) \left(c \sqrt{a+b x^2}\right)^{3/2}}{7 b}","\frac{2 c \left(a+b x^2\right)^{3/2} \sqrt{c \sqrt{a+b x^2}}}{7 b}",1,"(2*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2))/(7*b)","A",1
253,1,96,117,0.0378905,"\int \frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{x} \, dx","Integrate[(c*Sqrt[a + b*x^2])^(3/2)/x,x]","\frac{\left(c \sqrt{a+b x^2}\right)^{3/2} \left(3 a^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a+b x^2}}{\sqrt[4]{a}}\right)-3 a^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a+b x^2}}{\sqrt[4]{a}}\right)+2 \left(a+b x^2\right)^{3/4}\right)}{3 \left(a+b x^2\right)^{3/4}}","\frac{2}{3} \left(c \sqrt{a+b x^2}\right)^{3/2}+\frac{\left(c \sqrt{a+b x^2}\right)^{3/2} \tan ^{-1}\left(\sqrt[4]{\frac{b x^2}{a}+1}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2} \tanh ^{-1}\left(\sqrt[4]{\frac{b x^2}{a}+1}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}",1,"((c*Sqrt[a + b*x^2])^(3/2)*(2*(a + b*x^2)^(3/4) + 3*a^(3/4)*ArcTan[(a + b*x^2)^(1/4)/a^(1/4)] - 3*a^(3/4)*ArcTanh[(a + b*x^2)^(1/4)/a^(1/4)]))/(3*(a + b*x^2)^(3/4))","A",1
254,1,50,133,0.012954,"\int \frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{x^3} \, dx","Integrate[(c*Sqrt[a + b*x^2])^(3/2)/x^3,x]","\frac{2 b \left(a+b x^2\right) \left(c \sqrt{a+b x^2}\right)^{3/2} \, _2F_1\left(\frac{7}{4},2;\frac{11}{4};\frac{b x^2}{a}+1\right)}{7 a^2}","-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{2 x^2}+\frac{3 b \left(c \sqrt{a+b x^2}\right)^{3/2} \tan ^{-1}\left(\sqrt[4]{\frac{b x^2}{a}+1}\right)}{4 a \left(\frac{b x^2}{a}+1\right)^{3/4}}-\frac{3 b \left(c \sqrt{a+b x^2}\right)^{3/2} \tanh ^{-1}\left(\sqrt[4]{\frac{b x^2}{a}+1}\right)}{4 a \left(\frac{b x^2}{a}+1\right)^{3/4}}",1,"(2*b*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2)*Hypergeometric2F1[7/4, 2, 11/4, 1 + (b*x^2)/a])/(7*a^2)","C",1
255,1,68,152,0.0581288,"\int x^2 \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[x^2*(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{2 x \left(c \sqrt{a+b x^2}\right)^{3/2} \left(-\frac{a \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}+a+b x^2\right)}{9 b}","\frac{4 a^{3/2} \left(c \sqrt{a+b x^2}\right)^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{15 b^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4}}-\frac{4 a^2 x \left(c \sqrt{a+b x^2}\right)^{3/2}}{15 b \left(a+b x^2\right)}+\frac{2 a x \left(c \sqrt{a+b x^2}\right)^{3/2}}{15 b}+\frac{2}{9} x^3 \left(c \sqrt{a+b x^2}\right)^{3/2}",1,"(2*x*(c*Sqrt[a + b*x^2])^(3/2)*(a + b*x^2 - (a*Hypergeometric2F1[-3/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(3/4)))/(9*b)","C",1
256,1,52,119,0.0069553,"\int \left(c \sqrt{a+b x^2}\right)^{3/2} \, dx","Integrate[(c*Sqrt[a + b*x^2])^(3/2),x]","\frac{x \left(c \sqrt{a+b x^2}\right)^{3/2} \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right)}{\left(\frac{b x^2}{a}+1\right)^{3/4}}","\frac{2}{5} x \left(c \sqrt{a+b x^2}\right)^{3/2}+\frac{6 a x \left(c \sqrt{a+b x^2}\right)^{3/2}}{5 \left(a+b x^2\right)}-\frac{6 \sqrt{a} \left(c \sqrt{a+b x^2}\right)^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{5 \sqrt{b} \left(\frac{b x^2}{a}+1\right)^{3/4}}",1,"(x*(c*Sqrt[a + b*x^2])^(3/2)*Hypergeometric2F1[-3/4, 1/2, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(3/4)","C",1
257,1,55,115,0.0103043,"\int \frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{x^2} \, dx","Integrate[(c*Sqrt[a + b*x^2])^(3/2)/x^2,x]","-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2} \, _2F_1\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right)}{x \left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{x}+\frac{3 b x \left(c \sqrt{a+b x^2}\right)^{3/2}}{a+b x^2}-\frac{3 \sqrt{b} \left(c \sqrt{a+b x^2}\right)^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{\sqrt{a} \left(\frac{b x^2}{a}+1\right)^{3/4}}",1,"-(((c*Sqrt[a + b*x^2])^(3/2)*Hypergeometric2F1[-3/4, -1/2, 1/2, -((b*x^2)/a)])/(x*(1 + (b*x^2)/a)^(3/4)))","C",1
258,1,57,154,0.0131761,"\int \frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{x^4} \, dx","Integrate[(c*Sqrt[a + b*x^2])^(3/2)/x^4,x]","-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};-\frac{b x^2}{a}\right)}{3 x^3 \left(\frac{b x^2}{a}+1\right)^{3/4}}","-\frac{b^{3/2} \left(c \sqrt{a+b x^2}\right)^{3/2} E\left(\left.\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a}}\right)\right|2\right)}{2 a^{3/2} \left(\frac{b x^2}{a}+1\right)^{3/4}}+\frac{b^2 x \left(c \sqrt{a+b x^2}\right)^{3/2}}{2 a \left(a+b x^2\right)}-\frac{b \left(c \sqrt{a+b x^2}\right)^{3/2}}{2 a x}-\frac{\left(c \sqrt{a+b x^2}\right)^{3/2}}{3 x^3}",1,"-1/3*((c*Sqrt[a + b*x^2])^(3/2)*Hypergeometric2F1[-3/2, -3/4, -1/2, -((b*x^2)/a)])/(x^3*(1 + (b*x^2)/a)^(3/4))","C",1
259,1,106,71,0.1617641,"\int \sqrt{(b-x) (-a+x)} \, dx","Integrate[Sqrt[(b - x)*(-a + x)],x]","\frac{(a-x) \left((a-b)^{5/2} \sqrt{b-x} \sqrt{\frac{a-x}{a-b}} \sinh ^{-1}\left(\frac{\sqrt{b-x}}{\sqrt{a-b}}\right)-(a-x) (b-x) (a+b-2 x)\right)}{4 (x-a) \sqrt{(a-x) (x-b)}}","-\frac{1}{4} (a+b-2 x) \sqrt{x (a+b)-a b-x^2}-\frac{1}{8} (a-b)^2 \tan ^{-1}\left(\frac{a+b-2 x}{2 \sqrt{x (a+b)-a b-x^2}}\right)",1,"((a - x)*(-((a + b - 2*x)*(a - x)*(b - x)) + (a - b)^(5/2)*Sqrt[(a - x)/(a - b)]*Sqrt[b - x]*ArcSinh[Sqrt[b - x]/Sqrt[a - b]]))/(4*(-a + x)*Sqrt[(a - x)*(-b + x)])","A",1
260,1,59,48,0.0596281,"\int \sqrt{\left(1-x^2\right) \left(3+x^2\right)} \, dx","Integrate[Sqrt[(1 - x^2)*(3 + x^2)],x]","\frac{1}{3} \left(\sqrt{-x^4-2 x^2+3} x-4 i F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)-2 i E\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)\right)","\frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}-\frac{2 E\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(x*Sqrt[3 - 2*x^2 - x^4] - (2*I)*EllipticE[I*ArcSinh[x/Sqrt[3]], -3] - (4*I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3])/3","C",1
261,1,72,32,0.0266693,"\int \frac{1}{\sqrt{(b-x) (-a+x)}} \, dx","Integrate[1/Sqrt[(b - x)*(-a + x)],x]","-\frac{2 \sqrt{a-b} \sqrt{b-x} \sqrt{\frac{a-x}{a-b}} \sinh ^{-1}\left(\frac{\sqrt{b-x}}{\sqrt{a-b}}\right)}{\sqrt{(a-x) (x-b)}}","-\tan ^{-1}\left(\frac{a+b-2 x}{2 \sqrt{x (a+b)-a b-x^2}}\right)",1,"(-2*Sqrt[a - b]*Sqrt[(a - x)/(a - b)]*Sqrt[b - x]*ArcSinh[Sqrt[b - x]/Sqrt[a - b]])/Sqrt[(a - x)*(-b + x)]","B",1
262,1,18,12,0.017279,"\int \frac{1}{\sqrt{\left(1-x^2\right) \left(3+x^2\right)}} \, dx","Integrate[1/Sqrt[(1 - x^2)*(3 + x^2)],x]","-i F\left(\left.i \sinh ^{-1}\left(\frac{x}{\sqrt{3}}\right)\right|-3\right)","\frac{F\left(\sin ^{-1}(x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(-I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3]","C",1
263,1,198,244,0.5230466,"\int x^5 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-\frac{3 \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{a+b x^2}}-b \sqrt{d} \left(c+d x^2\right) \left(3 a^2 d^2-2 a b d \left(d x^2-2 c\right)+b^2 \left(-15 c^2+10 c d x^2-8 d^2 x^4\right)\right)\right)}{48 b^3 d^{7/2}}","\frac{\left(c+d x^2\right) \left(-a^2 d^2-2 a b c d+11 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{16 b^2 d^3}-\frac{\sqrt{e} (b c-a d) \left(a^2 d^2+2 a b c d+5 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{16 b^{5/2} d^{7/2}}-\frac{\left(c+d x^2\right)^2 (a d+3 b c) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 b d^3}+\frac{\left(c+d x^2\right)^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{6 b d^2 e}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(b*Sqrt[d]*(c + d*x^2)*(3*a^2*d^2 - 2*a*b*d*(-2*c + d*x^2) + b^2*(-15*c^2 + 10*c*d*x^2 - 8*d^2*x^4))) - (3*(b*c - a*d)^(3/2)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/Sqrt[a + b*x^2]))/(48*b^3*d^(7/2))","A",1
264,1,149,161,0.3808968,"\int x^3 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(b \sqrt{d} \left(c+d x^2\right) \left(a d-3 b c+2 b d x^2\right)+\frac{(a d+3 b c) (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{a+b x^2}}\right)}{8 b^2 d^{5/2}}","\frac{\sqrt{e} (b c-a d) (a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{8 b^{3/2} d^{5/2}}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 d^2}-\frac{\left(c+d x^2\right) (5 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 b d^2}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(b*Sqrt[d]*(c + d*x^2)*(-3*b*c + a*d + 2*b*d*x^2) + ((b*c - a*d)^(3/2)*(3*b*c + a*d)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/Sqrt[a + b*x^2]))/(8*b^2*d^(5/2))","A",1
265,1,143,103,0.2838828,"\int x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(b \sqrt{d} \left(a+b x^2\right) \left(c+d x^2\right)-\sqrt{a+b x^2} (b c-a d)^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)\right)}{2 b d^{3/2} \left(a+b x^2\right)}","\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 d}-\frac{\sqrt{e} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{2 \sqrt{b} d^{3/2}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(b*Sqrt[d]*(a + b*x^2)*(c + d*x^2) - (b*c - a*d)^(3/2)*Sqrt[a + b*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(2*b*d^(3/2)*(a + b*x^2))","A",1
266,1,173,112,0.1792337,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x,x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\sqrt{c} \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-\sqrt{a} \sqrt{d} \sqrt{c+d x^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)\right)}{\sqrt{c} \sqrt{d} \sqrt{a+b x^2}}","\frac{\sqrt{b} \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{\sqrt{d}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{c}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]] - Sqrt[a]*Sqrt[d]*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2])","A",1
267,1,133,127,0.1021513,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^3} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^3,x]","\frac{\sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{c x^2}\right)}{2 \sqrt{a+b x^2}}","\frac{(b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 c \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)}-\frac{\sqrt{e} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{2 \sqrt{a} c^{3/2}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-((Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^2)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*c^(3/2))))/(2*Sqrt[a + b*x^2])","A",1
268,1,174,208,0.1143186,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^5} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^5,x]","\frac{\sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(x^4 \left(-3 a^2 d^2+2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)+\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \sqrt{c+d x^2} \left(-2 a c+3 a d x^2-b c x^2\right)\right)}{8 a^{3/2} c^{5/2} x^4 \sqrt{a+b x^2}}","\frac{\sqrt{e} (3 a d+b c) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{8 a^{3/2} c^{5/2}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 c^2 \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)^2}+\frac{(b c-5 a d) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 a c^2 \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-2*a*c - b*c*x^2 + 3*a*d*x^2) + (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*x^4*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(8*a^(3/2)*c^(5/2)*x^4*Sqrt[a + b*x^2])","A",1
269,1,222,318,0.1763138,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^7} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^7,x]","\frac{\sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \sqrt{c+d x^2} \left(a^2 \left(-8 c^2+10 c d x^2-15 d^2 x^4\right)-2 a b c x^2 \left(c-2 d x^2\right)+3 b^2 c^2 x^4\right)-3 x^6 \left(-5 a^3 d^3+3 a^2 b c d^2+a b^2 c^2 d+b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)\right)}{48 a^{5/2} c^{7/2} x^6 \sqrt{a+b x^2}}","-\frac{\left(-11 a^2 d^2+2 a b c d+b^2 c^2\right) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{16 a^2 c^3 \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)}-\frac{\sqrt{e} \left(5 a^2 d^2+2 a b c d+b^2 c^2\right) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{16 a^{5/2} c^{7/2}}+\frac{(3 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 a c^3 \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)^2}+\frac{e^2 (b c-a d)^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{6 a c^2 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^3}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(3*b^2*c^2*x^4 - 2*a*b*c*x^2*(c - 2*d*x^2) + a^2*(-8*c^2 + 10*c*d*x^2 - 15*d^2*x^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*x^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(48*a^(5/2)*c^(7/2)*x^6*Sqrt[a + b*x^2])","A",1
270,1,255,357,0.4796073,"\int x^4 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[x^4*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2+7 a b c d-8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2+3 a b c d-8 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a d-4 b c+3 b d x^2\right)\right)}{15 b d^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{x \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{15 b^2 d^2}-\frac{\sqrt{c} \left(-2 a^2 d^2-3 a b c d+8 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^2 d^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{c^{3/2} (4 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b d^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{x \left(c+d x^2\right) (4 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{15 b d^2}+\frac{x^3 \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 d}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-4*b*c + a*d + 3*b*d*x^2) + I*c*(-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-8*b^2*c^2 + 7*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*b*Sqrt[b/a]*d^3*(a + b*x^2))","C",1
271,1,208,266,0.3141097,"\int x^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[x^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)+2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 d^2 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","-\frac{c^{3/2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 d^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{\sqrt{c} (2 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b d^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{x \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 d}-\frac{x (2 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 b d}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) - I*c*(-2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*Sqrt[b/a]*d^2*(a + b*x^2))","C",1
272,1,86,194,0.0554924,"\int \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{c+d x^2}{c}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}}}","x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}+\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[(a + b*x^2)/a])","A",1
273,1,111,239,0.2543246,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^2} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^2,x]","\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\frac{b \sqrt{\frac{b x^2}{a}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \left(a+b x^2\right) \sqrt{\frac{d x^2}{c}+1}}-\frac{1}{x}\right)}{c}","\frac{d x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c}-\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c x}+\frac{b \sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)*(-x^(-1) + (b*Sqrt[1 + (b*x^2)/a]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*(a + b*x^2)*Sqrt[1 + (d*x^2)/c])))/c","A",1
274,1,238,321,0.656027,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^4} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^4,x]","-\frac{\sqrt{\frac{b}{a}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a \left(c-2 d x^2\right)+b c x^2\right)+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 b c^2 x^3 \left(a+b x^2\right)}","-\frac{\sqrt{d} (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a c^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{d x (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 a c^2}-\frac{\left(c+d x^2\right) (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 a c^2 x}-\frac{b \sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a \sqrt{c} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 c x^3}",1,"-1/3*(Sqrt[b/a]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(b*c*x^2 + a*(c - 2*d*x^2)) - I*b*c*(-(b*c) + 2*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c^2*x^3*(a + b*x^2))","C",1
275,1,302,424,0.5732841,"\int \frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{x^6} \, dx","Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^6,x]","-\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-2 i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2-a b c d-b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(8 a^2 d^2-3 a b c d-2 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+\sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a^2 \left(3 c^2-4 c d x^2+8 d^2 x^4\right)+a b c x^2 \left(c-3 d x^2\right)-2 b^2 c^2 x^4\right)\right)}{15 a^2 c^3 x^5 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{\sqrt{d} \left(-8 a^2 d^2+3 a b c d+2 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 a^2 c^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{d x \left(-8 a^2 d^2+3 a b c d+2 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{15 a^2 c^3}+\frac{\left(c+d x^2\right) \left(-8 a^2 d^2+3 a b c d+2 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{15 a^2 c^3 x}-\frac{b \sqrt{d} (b c-4 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 a^2 c^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\left(c+d x^2\right) (b c-4 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{15 a c^2 x^3}-\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 c x^5}",1,"-1/15*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-2*b^2*c^2*x^4 + a*b*c*x^2*(c - 3*d*x^2) + a^2*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4)) + I*b*c*(-2*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*b*c*(-(b^2*c^2) - a*b*c*d + 2*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(a^2*Sqrt[b/a]*c^3*x^5*(a + b*x^2))","C",1
276,1,294,282,0.5405084,"\int x^5 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(b \sqrt{d} \sqrt{b c-a d} \left(3 a^3 d^2 \left(c+d x^2\right)+a^2 b d \left(-100 c^2-35 c d x^2+17 d^2 x^4\right)+a b^2 \left(105 c^3-65 c^2 d x^2-52 c d^2 x^4+22 d^3 x^6\right)+b^3 x^2 \left(105 c^3+35 c^2 d x^2-14 c d^2 x^4+8 d^3 x^6\right)\right)-3 \sqrt{a+b x^2} (b c-a d)^2 \left(-a^2 d^2-10 a b c d+35 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)\right)}{48 b^2 d^{9/2} \left(a+b x^2\right) \sqrt{b c-a d}}","\frac{e \left(c+d x^2\right) \left(-5 a^2 d^2-50 a b c d+79 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{48 b d^4}-\frac{e^{3/2} (b c-a d) \left(-a^2 d^2-10 a b c d+35 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{16 b^{3/2} d^{9/2}}+\frac{c^2 e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{d^4}-\frac{e \left(c+d x^2\right)^2 (a d+11 b c) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{24 d^4}+\frac{\left(c+d x^2\right)^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{5/2}}{6 b d^2 e}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(b*Sqrt[d]*Sqrt[b*c - a*d]*(3*a^3*d^2*(c + d*x^2) + a^2*b*d*(-100*c^2 - 35*c*d*x^2 + 17*d^2*x^4) + b^3*x^2*(105*c^3 + 35*c^2*d*x^2 - 14*c*d^2*x^4 + 8*d^3*x^6) + a*b^2*(105*c^3 - 65*c^2*d*x^2 - 52*c*d^2*x^4 + 22*d^3*x^6)) - 3*(b*c - a*d)^2*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(48*b^2*d^(9/2)*Sqrt[b*c - a*d]*(a + b*x^2))","A",1
277,1,191,199,0.6162734,"\int x^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^3*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(3 \sqrt{b c-a d} \left(a^2 d^2-6 a b c d+5 b^2 c^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)+b \sqrt{d} \sqrt{a+b x^2} \left(a d \left(13 c+5 d x^2\right)+b \left(-15 c^2-5 c d x^2+2 d^2 x^4\right)\right)\right)}{8 b d^{7/2} \sqrt{a+b x^2}}","\frac{3 e^{3/2} (b c-a d) (5 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{8 \sqrt{b} d^{7/2}}+\frac{b e \left(c+d x^2\right)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 d^3}-\frac{e \left(c+d x^2\right) (9 b c-5 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 d^3}-\frac{c e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{d^3}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(b*Sqrt[d]*Sqrt[a + b*x^2]*(a*d*(13*c + 5*d*x^2) + b*(-15*c^2 - 5*c*d*x^2 + 2*d^2*x^4)) + 3*Sqrt[b*c - a*d]*(5*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(8*b*d^(7/2)*Sqrt[a + b*x^2])","A",1
278,1,96,141,0.0591931,"\int x \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[x*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{e \left(a+b x^2\right)^2 \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{d \left(b x^2+a\right)}{a d-b c}\right)}{5 b c-5 a d}","-\frac{3 \sqrt{b} e^{3/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{2 d^{5/2}}+\frac{3 e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 d^2}+\frac{\left(c+d x^2\right) \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{2 d}",1,"(e*(a + b*x^2)^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*Hypergeometric2F1[3/2, 5/2, 7/2, (d*(a + b*x^2))/(-(b*c) + a*d)])/(5*b*c - 5*a*d)","C",1
279,1,193,151,1.277914,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x,x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\sqrt{d} \left(-\frac{a^{3/2} d \sqrt{c+d x^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{c^{3/2} \sqrt{a+b x^2}}+\frac{a d}{c}-b\right)+\frac{b \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{\sqrt{a+b x^2}}\right)}{d^{3/2}}","-\frac{a^{3/2} e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{c^{3/2}}+\frac{b^{3/2} e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{d^{3/2}}-\frac{e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*((b*Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/Sqrt[a + b*x^2] + Sqrt[d]*(-b + (a*d)/c - (a^(3/2)*d*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(c^(3/2)*Sqrt[a + b*x^2]))))/d^(3/2)","A",1
280,1,146,165,0.090834,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^3} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^3,x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(\sqrt{c} \sqrt{a+b x^2} \left(2 b c x^2-a \left(c+3 d x^2\right)\right)-3 \sqrt{a} x^2 \sqrt{c+d x^2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)\right)}{2 c^{5/2} x^2 \sqrt{a+b x^2}}","-\frac{3 \sqrt{a} e^{3/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{2 c^{5/2}}+\frac{3 e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 c^2}+\frac{(b c-a d) \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{2 c \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right)}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[c]*Sqrt[a + b*x^2]*(2*b*c*x^2 - a*(c + 3*d*x^2)) - 3*Sqrt[a]*(b*c - a*d)*x^2*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(2*c^(5/2)*x^2*Sqrt[a + b*x^2])","A",1
281,1,186,256,0.1286636,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^5} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^5,x]","-\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(3 x^4 \sqrt{c+d x^2} \left(5 a^2 d^2-6 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)+\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \left(a \left(2 c^2-5 c d x^2-15 d^2 x^4\right)+b c x^2 \left(5 c+13 d x^2\right)\right)\right)}{8 \sqrt{a} c^{7/2} x^4 \sqrt{a+b x^2}}","-\frac{3 e^{3/2} (b c-5 a d) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{8 \sqrt{a} c^{7/2}}-\frac{a e^3 (b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 c^3 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^2}+\frac{e^2 (5 b c-9 a d) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 c^3 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)}-\frac{d e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c^3}",1,"-1/8*(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*(b*c*x^2*(5*c + 13*d*x^2) + a*(2*c^2 - 5*c*d*x^2 - 15*d^2*x^4)) + 3*(b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*x^4*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(Sqrt[a]*c^(7/2)*x^4*Sqrt[a + b*x^2])","A",1
282,1,245,366,0.1911713,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^7} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(3 x^6 \sqrt{c+d x^2} \left(35 a^3 d^3-45 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)-\sqrt{a} \sqrt{c} \sqrt{a+b x^2} \left(a^2 \left(8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right)+2 a b c x^2 \left(7 c^2-19 c d x^2-50 d^2 x^4\right)+3 b^2 c^2 x^4 \left(c+d x^2\right)\right)\right)}{48 a^{3/2} c^{9/2} x^6 \sqrt{a+b x^2}}","-\frac{e^2 \left(-79 a^2 d^2+50 a b c d+5 b^2 c^2\right) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{48 a c^4 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)}+\frac{e^{3/2} \left(-35 a^2 d^2+10 a b c d+b^2 c^2\right) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{16 a^{3/2} c^{9/2}}+\frac{d^2 e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c^4}+\frac{e^3 (11 a d+b c) (b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{24 c^4 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^2}+\frac{e^2 (b c-a d)^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{5/2}}{6 a c^2 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^3}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*(3*b^2*c^2*x^4*(c + d*x^2) + 2*a*b*c*x^2*(7*c^2 - 19*c*d*x^2 - 50*d^2*x^4) + a^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6))) + 3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*x^6*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(48*a^(3/2)*c^(9/2)*x^6*Sqrt[a + b*x^2])","A",1
283,1,290,391,0.5970571,"\int x^4 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(8 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2-3 a b c d+2 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(a^2 d^2-16 a b c d+16 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a^2 d \left(7 c+2 d x^2\right)+a b \left(-8 c^2+5 c d x^2+3 d^2 x^4\right)+b^2 x^2 \left(-8 c^2-2 c d x^2+d^2 x^4\right)\right)\right)}{5 d^4 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","-\frac{\sqrt{c} e \left(a^2 d^2-16 a b c d+16 b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 b d^{7/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e x \left(-\frac{a^2 d}{b}+16 a c-\frac{16 b c^2}{d}\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 d^2}+\frac{c^{3/2} e (8 b c-7 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 d^{7/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e x \left(c+d x^2\right) (8 b c-7 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 d^3}+\frac{6 b e x^3 \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 d^2}-\frac{e x^3 \left(a+b x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{d}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a^2*d*(7*c + 2*d*x^2) + b^2*x^2*(-8*c^2 - 2*c*d*x^2 + d^2*x^4) + a*b*(-8*c^2 + 5*c*d*x^2 + 3*d^2*x^4)) - I*c*(16*b^2*c^2 - 16*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (8*I)*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(5*Sqrt[b/a]*d^4*(a + b*x^2))","C",1
284,1,235,310,0.4176693,"\int x^2 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","-\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-11 a b c d+8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(3 a d-b \left(4 c+d x^2\right)\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (7 a d-8 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 d^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","-\frac{\sqrt{c} e (4 b c-3 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 d^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{\sqrt{c} e (8 b c-7 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 d^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{4 b e x \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 d^2}-\frac{e x (8 b c-7 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 d^2}-\frac{e x \left(a+b x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{d}",1,"-1/3*(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a + b*x^2)*(3*a*d - b*(4*c + d*x^2)) + I*b*c*(-8*b*c + 7*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(8*b^2*c^2 - 11*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*d^3*(a + b*x^2))","C",1
285,1,206,262,0.3170374,"\int \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left((a d-b c) \left(d x \sqrt{\frac{b}{a}} \left(a+b x^2\right)-2 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{c d^2 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{b \sqrt{c} e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{d^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e (2 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} d^{3/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e x (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d}+\frac{e x (2 b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(I*b*c*(-2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-(b*c) + a*d)*(Sqrt[b/a]*d*x*(a + b*x^2) - (2*I)*b*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(Sqrt[b/a]*c*d^2*(a + b*x^2))","C",1
286,1,228,307,0.3576679,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^2} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^2,x]","-\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(d \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(a c+2 a d x^2-b c x^2\right)-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{c^2 d x \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{e (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{c^{3/2} \sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}+\frac{e \left(c+d x^2\right) (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c^2 d x}-\frac{e x (b c-2 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c^2}-\frac{e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d x}+\frac{b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}",1,"-((e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*(a + b*x^2)*(a*c - b*c*x^2 + 2*a*d*x^2) + I*b*c*(-(b*c) + 2*a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-(b*c) + a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*c^2*d*x*(a + b*x^2)))","C",1
287,1,275,383,0.4669499,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^4} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4,x]","\frac{e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-\sqrt{\frac{b}{a}} \left(a^2 \left(c^2-4 c d x^2-8 d^2 x^4\right)+a b x^2 \left(5 c^2+3 c d x^2-8 d^2 x^4\right)+b^2 c x^4 \left(4 c+7 d x^2\right)\right)-4 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (8 a d-7 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 c^3 x^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{b e (3 b c-4 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a c^{3/2} \sqrt{d} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{\sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 c^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e \left(c+d x^2\right) (7 b c-8 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 c^3 x}+\frac{d e x (7 b c-8 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 c^3}+\frac{e \left(c+d x^2\right) (3 b c-4 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{3 c^2 d x^3}-\frac{e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d x^3}",1,"(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[b/a]*(b^2*c*x^4*(4*c + 7*d*x^2) + a^2*(c^2 - 4*c*d*x^2 - 8*d^2*x^4) + a*b*x^2*(5*c^2 + 3*c*d*x^2 - 8*d^2*x^4))) + I*b*c*(-7*b*c + 8*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (4*I)*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*Sqrt[b/a]*c^3*x^3*(a + b*x^2))","C",1
288,1,357,480,0.668229,"\int \frac{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}{x^6} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^6,x]","-\frac{e \sqrt{\frac{b}{a}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(8 a^2 d^2-9 a b c d+b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+\sqrt{\frac{b}{a}} \left(a^3 \left(c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right)+a^2 b x^2 \left(3 c^3-11 c^2 d x^2-8 c d^2 x^4+16 d^3 x^6\right)+a b^2 c x^4 \left(3 c^2-8 c d x^2-16 d^2 x^4\right)+b^3 c^2 x^6 \left(c+d x^2\right)\right)\right)}{5 b c^4 x^5 \left(a+b x^2\right)}","-\frac{\sqrt{d} e \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 a c^{7/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e \left(c+d x^2\right) \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 a c^4 x}+\frac{d e x \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 a c^4}-\frac{b \sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 a c^{5/2} \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}}}-\frac{e \left(c+d x^2\right) (7 b c-8 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 c^3 x^3}+\frac{e \left(c+d x^2\right) (5 b c-6 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{5 c^2 d x^5}-\frac{e (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{c d x^5}",1,"-1/5*(Sqrt[b/a]*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(b^3*c^2*x^6*(c + d*x^2) + a*b^2*c*x^4*(3*c^2 - 8*c*d*x^2 - 16*d^2*x^4) + a^2*b*x^2*(3*c^3 - 11*c^2*d*x^2 - 8*c*d^2*x^4 + 16*d^3*x^6) + a^3*(c^3 - 2*c^2*d*x^2 + 8*c*d^2*x^4 + 16*d^3*x^6)) + I*b*c*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c^4*x^5*(a + b*x^2))","C",1
289,1,86,51,0.0320651,"\int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx","Integrate[x*Sqrt[(1 - x^2)/(1 + x^2)],x]","\frac{\sqrt{\frac{1-x^2}{x^2+1}} \sqrt{x^2+1} \left(\sqrt{x^2+1} \left(x^2-1\right)+2 \sqrt{1-x^2} \sin ^{-1}\left(\frac{\sqrt{1-x^2}}{\sqrt{2}}\right)\right)}{2 \left(x^2-1\right)}","\frac{1}{2} \sqrt{\frac{1-x^2}{x^2+1}} \left(x^2+1\right)-\tan ^{-1}\left(\sqrt{\frac{1-x^2}{x^2+1}}\right)",1,"(Sqrt[(1 - x^2)/(1 + x^2)]*Sqrt[1 + x^2]*((-1 + x^2)*Sqrt[1 + x^2] + 2*Sqrt[1 - x^2]*ArcSin[Sqrt[1 - x^2]/Sqrt[2]]))/(2*(-1 + x^2))","A",1
290,1,104,72,0.048102,"\int x \sqrt{\frac{5-7 x^2}{7+5 x^2}} \, dx","Integrate[x*Sqrt[(5 - 7*x^2)/(7 + 5*x^2)],x]","\frac{\sqrt{\frac{5-7 x^2}{5 x^2+7}} \sqrt{5 x^2+7} \left(35 \sqrt{5 x^2+7} \left(7 x^2-5\right)-74 \sqrt{35} \sqrt{7 x^2-5} \sinh ^{-1}\left(\sqrt{\frac{5}{74}} \sqrt{7 x^2-5}\right)\right)}{350 \left(7 x^2-5\right)}","\frac{1}{10} \sqrt{\frac{5-7 x^2}{5 x^2+7}} \left(5 x^2+7\right)-\frac{37 \tan ^{-1}\left(\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^2}{5 x^2+7}}\right)}{5 \sqrt{35}}",1,"(Sqrt[(5 - 7*x^2)/(7 + 5*x^2)]*Sqrt[7 + 5*x^2]*(35*Sqrt[7 + 5*x^2]*(-5 + 7*x^2) - 74*Sqrt[35]*Sqrt[-5 + 7*x^2]*ArcSinh[Sqrt[5/74]*Sqrt[-5 + 7*x^2]]))/(350*(-5 + 7*x^2))","A",1
291,1,86,53,0.0305335,"\int x^2 \sqrt{\frac{1-x^3}{1+x^3}} \, dx","Integrate[x^2*Sqrt[(1 - x^3)/(1 + x^3)],x]","\frac{\sqrt{\frac{1-x^3}{x^3+1}} \sqrt{x^3+1} \left(\sqrt{x^3+1} \left(x^3-1\right)+2 \sqrt{1-x^3} \sin ^{-1}\left(\frac{\sqrt{1-x^3}}{\sqrt{2}}\right)\right)}{3 \left(x^3-1\right)}","\frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left(x^3+1\right)-\frac{2}{3} \tan ^{-1}\left(\sqrt{\frac{1-x^3}{x^3+1}}\right)",1,"(Sqrt[(1 - x^3)/(1 + x^3)]*Sqrt[1 + x^3]*((-1 + x^3)*Sqrt[1 + x^3] + 2*Sqrt[1 - x^3]*ArcSin[Sqrt[1 - x^3]/Sqrt[2]]))/(3*(-1 + x^3))","A",1
292,1,98,113,0.0356362,"\int x^8 \sqrt{\frac{1-x^3}{1+x^3}} \, dx","Integrate[x^8*Sqrt[(1 - x^3)/(1 + x^3)],x]","\frac{\sqrt{\frac{1-x^3}{x^3+1}} \sqrt{x^3+1} \left(6 \sqrt{1-x^3} \sin ^{-1}\left(\frac{\sqrt{1-x^3}}{\sqrt{2}}\right)+\sqrt{x^3+1} \left(2 x^9-5 x^6+7 x^3-4\right)\right)}{18 \left(x^3-1\right)}","-\frac{1}{9} \left(\frac{1-x^3}{x^3+1}\right)^{3/2} \left(x^3+1\right)^3-\frac{1}{6} \sqrt{\frac{1-x^3}{x^3+1}} \left(x^3+1\right)^2+\frac{1}{2} \sqrt{\frac{1-x^3}{x^3+1}} \left(x^3+1\right)-\frac{1}{3} \tan ^{-1}\left(\sqrt{\frac{1-x^3}{x^3+1}}\right)",1,"(Sqrt[(1 - x^3)/(1 + x^3)]*Sqrt[1 + x^3]*(Sqrt[1 + x^3]*(-4 + 7*x^3 - 5*x^6 + 2*x^9) + 6*Sqrt[1 - x^3]*ArcSin[Sqrt[1 - x^3]/Sqrt[2]]))/(18*(-1 + x^3))","A",1
293,1,109,106,0.0499167,"\int x^9 \sqrt{\frac{5-7 x^5}{7+5 x^5}} \, dx","Integrate[x^9*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)],x]","\frac{\sqrt{\frac{5-7 x^5}{5 x^5+7}} \sqrt{5 x^5+7} \left(4514 \sqrt{35} \sqrt{7 x^5-5} \sinh ^{-1}\left(\sqrt{\frac{5}{74}} \sqrt{7 x^5-5}\right)+35 \sqrt{5 x^5+7} \left(245 x^{10}-777 x^5+430\right)\right)}{61250 \left(7 x^5-5\right)}","\frac{1}{250} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left(5 x^5+7\right)^2-\frac{27}{350} \sqrt{\frac{5-7 x^5}{5 x^5+7}} \left(5 x^5+7\right)+\frac{2257 \tan ^{-1}\left(\sqrt{\frac{5}{7}} \sqrt{\frac{5-7 x^5}{5 x^5+7}}\right)}{875 \sqrt{35}}",1,"(Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*Sqrt[7 + 5*x^5]*(35*Sqrt[7 + 5*x^5]*(430 - 777*x^5 + 245*x^10) + 4514*Sqrt[35]*Sqrt[-5 + 7*x^5]*ArcSinh[Sqrt[5/74]*Sqrt[-5 + 7*x^5]]))/(61250*(-5 + 7*x^5))","A",1
294,1,49,52,0.0169463,"\int \frac{\sqrt{\frac{x^2}{-1+x^2}}}{1+x^2} \, dx","Integrate[Sqrt[x^2/(-1 + x^2)]/(1 + x^2),x]","\frac{\sqrt{\frac{x^2}{x^2-1}} \sqrt{x^2-1} \tan ^{-1}\left(\frac{\sqrt{x^2-1}}{\sqrt{2}}\right)}{\sqrt{2} x}","\frac{\sqrt{-\frac{x^2}{1-x^2}} \sqrt{x^2-1} \tan ^{-1}\left(\frac{\sqrt{x^2-1}}{\sqrt{2}}\right)}{\sqrt{2} x}",1,"(Sqrt[x^2/(-1 + x^2)]*Sqrt[-1 + x^2]*ArcTan[Sqrt[-1 + x^2]/Sqrt[2]])/(Sqrt[2]*x)","A",1
295,1,65,68,0.0249522,"\int \frac{\sqrt{\frac{x^2}{-1+a+(1+a) x^2}}}{1+x^2} \, dx","Integrate[Sqrt[x^2/(-1 + a + (1 + a)*x^2)]/(1 + x^2),x]","\frac{\sqrt{a x^2+a+x^2-1} \sqrt{\frac{x^2}{(a+1) x^2+a-1}} \tan ^{-1}\left(\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right)}{\sqrt{2} x}","\frac{\sqrt{-\frac{x^2}{-\left((a+1) x^2\right)-a+1}} \sqrt{(a+1) x^2+a-1} \tan ^{-1}\left(\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right)}{\sqrt{2} x}",1,"(Sqrt[-1 + a + x^2 + a*x^2]*Sqrt[x^2/(-1 + a + (1 + a)*x^2)]*ArcTan[Sqrt[-1 + a + (1 + a)*x^2]/Sqrt[2]])/(Sqrt[2]*x)","A",1
296,1,224,281,0.3863656,"\int \frac{x^5}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[x^5/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{a+b x^2} \left(3 \sqrt{b c-a d} \left(5 a^2 d^2+2 a b c d+b^2 c^2\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)+\sqrt{d} \sqrt{a+b x^2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \left(15 a^2 d^2-2 a b d \left(2 c+5 d x^2\right)+b^2 \left(-3 c^2+2 c d x^2+8 d^2 x^4\right)\right)\right)}{48 b^3 d^{5/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{(b c-a d) \left(5 a^2 d^2+2 a b c d+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{16 b^{7/2} d^{5/2} \sqrt{e}}+\frac{\left(c+d x^2\right) \left(5 a^2 d^2+2 a b c d+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{16 b^3 d^2 e}-\frac{\left(c+d x^2\right)^2 (5 a d+3 b c) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{24 b^2 d^2 e}-\frac{\left(c+d x^2\right)^3 \left(a-\frac{c \left(a+b x^2\right)}{c+d x^2}\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{6 b d e (b c-a d)}",1,"(Sqrt[a + b*x^2]*(Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(15*a^2*d^2 - 2*a*b*d*(2*c + 5*d*x^2) + b^2*(-3*c^2 + 2*c*d*x^2 + 8*d^2*x^4)) + 3*Sqrt[b*c - a*d]*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(48*b^3*d^(5/2)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
297,1,172,169,0.3416757,"\int \frac{x^3}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[x^3/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{d} \left(a+b x^2\right) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \left(b \left(c+2 d x^2\right)-3 a d\right)-\sqrt{a+b x^2} \sqrt{b c-a d} (3 a d+b c) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{8 b^2 d^{3/2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{8 b^{5/2} d^{3/2} \sqrt{e}}-\frac{\left(c+d x^2\right) (3 a d+b c) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 b^2 d e}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 b d e}",1,"(Sqrt[d]*(a + b*x^2)*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(-3*a*d + b*(c + 2*d*x^2)) - Sqrt[b*c - a*d]*(b*c + 3*a*d)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(8*b^2*d^(3/2)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
298,1,152,106,0.1319746,"\int \frac{x}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[x/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{a+b x^2} \left(\sqrt{d} \sqrt{a+b x^2} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}}+\sqrt{b c-a d} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)\right)}{2 b \sqrt{d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{2 b^{3/2} \sqrt{d} \sqrt{e}}+\frac{\left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 b e}",1,"(Sqrt[a + b*x^2]*(Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)] + Sqrt[b*c - a*d]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]))/(2*b*Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
299,1,190,112,0.2354421,"\int \frac{1}{x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/(x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]","\frac{\sqrt{a+b x^2} \left(\sqrt{a} \sqrt{d} \sqrt{c+d x^2} \sqrt{b c-a d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-b \sqrt{c} \left(c+d x^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)\right)}{\sqrt{a} b \left(c+d x^2\right)^{3/2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{\sqrt{b} \sqrt{e}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{\sqrt{a} \sqrt{e}}",1,"(Sqrt[a + b*x^2]*(Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[c + d*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]] - b*Sqrt[c]*(c + d*x^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(Sqrt[a]*b*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)^(3/2))","A",1
300,1,133,130,0.1233032,"\int \frac{1}{x^3 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/(x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]","\frac{\sqrt{a+b x^2} \left(-\frac{(a d-b c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{a x^2}\right)}{2 \sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{2 a^{3/2} \sqrt{c} \sqrt{e}}+\frac{(b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{2 a \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)}",1,"(Sqrt[a + b*x^2]*(-((Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*x^2)) - ((-(b*c) + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(3/2)*Sqrt[c])))/(2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])","A",1
301,1,173,218,0.1587669,"\int \frac{1}{x^5 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/(x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]","\frac{\sqrt{a} \sqrt{c} \left(a+b x^2\right) \sqrt{c+d x^2} \left(3 b c x^2-a \left(2 c+d x^2\right)\right)-x^4 \sqrt{a+b x^2} \left(-a^2 d^2-2 a b c d+3 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{5/2} c^{3/2} x^4 \sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{(a d+3 b c) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{8 a^{5/2} c^{3/2} \sqrt{e}}-\frac{(a d+3 b c) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 a^2 c \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)}-\frac{e (b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 a c \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^2}",1,"(Sqrt[a]*Sqrt[c]*(a + b*x^2)*Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(2*c + d*x^2)) - (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*x^4*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(3/2)*x^4*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])","A",1
302,1,258,403,0.4803406,"\int \frac{x^4}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[x^4/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(2 a^2 d^2-a b c d-b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(8 a^2 d^2-3 a b c d-2 b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \left(-\sqrt{\frac{b}{a}}\right) \left(a+b x^2\right) \left(c+d x^2\right) \left(4 a d-b \left(c+3 d x^2\right)\right)}{15 a^2 d^2 \left(\frac{b}{a}\right)^{5/2} \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{x \left(a+b x^2\right) \left(-8 a^2 d^2+3 a b c d+2 b^2 c^2\right)}{15 b^3 d \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\sqrt{c} \left(a+b x^2\right) \left(-8 a^2 d^2+3 a b c d+2 b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^3 d^{3/2} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{c^{3/2} \left(a+b x^2\right) (b c-4 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{15 b^2 d^{3/2} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{x \left(a+b x^2\right) (b c-4 a d)}{15 b^2 d \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{x^3 \left(a+b x^2\right)}{5 b \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*d - b*(c + 3*d*x^2))) - I*c*(-2*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*c*(-(b^2*c^2) - a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))","C",1
303,1,212,312,0.2796097,"\int \frac{x^2}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[x^2/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{d x \sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 b d \sqrt{\frac{b}{a}} \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{x \left(a+b x^2\right) (b c-2 a d)}{3 b^2 \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \left(a+b x^2\right) (b c-2 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b^2 \sqrt{d} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{c^{3/2} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b \sqrt{d} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{x \left(a+b x^2\right)}{3 b \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + I*c*(-(b*c) + 2*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*b*Sqrt[b/a]*d*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))","C",1
304,1,86,252,0.0563874,"\int \frac{1}{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]","\frac{\sqrt{\frac{a+b x^2}{a}} E\left(\sin ^{-1}\left(\sqrt{-\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{\sqrt{-\frac{b}{a}} \sqrt{\frac{c+d x^2}{c}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{c^{3/2} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \sqrt{d} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{d x \left(a+b x^2\right)}{b \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{b \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(a + b*x^2)/a]*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(Sqrt[-(b/a)]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(c + d*x^2)/c])","A",1
305,1,111,289,0.2616241,"\int \frac{1}{x^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/(x^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]","\frac{\left(a+b x^2\right) \left(\frac{d \sqrt{\frac{d x^2}{c}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{b c}{a d}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{b x^2}{a}+1} \left(c+d x^2\right)}-\frac{1}{x}\right)}{a \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{a+b x^2}{a x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{d x \left(a+b x^2\right)}{a \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"((a + b*x^2)*(-x^(-1) + (d*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[1 + (b*x^2)/a]*(c + d*x^2))))/(a*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])","A",1
306,1,238,375,0.4063244,"\int \frac{1}{x^4 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}} \, dx","Integrate[1/(x^4*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]","\frac{-\sqrt{\frac{b}{a}} \left(a+b x^2\right) \left(c+d x^2\right) \left(a \left(c+d x^2\right)-2 b c x^2\right)+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)}{3 a^2 c x^3 \sqrt{\frac{b}{a}} \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{\left(a+b x^2\right) (2 b c-a d)}{3 a^2 c x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{d x \left(a+b x^2\right) (2 b c-a d)}{3 a^2 c \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{b \sqrt{c} \sqrt{d} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^2 \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\sqrt{d} \left(a+b x^2\right) (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^2 \sqrt{c} \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{-a-b x^2}{3 a x^3 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-2*b*c*x^2 + a*(c + d*x^2))) - I*b*c*(-2*b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*c*x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))","C",1
307,1,247,354,0.4751776,"\int \frac{x^5}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{\sqrt{d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \left(105 a^3 d^2+5 a^2 b d \left(7 d x^2-20 c\right)+a b^2 \left(3 c^2-38 c d x^2-14 d^2 x^4\right)+b^3 x^2 \left(3 c^2+14 c d x^2+8 d^2 x^4\right)\right)-3 \sqrt{a+b x^2} \sqrt{b c-a d} \left(-35 a^2 d^2+10 a b c d+b^2 c^2\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{48 b^4 d^{3/2} e \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{\left(c+d x^2\right)^3 \left(7 a^2 d^2-2 a b c d+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{6 b^2 d e^2 (b c-a d)^2}-\frac{a^2 \left(c+d x^2\right)^3}{b e (b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{(b c-a d) \left(5 a d (2 b c-7 a d)+b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{16 b^{9/2} d^{3/2} e^{3/2}}-\frac{\left(c+d x^2\right) \left(5 a d (2 b c-7 a d)+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{16 b^4 d e^2}-\frac{\left(c+d x^2\right)^2 \left(5 a d (2 b c-7 a d)+b^2 c^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{24 b^3 d e^2 (b c-a d)}",1,"(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x^2) + a*b^2*(3*c^2 - 38*c*d*x^2 - 14*d^2*x^4) + b^3*x^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4)) - 3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^4*d^(3/2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
308,1,190,202,0.316521,"\int \frac{x^3}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^3/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{\sqrt{d} \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \left(-15 a^2 d+a b \left(13 c-5 d x^2\right)+b^2 x^2 \left(5 c+2 d x^2\right)\right)+3 \sqrt{a+b x^2} (b c-5 a d) \sqrt{b c-a d} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)}{8 b^3 \sqrt{d} e \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{8 b^{7/2} \sqrt{d} e^{3/2}}+\frac{\left(c+d x^2\right) (3 b c-7 a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 b^3 e^2}+\frac{a (b c-a d)}{b^3 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 b^2 e^2}",1,"(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(-15*a^2*d + a*b*(13*c - 5*d*x^2) + b^2*x^2*(5*c + 2*d*x^2)) + 3*(b*c - 5*a*d)*Sqrt[b*c - a*d]*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(8*b^3*Sqrt[d]*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])","A",1
309,1,86,146,0.0674031,"\int \frac{x}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","-\frac{\left(a+b x^2\right) \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{d \left(b x^2+a\right)}{a d-b c}\right)}{b \left(\frac{b \left(c+d x^2\right)}{b c-a d}\right)^{3/2} \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}}","\frac{3 \sqrt{d} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{2 b^{5/2} e^{3/2}}-\frac{3 (b c-a d)}{2 b^2 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{c+d x^2}{2 b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"-(((a + b*x^2)*Hypergeometric2F1[-3/2, -1/2, 1/2, (d*(a + b*x^2))/(-(b*c) + a*d)])/(b*((e*(a + b*x^2))/(c + d*x^2))^(3/2)*((b*(c + d*x^2))/(b*c - a*d))^(3/2)))","C",1
310,1,253,152,0.3488973,"\int \frac{1}{x \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]","\frac{-a^{3/2} d^{3/2} \sqrt{a+b x^2} \sqrt{c+d x^2} (a d-b c) \sqrt{\frac{b \left(c+d x^2\right)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right)-b \left(c+d x^2\right) \sqrt{b c-a d} \left(b c^{3/2} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)+\sqrt{a} \sqrt{c+d x^2} (a d-b c)\right)}{a^{3/2} b^2 e \left(c+d x^2\right)^{3/2} \sqrt{b c-a d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{a^{3/2} e^{3/2}}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{b} \sqrt{e}}\right)}{b^{3/2} e^{3/2}}+\frac{b c-a d}{a b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(-(a^(3/2)*d^(3/2)*(-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]]) - b*Sqrt[b*c - a*d]*(c + d*x^2)*(Sqrt[a]*(-(b*c) + a*d)*Sqrt[c + d*x^2] + b*c^(3/2)*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(a^(3/2)*b^2*Sqrt[b*c - a*d]*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)^(3/2))","A",1
311,1,148,170,0.0865684,"\int \frac{1}{x^3 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^3*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]","\frac{3 \sqrt{c} x^2 \sqrt{a+b x^2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)-\sqrt{a} \sqrt{c+d x^2} \left(a \left(c-2 d x^2\right)+3 b c x^2\right)}{2 a^{5/2} e x^2 \sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","\frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{2 a^{5/2} e^{3/2}}-\frac{3 (b c-a d)}{2 a^2 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{b c-a d}{2 a \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)}",1,"(-(Sqrt[a]*Sqrt[c + d*x^2]*(3*b*c*x^2 + a*(c - 2*d*x^2))) + 3*Sqrt[c]*(b*c - a*d)*x^2*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(5/2)*e*x^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])","A",1
312,1,189,255,0.1203775,"\int \frac{1}{x^5 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]","\frac{\sqrt{a} \sqrt{c} \sqrt{c+d x^2} \left(-a^2 \left(2 c+5 d x^2\right)+a b x^2 \left(5 c-13 d x^2\right)+15 b^2 c x^4\right)-3 x^4 \sqrt{a+b x^2} \left(a^2 d^2-6 a b c d+5 b^2 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{8 a^{7/2} \sqrt{c} e x^4 \sqrt{c+d x^2} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}","-\frac{3 (5 b c-a d) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right)}{8 a^{7/2} \sqrt{c} e^{3/2}}-\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{8 a^3 \left(a e^2-\frac{c e^2 \left(a+b x^2\right)}{c+d x^2}\right)}+\frac{b (b c-a d)}{a^3 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}{4 a^2 \left(a e-\frac{c e \left(a+b x^2\right)}{c+d x^2}\right)^2}",1,"(Sqrt[a]*Sqrt[c]*Sqrt[c + d*x^2]*(15*b^2*c*x^4 + a*b*x^2*(5*c - 13*d*x^2) - a^2*(2*c + 5*d*x^2)) - 3*(5*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x^4*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(7/2)*Sqrt[c]*e*x^4*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])","A",1
313,1,271,453,0.5015019,"\int \frac{x^4}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^4/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(8 a^2 d^2-9 a b c d+b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+d x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(-8 a^2 d+a b \left(7 c-2 d x^2\right)+b^2 x^2 \left(2 c+d x^2\right)\right)\right)}{5 b^3 d e^2 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{x \left(a+b x^2\right) \left(16 a^2 d^2-16 a b c d+b^2 c^2\right)}{5 b^4 e \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \left(a+b x^2\right) \left(16 a^2 d^2-16 a b c d+b^2 c^2\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 b^4 \sqrt{d} e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{c^{3/2} \left(a+b x^2\right) (7 b c-8 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{5 b^3 \sqrt{d} e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{x \left(a+b x^2\right) (7 b c-8 a d)}{5 b^3 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{6 d x^3 \left(a+b x^2\right)}{5 b^2 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{x^3 \left(c+d x^2\right)}{b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(c + d*x^2)*(-8*a^2*d + a*b*(7*c - 2*d*x^2) + b^2*x^2*(2*c + d*x^2)) - I*c*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(5*b^3*Sqrt[b/a]*d*e^2*(a + b*x^2))","C",1
314,1,219,378,0.410053,"\int \frac{x^2}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^2/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(x \sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(4 a d-3 b c+b d x^2\right)-4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (8 a d-7 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 a^2 e^2 \left(\frac{b}{a}\right)^{5/2} \left(a+b x^2\right)}","\frac{d x \left(a+b x^2\right) (7 b c-8 a d)}{3 b^3 e \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) (7 b c-8 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 b^3 e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{c^{3/2} \left(a+b x^2\right) (3 b c-4 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a b^2 \sqrt{d} e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{4 d x \left(a+b x^2\right)}{3 b^2 e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{x \left(c+d x^2\right)}{b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*x*(c + d*x^2)*(-3*b*c + 4*a*d + b*d*x^2) + I*c*(-7*b*c + 8*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (4*I)*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^2*(b/a)^(5/2)*e^2*(a + b*x^2))","C",1
315,1,203,327,0.4719275,"\int \frac{1}{\left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[((e*(a + b*x^2))/(c + d*x^2))^(-3/2),x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left((b c-a d) \left(x \sqrt{\frac{b}{a}} \left(c+d x^2\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (2 a d-b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{a^2 e^2 \left(\frac{b}{a}\right)^{3/2} \left(a+b x^2\right)}","-\frac{d x \left(a+b x^2\right) (b c-2 a d)}{a b^2 e \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) (b c-2 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a b^2 e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{c^{3/2} \sqrt{d} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a b e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{x (b c-a d)}{a b e \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*((-I)*c*(-(b*c) + 2*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b*c - a*d)*(Sqrt[b/a]*x*(c + d*x^2) - I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(a^2*(b/a)^(3/2)*e^2*(a + b*x^2))","C",1
316,1,223,380,0.3693742,"\int \frac{1}{x^2 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-\sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(a c-a d x^2+2 b c x^2\right)-2 i c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)+i c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{a^2 e^2 x \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{c^{3/2} \sqrt{d} \left(a+b x^2\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a^2 e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\left(a+b x^2\right) (2 b c-a d)}{a^2 b e x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{d x \left(a+b x^2\right) (2 b c-a d)}{a^2 b e \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) (2 b c-a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{a^2 b e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{b c-a d}{a b e x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[b/a]*(c + d*x^2)*(a*c + 2*b*c*x^2 - a*d*x^2)) + I*c*(-2*b*c + a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*c*(-(b*c) + a*d)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(a^2*Sqrt[b/a]*e^2*x*(a + b*x^2))","C",1
317,1,266,444,0.4792767,"\int \frac{1}{x^4 \left(\frac{e \left(a+b x^2\right)}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]","\frac{\sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}} \left(-i x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left(3 a^2 d^2-11 a b c d+8 b^2 c^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)-\sqrt{\frac{b}{a}} \left(c+d x^2\right) \left(a^2 \left(c+4 d x^2\right)+a b \left(7 d x^4-4 c x^2\right)-8 b^2 c x^4\right)-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (7 a d-8 b c) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b}{a}} x\right)|\frac{a d}{b c}\right)\right)}{3 a^3 e^2 x^3 \sqrt{\frac{b}{a}} \left(a+b x^2\right)}","\frac{\left(a+b x^2\right) (8 b c-7 a d)}{3 a^3 e x \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{d x \left(a+b x^2\right) (8 b c-7 a d)}{3 a^3 e \left(c+d x^2\right) \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) (4 b c-3 a d) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^3 e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left(a+b x^2\right) (8 b c-7 a d) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|1-\frac{b c}{a d}\right)}{3 a^3 e \left(c+d x^2\right) \sqrt{\frac{c \left(a+b x^2\right)}{a \left(c+d x^2\right)}} \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}-\frac{\left(a+b x^2\right) (4 b c-3 a d)}{3 a^2 b e x^3 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}+\frac{b c-a d}{a b e x^3 \sqrt{\frac{e \left(a+b x^2\right)}{c+d x^2}}}",1,"(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[b/a]*(c + d*x^2)*(-8*b^2*c*x^4 + a^2*(c + 4*d*x^2) + a*b*(-4*c*x^2 + 7*d*x^4))) - I*b*c*(-8*b*c + 7*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(8*b^2*c^2 - 11*a*b*c*d + 3*a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^3*Sqrt[b/a]*e^2*x^3*(a + b*x^2))","C",1
318,1,137,216,0.3186964,"\int x^5 \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[x^5*Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{a} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(8 a^2 \left(c^2-c d x^2+d^2 x^4\right)+2 a b \left(d x^2-5 c\right)-3 b^2\right)+3 b \left(8 a^2 c^2+4 a b c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{48 a^{5/2} d^3}","-\frac{\left(-8 a^2 c^2+4 a b c+b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^2 d^3}+\frac{b \left(8 a^2 c^2+4 a b c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{16 a^{5/2} d^3}+\frac{\left(c+d x^2\right)^3 \left(\frac{a c+a d x^2+b}{c+d x^2}\right)^{3/2}}{6 a d^3}-\frac{(4 a c+b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 a d^3}",1,"(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-3*b^2 + 2*a*b*(-5*c + d*x^2) + 8*a^2*(c^2 - c*d*x^2 + d^2*x^4)) + 3*b*(b^2 + 4*a*b*c + 8*a^2*c^2)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(48*a^(5/2)*d^3)","A",1
319,1,97,141,0.1799984,"\int x^3 \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[x^3*Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{a} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 a c+2 a d x^2+b\right)-b (4 a c+b) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{8 a^{3/2} d^2}","-\frac{b (4 a c+b) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{8 a^{3/2} d^2}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 d^2}+\frac{(b-4 a c) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 a d^2}",1,"(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b - 2*a*c + 2*a*d*x^2) - b*(b + 4*a*c)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(8*a^(3/2)*d^2)","A",1
320,1,77,69,0.0886724,"\int x \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[x*Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{a} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}+b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 \sqrt{a} d}","\frac{\left(c+d x^2\right) \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 \sqrt{a} d}",1,"(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)] + b*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(2*Sqrt[a]*d)","A",1
321,1,80,96,0.1356912,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x,x]","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)-\frac{\sqrt{a c+b} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{\sqrt{c}}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)-\frac{\sqrt{a c+b} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{\sqrt{c}}",1,"Sqrt[a]*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]] - (Sqrt[b + a*c]*ArcTanh[(Sqrt[c]*Sqrt[a + b/(c + d*x^2)])/Sqrt[b + a*c]])/Sqrt[c]","A",1
322,1,212,104,0.4690794,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^3} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^3,x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(a c+a d x^2+b\right)-2 b d x^2 \log (x) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+b d x^2 \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{4 c x^2 \sqrt{c (a c+b)} \left(a \left(c+d x^2\right)+b\right)}","\frac{b d \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{2 c^{3/2} \sqrt{a c+b}}-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{2 c x^2}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b + a*c + a*d*x^2) - 2*b*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + b*d*x^2*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(4*c*Sqrt[c*(b + a*c)]*x^2*(b + a*(c + d*x^2)))","B",1
323,1,278,174,0.4824586,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^5} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^5,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(2 a^2 c \left(c^2-d^2 x^4\right)+a b \left(4 c^2-3 c d x^2-3 d^2 x^4\right)+b^2 \left(2 c-3 d x^2\right)\right)-2 b d^2 x^4 \log (x) (4 a c+3 b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+b d^2 x^4 (4 a c+3 b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{16 c x^4 (c (a c+b))^{3/2} \left(a \left(c+d x^2\right)+b\right)}","-\frac{b d^2 (4 a c+3 b) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{8 c^{5/2} (a c+b)^{3/2}}+\frac{d (4 a c+5 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 c^2 x^4}",1,"-1/16*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b^2*(2*c - 3*d*x^2) + a*b*(4*c^2 - 3*c*d*x^2 - 3*d^2*x^4) + 2*a^2*c*(c^2 - d^2*x^4)) - 2*b*(3*b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + b*(3*b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(c*(c*(b + a*c))^(3/2)*x^4*(b + a*(c + d*x^2)))","A",1
324,1,245,265,0.5005561,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^7} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^7,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\frac{3 b d^3 \left(8 a^2 c^2+12 a b c+5 b^2\right) \left(c+d x^2\right) \left(2 \log (x)-\log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{\sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}}+2 d^3 \left(8 a^2 c^2+26 a b c+15 b^2\right)+\frac{16 c^3 (a c+b)^2}{x^6}-\frac{4 b c^2 d (a c+b)}{x^4}+\frac{2 b c d^2 (8 a c+5 b)}{x^2}\right)}{96 c^3 (a c+b)^2}","\frac{b d^3 \left(8 a^2 c^2+12 a b c+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{16 c^{7/2} (a c+b)^{5/2}}-\frac{d^2 \left(8 a^2 c^2+20 a b c+11 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 c^3 x^2 (a c+b)^2}+\frac{d (4 a c+3 b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c^3 x^4 (a c+b)}-\frac{\left(c+d x^2\right)^3 \left(\frac{a c+a d x^2+b}{c+d x^2}\right)^{3/2}}{6 c^2 x^6 (a c+b)}",1,"-1/96*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*(15*b^2 + 26*a*b*c + 8*a^2*c^2)*d^3 + (16*c^3*(b + a*c)^2)/x^6 - (4*b*c^2*(b + a*c)*d)/x^4 + (2*b*c*(5*b + 8*a*c)*d^2)/x^2 + (3*b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*d^3*(c + d*x^2)*(2*Log[x] - Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))])))/(c^3*(b + a*c)^2)","A",1
325,1,293,368,0.925779,"\int x^4 \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[x^4*Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(-3 a^2 \left(c^2-d^2 x^4\right)-2 a b \left(c-2 d x^2\right)+b^2\right)+i c \left(-3 a^2 c^2+7 a b c+2 b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i b c (9 a c+b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{15 a d^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{\sqrt{c} \left(-3 a^2 c^2+7 a b c+2 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 a^2 d^{5/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{x \left(-3 a^2 c^2+7 a b c+2 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 a^2 d^2}-\frac{c^{3/2} (b-3 a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 a d^{5/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x (b-3 a c) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 a d^2}+\frac{x^3 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 d}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(b^2 - 2*a*b*(c - 2*d*x^2) - 3*a^2*(c^2 - d^2*x^4)) + I*c*(2*b^2 + 7*a*b*c - 3*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*b*c*(b + 9*a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(15*a*d^2*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
326,1,250,282,0.575517,"\int x^2 \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[x^2*Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a c+a d x^2+b\right)+2 i b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i c (a c-b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 d \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{c^{3/2} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 d^{3/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{c} (b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 a d^{3/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x (b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 a d}+\frac{x \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 d}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(b + a*c + a*d*x^2) + I*c*(-b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + (2*I)*b*c*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(3*d*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
327,1,98,213,0.0625848,"\int \sqrt{a+\frac{b}{c+d x^2}} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{\frac{c+d x^2}{c}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{a c}{b+a c}\right)}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a c+a d x^2+b}{a c+b}}}","x \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}+\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{d} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{d} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"(Sqrt[(c + d*x^2)/c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (a*c)/(b + a*c)])/(Sqrt[-(d/c)]*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)])","A",1
328,1,141,265,0.5737768,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^2} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^2,x]","\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-\frac{i a d \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)}{\sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}-\frac{d x}{c}-\frac{1}{x}\right)","\frac{d x \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x}+\frac{a \sqrt{c} \sqrt{d} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{(a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{d} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{c} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-x^(-1) - (d*x)/c - (I*a*d*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)])/(Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2))))","C",1
329,1,314,362,0.9924024,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^4} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^4,x]","-\frac{\sqrt{\frac{a d}{a c+b}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a^2 c \left(c^2-d^2 x^4\right)+2 a b \left(c^2-c d x^2-d^2 x^4\right)+b^2 \left(c-2 d x^2\right)\right)+i a b c d^2 x^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i a c d^2 x^3 (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 a c^2 d x^3 \left(a \left(c+d x^2\right)+b\right)}","\frac{d^{3/2} (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{d^2 x (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 (a c+b)}+\frac{d (a c+2 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 x (a c+b)}-\frac{a d^{3/2} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 \sqrt{c} (a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c x^3}",1,"-1/3*(Sqrt[(a*d)/(b + a*c)]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b^2*(c - 2*d*x^2) + a^2*c*(c^2 - d^2*x^4) + 2*a*b*(c^2 - c*d*x^2 - d^2*x^4)) - I*a*c*(2*b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + I*a*b*c*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(a*c^2*d*x^3*(b + a*(c + d*x^2)))","C",1
330,1,402,466,1.0888589,"\int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^6} \, dx","Integrate[Sqrt[a + b/(c + d*x^2)]/x^6,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(i a c d^3 x^5 \left(3 a^2 c^2+13 a b c+8 b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+\left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(3 a^3 c^2 \left(c^3+d^3 x^6\right)+a^2 b c \left(9 c^3-4 c^2 d x^2+9 c d^2 x^4+13 d^3 x^6\right)+a b^2 \left(9 c^3-8 c^2 d x^2+17 c d^2 x^4+8 d^3 x^6\right)+b^3 \left(3 c^2-4 c d x^2+8 d^2 x^4\right)\right)-2 i a b c d^3 x^5 (3 a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{15 c^3 x^5 (a c+b)^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{d^{5/2} \left(3 a^2 c^2+13 a b c+8 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 c^{5/2} (a c+b)^2 \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d^3 x \left(3 a^2 c^2+13 a b c+8 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^3 (a c+b)^2}-\frac{d^2 \left(3 a^2 c^2+13 a b c+8 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^3 x (a c+b)^2}+\frac{a d^{5/2} (3 a c+4 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 c^{3/2} (a c+b)^2 \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d (3 a c+4 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{15 c^2 x^3 (a c+b)}-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c x^5}",1,"-1/15*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b^3*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4) + 3*a^3*c^2*(c^3 + d^3*x^6) + a*b^2*(9*c^3 - 8*c^2*d*x^2 + 17*c*d^2*x^4 + 8*d^3*x^6) + a^2*b*c*(9*c^3 - 4*c^2*d*x^2 + 9*c*d^2*x^4 + 13*d^3*x^6)) + I*a*c*(8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - (2*I)*a*b*c*(2*b + 3*a*c)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(c^3*(b + a*c)^2*Sqrt[(a*d)/(b + a*c)]*x^5*(b + a*(c + d*x^2)))","C",1
331,1,142,249,0.4085607,"\int x^5 \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^5*(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{a} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(8 a^2 \left(c^3+d^3 x^6\right)-2 a b \left(47 c^2+16 c d x^2-7 d^2 x^4\right)+3 b^2 \left(c+d x^2\right)\right)-3 b \left(-24 a^2 c^2+12 a b c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{48 a^{3/2} d^3}","-\frac{\left(-24 a^2 c^2+60 a b c+5 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{48 a d^3}-\frac{b \left(-24 a^2 c^2+12 a b c+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{16 a^{3/2} d^3}-\frac{b c^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{d^3}+\frac{\left(c+d x^2\right)^3 \left(\frac{a c+a d x^2+b}{c+d x^2}\right)^{5/2}}{6 a d^3}-\frac{(12 a c+b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 d^3}",1,"(Sqrt[a]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(3*b^2*(c + d*x^2) - 2*a*b*(47*c^2 + 16*c*d*x^2 - 7*d^2*x^4) + 8*a^2*(c^3 + d^3*x^6)) - 3*b*(b^2 + 12*a*b*c - 24*a^2*c^2)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(48*a^(3/2)*d^3)","A",1
332,1,104,172,0.2439644,"\int x^3 \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^3*(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{a} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 a c^2+2 a d^2 x^4+13 b c+5 b d x^2\right)+3 b (b-4 a c) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{8 \sqrt{a} d^2}","\frac{a \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 d^2}+\frac{(5 b-4 a c) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 d^2}+\frac{b c \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{d^2}+\frac{3 b (b-4 a c) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{8 \sqrt{a} d^2}",1,"(Sqrt[a]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(13*b*c - 2*a*c^2 + 5*b*d*x^2 + 2*a*d^2*x^4) + 3*b*(b - 4*a*c)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(8*Sqrt[a]*d^2)","A",1
333,1,79,94,0.1018441,"\int x \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[x*(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(a \left(c+d x^2\right)-2 b\right)+3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 d}","\frac{\left(c+d x^2\right) \left(a+\frac{b}{c+d x^2}\right)^{3/2}}{2 d}-\frac{3 b \sqrt{a+\frac{b}{c+d x^2}}}{2 d}+\frac{3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 d}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*b + a*(c + d*x^2)) + 3*Sqrt[a]*b*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(2*d)","A",1
334,1,118,126,0.2259826,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x,x]","\frac{a^{3/2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)+b \sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}-(a c+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{c^{3/2}}","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)-\frac{(a c+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{c^{3/2}}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}",1,"(b*Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)] + a^(3/2)*c^(3/2)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]] - (b + a*c)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b/(c + d*x^2)])/Sqrt[b + a*c]])/c^(3/2)","A",1
335,1,256,138,0.5803738,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^3} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^3,x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 \sqrt{c (a c+b)} \left(a^2 c \left(c+d x^2\right)^2+a b \left(2 c^2+5 c d x^2+3 d^2 x^4\right)+b^2 \left(c+3 d x^2\right)\right)-6 b d x^2 \log (x) (a c+b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+3 b d x^2 (a c+b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{4 c^2 x^2 \sqrt{c (a c+b)} \left(a \left(c+d x^2\right)+b\right)}","\frac{3 b d \sqrt{a c+b} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{2 c^{5/2}}-\frac{3 b d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{2 c^2}-\frac{\left(c+d x^2\right) \left(\frac{a c+a d x^2+b}{c+d x^2}\right)^{3/2}}{2 c x^2}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(a^2*c*(c + d*x^2)^2 + b^2*(c + 3*d*x^2) + a*b*(2*c^2 + 5*c*d*x^2 + 3*d^2*x^4)) - 6*b*(b + a*c)*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + 3*b*(b + a*c)*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(4*c^2*Sqrt[c*(b + a*c)]*x^2*(b + a*(c + d*x^2)))","A",1
336,1,190,205,0.3258864,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^5} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^5,x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-\frac{2 c^2 (a c+b)}{x^4}+\frac{3 b d^2 (4 a c+5 b) \left(c+d x^2\right) \left(2 \log (x)-\log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}}+d^2 (2 a c+15 b)+\frac{5 b c d}{x^2}\right)}{8 c^3}","-\frac{3 b d^2 (4 a c+5 b) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{8 c^{7/2} \sqrt{a c+b}}+\frac{b d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^3}+\frac{d (4 a c+9 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c^3 x^2}-\frac{(a c+b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 c^3 x^4}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((15*b + 2*a*c)*d^2 - (2*c^2*(b + a*c))/x^4 + (5*b*c*d)/x^2 + (3*b*(5*b + 4*a*c)*d^2*(c + d*x^2)*(2*Log[x] - Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))])))/(8*c^3)","A",1
337,1,245,292,0.5185293,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^7} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^7,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\frac{3 b d^3 \left(24 a^2 c^2+60 a b c+35 b^2\right) \left(c+d x^2\right) \left(2 \log (x)-\log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{\sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}}+2 d^3 \left(8 a^2 c^2+110 a b c+105 b^2\right)+\frac{16 c^3 (a c+b)^2}{x^6}-\frac{28 b c^2 d (a c+b)}{x^4}+\frac{2 b c d^2 (32 a c+35 b)}{x^2}\right)}{96 c^4 (a c+b)}","\frac{b d^3 \left(24 a^2 c^2+60 a b c+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{16 c^{9/2} (a c+b)^{3/2}}-\frac{d^2 \left(24 a^2 c^2+108 a b c+79 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{48 c^4 x^2 (a c+b)}-\frac{b d^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^4}+\frac{d (12 a c+11 b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 c^4 x^4}-\frac{\left(c+d x^2\right)^3 \left(\frac{a c+a d x^2+b}{c+d x^2}\right)^{5/2}}{6 c^2 x^6 (a c+b)}",1,"-1/96*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*(105*b^2 + 110*a*b*c + 8*a^2*c^2)*d^3 + (16*c^3*(b + a*c)^2)/x^6 - (28*b*c^2*(b + a*c)*d)/x^4 + (2*b*c*(35*b + 32*a*c)*d^2)/x^2 + (3*b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*d^3*(c + d*x^2)*(2*Log[x] - Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))])))/(c^4*(b + a*c))","A",1
338,1,308,405,0.8541499,"\int x^4 \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^4*(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \sqrt{\frac{a d}{a c+b}} \left(-a^2 \left(c-d x^2\right) \left(c+d x^2\right)^2+3 a b \left(2 c^2+3 c d x^2+d^2 x^4\right)+b^2 \left(7 c+2 d x^2\right)\right)-i c \left(a^2 c^2-14 a b c+b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+8 i b c (b-a c) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{5 d^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{\sqrt{c} \left(a^2 c^2-14 a b c+b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 a d^{5/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x \left(a^2 c^2-14 a b c+b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 a d^2}-\frac{c^{3/2} (7 b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 d^{5/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x (7 b-a c) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 d^2}+\frac{6 a x^3 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 d}-\frac{x^3 \left(a c+a d x^2+b\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{d}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(-(a^2*(c - d*x^2)*(c + d*x^2)^2) + b^2*(7*c + 2*d*x^2) + 3*a*b*(2*c^2 + 3*c*d*x^2 + d^2*x^4)) - I*c*(b^2 - 14*a*b*c + a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + (8*I)*b*c*(b - a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(5*d^2*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
339,1,270,331,0.7765769,"\int x^2 \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[x^2*(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \sqrt{\frac{a d}{a c+b}} \left(a^2 \left(c+d x^2\right)^2-2 a b \left(c+d x^2\right)-3 b^2\right)+i b (5 a c-3 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i a c (a c-7 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 d \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{\sqrt{c} (3 b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 d^{3/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{c} (7 b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 d^{3/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{4 a x \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 d}-\frac{x \left(a c+a d x^2+b\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{d}+\frac{x (7 b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 d}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(-3*b^2 - 2*a*b*(c + d*x^2) + a^2*(c + d*x^2)^2) + I*a*c*(-7*b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + I*b*(-3*b + 5*a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(3*d*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
340,1,243,260,0.5476266,"\int \left(a+\frac{b}{c+d x^2}\right)^{3/2} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(b x \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)-2 i a b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i a c (a c-b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{c \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{x (b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}+\frac{b x \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c}+\frac{a \sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{d} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{(b-a c) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{c} \sqrt{d} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*Sqrt[(a*d)/(b + a*c)]*x*(b + a*(c + d*x^2)) - I*a*c*(-b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - (2*I)*a*b*c*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(c*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
341,1,278,312,0.76437,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^2} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^2,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\sqrt{\frac{a d}{a c+b}} \left(a^2 c \left(c+d x^2\right)^2+2 a b \left(c+d x^2\right)^2+b^2 \left(c+2 d x^2\right)\right)-i a b c d x \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i a c d x (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{c^2 x \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{\sqrt{d} (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{c^{3/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d x (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^2}-\frac{(a c+2 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^2 x}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x}+\frac{a \sqrt{d} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{c} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"-((Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(2*a*b*(c + d*x^2)^2 + a^2*c*(c + d*x^2)^2 + b^2*(c + 2*d*x^2)) + I*a*c*(2*b + a*c)*d*x*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*a*b*c*d*x*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(c^2*Sqrt[(a*d)/(b + a*c)]*x*(b + a*(c + d*x^2))))","C",1
342,1,329,388,0.8344629,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^4} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^4,x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\sqrt{\frac{a d}{a c+b}} \left(a^2 c \left(c-d x^2\right) \left(c+d x^2\right)^2+a b \left(2 c^3-3 c^2 d x^2-13 c d^2 x^4-8 d^3 x^6\right)+b^2 \left(c^2-4 c d x^2-8 d^2 x^4\right)\right)+4 i a b c d^2 x^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i a c d^2 x^3 (a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 c^3 x^3 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{a d^{3/2} (a c+4 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d^{3/2} (a c+8 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 c^{5/2} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{d^2 x (a c+8 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^3}+\frac{d (a c+8 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^3 x}-\frac{(a c+4 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 x^3}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x^3}",1,"-1/3*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(a^2*c*(c - d*x^2)*(c + d*x^2)^2 + b^2*(c^2 - 4*c*d*x^2 - 8*d^2*x^4) + a*b*(2*c^3 - 3*c^2*d*x^2 - 13*c*d^2*x^4 - 8*d^3*x^6)) - I*a*c*(8*b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + (4*I)*a*b*c*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(c^3*Sqrt[(a*d)/(b + a*c)]*x^3*(b + a*(c + d*x^2)))","C",1
343,1,430,494,1.109933,"\int \frac{\left(a+\frac{b}{c+d x^2}\right)^{3/2}}{x^6} \, dx","Integrate[(a + b/(c + d*x^2))^(3/2)/x^6,x]","-\frac{\sqrt{\frac{a d}{a c+b}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(i a c d^3 x^5 \left(a^2 c^2+16 a b c+16 b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+\sqrt{\frac{a d}{a c+b}} \left(a^3 c^2 \left(c^4+c^3 d x^2+c d^3 x^6+d^4 x^8\right)+a^2 b c \left(3 c^4+5 c^2 d^2 x^4+24 c d^3 x^6+16 d^4 x^8\right)+a b^2 \left(3 c^4-3 c^3 d x^2+13 c^2 d^2 x^4+40 c d^3 x^6+16 d^4 x^8\right)+b^3 \left(c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right)\right)-i a b c d^3 x^5 (7 a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{5 a c^4 d x^5 \left(a \left(c+d x^2\right)+b\right)}","-\frac{d^{5/2} \left(a^2 c^2+16 a b c+16 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 c^{7/2} (a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d^3 x \left(a^2 c^2+16 a b c+16 b^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c^4 (a c+b)}-\frac{d^2 \left(a^2 c^2+16 a b c+16 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c^4 x (a c+b)}+\frac{a d^{5/2} (a c+8 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 c^{5/2} (a c+b) \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d (a c+8 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c^3 x^3}-\frac{(a c+6 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{5 c^2 x^5}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x^5}",1,"-1/5*(Sqrt[(a*d)/(b + a*c)]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(b^3*(c^3 - 2*c^2*d*x^2 + 8*c*d^2*x^4 + 16*d^3*x^6) + a^3*c^2*(c^4 + c^3*d*x^2 + c*d^3*x^6 + d^4*x^8) + a^2*b*c*(3*c^4 + 5*c^2*d^2*x^4 + 24*c*d^3*x^6 + 16*d^4*x^8) + a*b^2*(3*c^4 - 3*c^3*d*x^2 + 13*c^2*d^2*x^4 + 40*c*d^3*x^6 + 16*d^4*x^8)) + I*a*c*(16*b^2 + 16*a*b*c + a^2*c^2)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*a*b*c*(8*b + 7*a*c)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(a*c^4*d*x^5*(b + a*(c + d*x^2)))","C",1
344,1,140,225,0.2820123,"\int \frac{x^5}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[x^5/Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{a} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(8 a^2 \left(c^2-c d x^2+d^2 x^4\right)+2 a b \left(13 c-5 d x^2\right)+15 b^2\right)-3 b \left(8 a^2 c^2+12 a b c+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{48 a^{7/2} d^3}","-\frac{(8 a c+5 b) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 a^2 d^3}-\frac{b \left(8 a^2 c^2+12 a b c+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{16 a^{7/2} d^3}+\frac{\left(8 a^2 c^2+12 a b c+5 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^3 d^3}+\frac{x^2 \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{6 a d^2}",1,"(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(15*b^2 + 2*a*b*(13*c - 5*d*x^2) + 8*a^2*(c^2 - c*d*x^2 + d^2*x^4)) - 3*b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(48*a^(7/2)*d^3)","A",1
345,1,101,148,0.1680197,"\int \frac{x^3}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[x^3/Sqrt[a + b/(c + d*x^2)],x]","\frac{b (4 a c+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)-\sqrt{a} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(2 a \left(c-d x^2\right)+3 b\right)}{8 a^{5/2} d^2}","\frac{b (4 a c+3 b) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{8 a^{5/2} d^2}-\frac{(4 a c+3 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 a^2 d^2}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 a d^2}",1,"(-(Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(3*b + 2*a*(c - d*x^2))) + b*(3*b + 4*a*c)*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(8*a^(5/2)*d^2)","A",1
346,1,70,72,0.0789319,"\int \frac{x}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[x/Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{a} \left(c+d x^2\right) \sqrt{a+\frac{b}{c+d x^2}}-b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 a^{3/2} d}","\frac{\left(c+d x^2\right) \sqrt{a+\frac{b}{c+d x^2}}}{2 a d}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 a^{3/2} d}",1,"(Sqrt[a]*(c + d*x^2)*Sqrt[a + b/(c + d*x^2)] - b*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(2*a^(3/2)*d)","A",1
347,1,80,96,0.1217741,"\int \frac{1}{x \sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/(x*Sqrt[a + b/(c + d*x^2)]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{\sqrt{a c+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{\sqrt{a c+b}}",1,"ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]]/Sqrt[a] - (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b/(c + d*x^2)])/Sqrt[b + a*c]])/Sqrt[b + a*c]","A",1
348,1,210,108,0.3000053,"\int \frac{1}{x^3 \sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/(x^3*Sqrt[a + b/(c + d*x^2)]),x]","-\frac{c \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(a c+a d x^2+b\right)-2 b d x^2 \log (x) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+b d x^2 \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{4 x^2 (c (a c+b))^{3/2} \left(a \left(c+d x^2\right)+b\right)}","-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{2 x^2 (a c+b)}-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{2 \sqrt{c} (a c+b)^{3/2}}",1,"-1/4*(c*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b + a*c + a*d*x^2) - 2*b*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + b*d*x^2*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/((c*(b + a*c))^(3/2)*x^2*(b + a*(c + d*x^2)))","A",1
349,1,269,177,0.4794265,"\int \frac{1}{x^5 \sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/(x^5*Sqrt[a + b/(c + d*x^2)]),x]","\frac{c \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(2 a^2 c \left(c^2-d^2 x^4\right)+a b \left(4 c^2+c d x^2+d^2 x^4\right)+b^2 \left(2 c+d x^2\right)\right)-2 b d^2 x^4 \log (x) (4 a c+b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+b d^2 x^4 (4 a c+b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{16 x^4 (c (a c+b))^{5/2} \left(a \left(c+d x^2\right)+b\right)}","\frac{b d^2 (4 a c+b) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{8 c^{3/2} (a c+b)^{5/2}}+\frac{d (4 a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 c x^2 (a c+b)^2}-\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 c x^4 (a c+b)}",1,"(c*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b^2*(2*c + d*x^2) + 2*a^2*c*(c^2 - d^2*x^4) + a*b*(4*c^2 + c*d*x^2 + d^2*x^4)) - 2*b*(b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + b*(b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(16*(c*(b + a*c))^(5/2)*x^4*(b + a*(c + d*x^2)))","A",1
350,1,297,443,0.7938878,"\int \frac{x^4}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[x^4/Sqrt[a + b/(c + d*x^2)],x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(3 a^2 \left(c^2-d^2 x^4\right)+a b \left(7 c+d x^2\right)+4 b^2\right)+i c \left(3 a^2 c^2+13 a b c+8 b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-2 i b c (3 a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{15 a^2 d^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{c^{3/2} (3 a c+4 b) \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 a^2 d^{5/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{x (3 a c+4 b) \left(a c+a d x^2+b\right)}{15 a^2 d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} \left(3 a^2 c^2+13 a b c+8 b^2\right) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{15 a^3 d^{5/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x \left(3 a^2 c^2+13 a b c+8 b^2\right) \left(a c+a d x^2+b\right)}{15 a^3 d^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{x^3 \left(a c+a d x^2+b\right)}{5 a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"-1/15*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(4*b^2 + a*b*(7*c + d*x^2) + 3*a^2*(c^2 - d^2*x^4)) + I*c*(8*b^2 + 13*a*b*c + 3*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - (2*I)*b*c*(2*b + 3*a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(a^2*d^2*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
351,1,253,354,0.6074546,"\int \frac{x^2}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[x^2/Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a c+a d x^2+b\right)-i b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i c (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 a d \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{\sqrt{c} (a c+2 b) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 a^2 d^{3/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{x (a c+2 b) \left(a c+a d x^2+b\right)}{3 a^2 d \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{c^{3/2} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 a d^{3/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x \left(a c+a d x^2+b\right)}{3 a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(b + a*c + a*d*x^2) + I*c*(2*b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*b*c*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(3*a*d*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
352,1,107,286,0.1117237,"\int \frac{1}{\sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/Sqrt[a + b/(c + d*x^2)],x]","\frac{\sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(\sin ^{-1}\left(\sqrt{-\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)}{\sqrt{\frac{d x^2}{c}+1} \sqrt{-\frac{a d}{a c+b}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}","\frac{c^{3/2} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{\sqrt{d} (a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x \left(a c+a d x^2+b\right)}{a \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{a \sqrt{d} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*EllipticE[ArcSin[Sqrt[-((a*d)/(b + a*c))]*x], 1 + b/(a*c)])/(Sqrt[-((a*d)/(b + a*c))]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*Sqrt[1 + (d*x^2)/c])","A",1
353,1,151,343,0.1792044,"\int \frac{1}{x^2 \sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/(x^2*Sqrt[a + b/(c + d*x^2)]),x]","\frac{d \sqrt{\frac{c+d x^2}{c}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left(\sin ^{-1}\left(\sqrt{-\frac{d}{c}} x\right)|\frac{a c}{b+a c}\right)}{\sqrt{-\frac{d}{c}} (a c+b) \sqrt{\frac{a c+a d x^2+b}{a c+b}}}-\frac{\left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{x (a c+b)}","-\frac{a c+a d x^2+b}{x (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{d x \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{(a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{c} \sqrt{d} \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{(a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"-(((c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/((b + a*c)*x)) + (d*Sqrt[(c + d*x^2)/c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], (a*c)/(b + a*c)])/((b + a*c)*Sqrt[-(d/c)]*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)])","A",1
354,1,314,435,1.016376,"\int \frac{1}{x^4 \sqrt{a+\frac{b}{c+d x^2}}} \, dx","Integrate[1/(x^4*Sqrt[a + b/(c + d*x^2)]),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-\left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a^2 c \left(c^2-d^2 x^4\right)+a b \left(2 c^2+c d x^2+d^2 x^4\right)+b^2 \left(c+d x^2\right)\right)+2 i a b c d^2 x^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i a c d^2 x^3 (a c-b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 c x^3 (a c+b)^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","-\frac{a \sqrt{c} d^{3/2} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 (a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{d^{3/2} (b-a c) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 \sqrt{c} (a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d^2 x (b-a c) \left(a c+a d x^2+b\right)}{3 c (a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{d (b-a c) \left(a c+a d x^2+b\right)}{3 c x (a c+b)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{-a c-a d x^2-b}{3 x^3 (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b^2*(c + d*x^2) + a^2*c*(c^2 - d^2*x^4) + a*b*(2*c^2 + c*d*x^2 + d^2*x^4))) + I*a*c*(-b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + (2*I)*a*b*c*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(3*c*(b + a*c)^2*Sqrt[(a*d)/(b + a*c)]*x^3*(b + a*(c + d*x^2)))","C",1
355,1,1215,310,11.6465779,"\int \frac{x^5}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^5/(a + b/(c + d*x^2))^(3/2),x]","\frac{b \left(-344 c^2 \, _4F_3\left(\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^5-192 c^2 \, _5F_4\left(\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^5-32 c^2 \, _6F_5\left(\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^5-105 a c^2 \left(a+\frac{b}{d x^2+c}\right)^4+105 a c^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^4+\frac{120 a c^2 \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^4}{\sqrt{\frac{b}{a d x^2+a c}+1}}+\frac{60 c (b+a c) \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^4}{\left(\frac{b}{a d x^2+a c}+1\right)^{3/2}}+1040 c (b+a c) \, _4F_3\left(\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^4+448 c (b+a c) \, _5F_4\left(\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^4+64 c (b+a c) \, _6F_5\left(\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^4+765 a^2 c^2 \left(a+\frac{b}{d x^2+c}\right)^3+300 a c (b+a c) \left(a+\frac{b}{d x^2+c}\right)^3-300 a c (b+a c) \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^3+\frac{300 (b+a c)^2 \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^3}{\left(\frac{b}{a d x^2+a c}+1\right)^{3/2}}-760 (b+a c)^2 \, _4F_3\left(\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^3-256 (b+a c)^2 \, _5F_4\left(\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^3-32 (b+a c)^2 \, _6F_5\left(\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right) \left(a+\frac{b}{d x^2+c}\right)^3+1365 a (b+a c)^2 \left(a+\frac{b}{d x^2+c}\right)^2-3240 a^2 c (b+a c) \left(a+\frac{b}{d x^2+c}\right)^2-765 a^3 c^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^2-1365 a (b+a c)^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right) \left(a+\frac{b}{d x^2+c}\right)^2+2835 a^2 (b+a c)^2 \left(a+\frac{b}{d x^2+c}\right)+3240 a^4 c (b+a c) \left(\frac{b}{a d x^2+a c}+1\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right)-2835 a^3 (b+a c)^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left(\sqrt{\frac{b}{a d x^2+a c}+1}\right)\right)}{720 a^5 d^3 \left(a+\frac{b}{d x^2+c}\right)^{5/2}}","\frac{\left(6 a^2 c^2+12 a b c+7 b^2\right) \left(c+d x^2\right)^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{6 a^2 b^2 d^3}-\frac{b \left(24 a^2 c^2+60 a b c+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{16 a^{9/2} d^3}+\frac{\left(24 a^2 c^2+60 a b c+35 b^2\right) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^4 d^3}-\frac{\left(24 a^2 c^2+60 a b c+35 b^2\right) \left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 a^3 b d^3}-\frac{(a c+b)^2 \left(c+d x^2\right)^3}{a b^2 d^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"(b*(2835*a^2*(b + a*c)^2*(a + b/(c + d*x^2)) - 3240*a^2*c*(b + a*c)*(a + b/(c + d*x^2))^2 + 1365*a*(b + a*c)^2*(a + b/(c + d*x^2))^2 + 765*a^2*c^2*(a + b/(c + d*x^2))^3 + 300*a*c*(b + a*c)*(a + b/(c + d*x^2))^3 - 105*a*c^2*(a + b/(c + d*x^2))^4 + (300*(b + a*c)^2*(a + b/(c + d*x^2))^3*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/(1 + b/(a*c + a*d*x^2))^(3/2) + (60*c*(b + a*c)*(a + b/(c + d*x^2))^4*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/(1 + b/(a*c + a*d*x^2))^(3/2) + (120*a*c^2*(a + b/(c + d*x^2))^4*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/Sqrt[1 + b/(a*c + a*d*x^2)] - 2835*a^3*(b + a*c)^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 765*a^3*c^2*(a + b/(c + d*x^2))^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 1365*a*(b + a*c)^2*(a + b/(c + d*x^2))^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 300*a*c*(b + a*c)*(a + b/(c + d*x^2))^3*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] + 105*a*c^2*(a + b/(c + d*x^2))^4*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] + 3240*a^4*c*(b + a*c)*(1 + b/(a*c + a*d*x^2))^(3/2)*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 760*(b + a*c)^2*(a + b/(c + d*x^2))^3*HypergeometricPFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 1040*c*(b + a*c)*(a + b/(c + d*x^2))^4*HypergeometricPFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 344*c^2*(a + b/(c + d*x^2))^5*HypergeometricPFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 256*(b + a*c)^2*(a + b/(c + d*x^2))^3*HypergeometricPFQ[{1/2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 448*c*(b + a*c)*(a + b/(c + d*x^2))^4*HypergeometricPFQ[{1/2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 192*c^2*(a + b/(c + d*x^2))^5*HypergeometricPFQ[{1/2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 32*(b + a*c)^2*(a + b/(c + d*x^2))^3*HypergeometricPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 64*c*(b + a*c)*(a + b/(c + d*x^2))^4*HypergeometricPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 32*c^2*(a + b/(c + d*x^2))^5*HypergeometricPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)]))/(720*a^5*d^3*(a + b/(c + d*x^2))^(5/2))","C",0
356,1,133,187,0.2663228,"\int \frac{x^3}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^3/(a + b/(c + d*x^2))^(3/2),x]","\frac{3 b (4 a c+5 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)-\sqrt{a} \left(2 a^2 \left(c^2-d^2 x^4\right)+a b \left(17 c+5 d x^2\right)+15 b^2\right)}{8 a^{7/2} d^2 \sqrt{a+\frac{b}{c+d x^2}}}","\frac{3 b (4 a c+5 b) \tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{8 a^{7/2} d^2}-\frac{(4 a c+7 b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 a^3 d^2}-\frac{b (a c+b)}{a^3 d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 a^2 d^2}",1,"(-(Sqrt[a]*(15*b^2 + a*b*(17*c + 5*d*x^2) + 2*a^2*(c^2 - d^2*x^4))) + 3*b*(5*b + 4*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]])/(8*a^(7/2)*d^2*Sqrt[a + b/(c + d*x^2)])","A",1
357,1,50,100,0.0557555,"\int \frac{x}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x/(a + b/(c + d*x^2))^(3/2),x]","\frac{b \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\frac{a+\frac{b}{d x^2+c}}{a}\right)}{a^2 d \sqrt{a+\frac{b}{c+d x^2}}}","-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{2 a^{5/2} d}+\frac{3 b}{2 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c+d x^2}{2 a d \sqrt{a+\frac{b}{c+d x^2}}}",1,"(b*Hypergeometric2F1[-1/2, 2, 1/2, (a + b/(c + d*x^2))/a])/(a^2*d*Sqrt[a + b/(c + d*x^2)])","C",1
358,1,110,134,0.4645865,"\int \frac{1}{x \left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x*(a + b/(c + d*x^2))^(3/2)),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{(a c+b)^{3/2}}-\frac{b}{a (a c+b) \sqrt{a+\frac{b}{c+d x^2}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{(a c+b)^{3/2}}-\frac{b}{a (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"-(b/(a*(b + a*c)*Sqrt[a + b/(c + d*x^2)])) + ArcTanh[Sqrt[a + b/(c + d*x^2)]/Sqrt[a]]/a^(3/2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b/(c + d*x^2)])/Sqrt[b + a*c]])/(b + a*c)^(3/2)","A",1
359,1,229,146,0.4180902,"\int \frac{1}{x^3 \left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^3*(a + b/(c + d*x^2))^(3/2)),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(-2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(a c \left(c+d x^2\right)+b \left(c-2 d x^2\right)\right)+6 b c d x^2 \log (x) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}-3 b c d x^2 \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{4 x^2 (a c+b)^2 \sqrt{c (a c+b)} \left(a \left(c+d x^2\right)+b\right)}","\frac{3 b d}{2 (a c+b)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{c+d x^2}{2 x^2 (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{3 b \sqrt{c} d \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{2 (a c+b)^{5/2}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b*(c - 2*d*x^2) + a*c*(c + d*x^2)) + 6*b*c*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] - 3*b*c*d*x^2*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/(4*(b + a*c)^2*Sqrt[c*(b + a*c)]*x^2*(b + a*(c + d*x^2)))","A",1
360,1,281,212,0.5680051,"\int \frac{1}{x^5 \left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^5*(a + b/(c + d*x^2))^(3/2)),x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(2 \sqrt{c (a c+b)} \left(c+d x^2\right) \left(2 a^2 c \left(c^2-d^2 x^4\right)+a b \left(4 c^2+5 c d x^2+13 d^2 x^4\right)+b^2 \left(2 c+5 d x^2\right)\right)+6 b d^2 x^4 \log (x) (4 a c-b) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)}+3 b d^2 x^4 (b-4 a c) \sqrt{\left(c+d x^2\right) \left(a \left(c+d x^2\right)+b\right)} \log \left(2 \sqrt{c (a c+b)} \sqrt{\left(c+d x^2\right) \left(a c+a d x^2+b\right)}+2 a c \left(c+d x^2\right)+b \left(2 c+d x^2\right)\right)\right)}{16 x^4 (a c+b)^3 \sqrt{c (a c+b)} \left(a \left(c+d x^2\right)+b\right)}","-\frac{a b d^2}{(a c+b)^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{3 b d^2 (b-4 a c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a c+b}}\right)}{8 \sqrt{c} (a c+b)^{7/2}}-\frac{d (3 b-4 a c) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{8 x^2 (a c+b)^3}-\frac{\left(c+d x^2\right)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{4 x^4 (a c+b)^2}",1,"-1/16*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(2*Sqrt[c*(b + a*c)]*(c + d*x^2)*(b^2*(2*c + 5*d*x^2) + 2*a^2*c*(c^2 - d^2*x^4) + a*b*(4*c^2 + 5*c*d*x^2 + 13*d^2*x^4)) + 6*b*(-b + 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[x] + 3*b*(b - 4*a*c)*d^2*x^4*Sqrt[(c + d*x^2)*(b + a*(c + d*x^2))]*Log[2*a*c*(c + d*x^2) + b*(2*c + d*x^2) + 2*Sqrt[c*(b + a*c)]*Sqrt[(c + d*x^2)*(b + a*c + a*d*x^2)]]))/((b + a*c)^3*Sqrt[c*(b + a*c)]*x^4*(b + a*(c + d*x^2)))","A",1
361,1,296,482,0.822024,"\int \frac{x^4}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^4/(a + b/(c + d*x^2))^(3/2),x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a^2 \left(c^2-d^2 x^4\right)+a b \left(9 c+2 d x^2\right)+8 b^2\right)+i c \left(a^2 c^2+16 a b c+16 b^2\right) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i b c (7 a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{5 a^3 d^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{c^{3/2} (a c+8 b) \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 a^3 d^{5/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{x (a c+8 b) \left(a c+a d x^2+b\right)}{5 a^3 d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{6 x^3 \left(a c+a d x^2+b\right)}{5 a^2 d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} \left(a^2 c^2+16 a b c+16 b^2\right) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{5 a^4 d^{5/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{x \left(a^2 c^2+16 a b c+16 b^2\right) \left(a c+a d x^2+b\right)}{5 a^4 d^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{x^3 \left(c+d x^2\right)}{a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"-1/5*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(8*b^2 + a*b*(9*c + 2*d*x^2) + a^2*(c^2 - d^2*x^4)) + I*c*(16*b^2 + 16*a*b*c + a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*b*c*(8*b + 7*a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(a^3*d^2*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
362,1,255,409,0.6020756,"\int \frac{x^2}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[x^2/(a + b/(c + d*x^2))^(3/2),x]","\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a c+a d x^2+4 b\right)-4 i b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i c (a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 a^2 d \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{\sqrt{c} (a c+8 b) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 a^3 d^{3/2} \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{x (a c+8 b) \left(a c+a d x^2+b\right)}{3 a^3 d \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{c^{3/2} (a c+4 b) \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 a^2 d^{3/2} (a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{4 x \left(a c+a d x^2+b\right)}{3 a^2 d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{x \left(c+d x^2\right)}{a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2)*(4*b + a*c + a*d*x^2) + I*c*(8*b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - (4*I)*b*c*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(3*a^2*d*Sqrt[(a*d)/(b + a*c)]*(b + a*(c + d*x^2)))","C",1
363,1,241,356,0.5160441,"\int \frac{1}{\left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[(a + b/(c + d*x^2))^(-3/2),x]","-\frac{\sqrt{\frac{a d}{a c+b}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(b x \left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}}-i b c \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i c (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{a^2 d \left(a \left(c+d x^2\right)+b\right)}","\frac{x (a c+2 b) \left(a c+a d x^2+b\right)}{a^2 (a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} (a c+2 b) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{a^2 \sqrt{d} (a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{c^{3/2} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{a \sqrt{d} (a c+b) \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{b x}{a (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"-((Sqrt[(a*d)/(b + a*c)]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b*Sqrt[(a*d)/(b + a*c)]*x*(c + d*x^2) + I*c*(2*b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] - I*b*c*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/(a^2*d*(b + a*(c + d*x^2))))","C",1
364,1,268,410,0.653552,"\int \frac{1}{x^2 \left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b/(c + d*x^2))^(3/2)),x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a c \left(c+d x^2\right)+b \left(c-d x^2\right)\right)+2 i b c d x \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)+i c d x (a c-b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{x (a c+b)^2 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{c^{3/2} \sqrt{d} \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{(a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{(b-a c) \left(a c+a d x^2+b\right)}{a x (a c+b)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{d x (b-a c) \left(a c+a d x^2+b\right)}{a (a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{b}{a x (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} (b-a c) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{a (a c+b)^2 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}",1,"-((Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b*(c - d*x^2) + a*c*(c + d*x^2)) + I*c*(-b + a*c)*d*x*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + (2*I)*b*c*d*x*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/((b + a*c)^2*Sqrt[(a*d)/(b + a*c)]*x*(b + a*(c + d*x^2))))","C",1
365,1,319,490,0.9043616,"\int \frac{1}{x^4 \left(a+\frac{b}{c+d x^2}\right)^{3/2}} \, dx","Integrate[1/(x^4*(a + b/(c + d*x^2))^(3/2)),x]","-\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left(\left(c+d x^2\right) \sqrt{\frac{a d}{a c+b}} \left(a^2 c \left(c^2-d^2 x^4\right)+a b \left(2 c^2+4 c d x^2+7 d^2 x^4\right)+b^2 \left(c+4 d x^2\right)\right)+i b d^2 x^3 (3 b-5 a c) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)-i a c d^2 x^3 (a c-7 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{a d}{b+a c}} x\right)|\frac{b}{a c}+1\right)\right)}{3 x^3 (a c+b)^3 \sqrt{\frac{a d}{a c+b}} \left(a \left(c+d x^2\right)+b\right)}","\frac{\sqrt{c} d^{3/2} (3 b-a c) \left(a c+a d x^2+b\right) F\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 (a c+b)^3 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}-\frac{\sqrt{c} d^{3/2} (7 b-a c) \left(a c+a d x^2+b\right) E\left(\tan ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{c}}\right)|\frac{b}{b+a c}\right)}{3 (a c+b)^3 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left(a c+a d x^2+b\right)}{(a c+b) \left(c+d x^2\right)}}}+\frac{d^2 x (7 b-a c) \left(a c+a d x^2+b\right)}{3 (a c+b)^3 \left(c+d x^2\right) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{d (7 b-a c) \left(a c+a d x^2+b\right)}{3 x (a c+b)^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{(3 b-a c) \left(a c+a d x^2+b\right)}{3 a x^3 (a c+b)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{b}{a x^3 (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}",1,"-1/3*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[(a*d)/(b + a*c)]*(c + d*x^2)*(b^2*(c + 4*d*x^2) + a^2*c*(c^2 - d^2*x^4) + a*b*(2*c^2 + 4*c*d*x^2 + 7*d^2*x^4)) - I*a*c*(-7*b + a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)] + I*b*(3*b - 5*a*c)*d^2*x^3*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[(a*d)/(b + a*c)]*x], 1 + b/(a*c)]))/((b + a*c)^3*Sqrt[(a*d)/(b + a*c)]*x^3*(b + a*(c + d*x^2)))","C",1
366,1,49,75,0.0213299,"\int \frac{\sqrt{a x^{23}}}{\sqrt{1+x^5}} \, dx","Integrate[Sqrt[a*x^23]/Sqrt[1 + x^5],x]","\frac{\sqrt{a x^{23}} \left(3 \sinh ^{-1}\left(x^{5/2}\right)+\sqrt{x^5+1} \left(2 x^5-3\right) x^{5/2}\right)}{20 x^{23/2}}","\frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left(x^{5/2}\right)}{20 x^{23/2}}-\frac{3 \sqrt{x^5+1} \sqrt{a x^{23}}}{20 x^9}+\frac{\sqrt{x^5+1} \sqrt{a x^{23}}}{10 x^4}",1,"(Sqrt[a*x^23]*(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5) + 3*ArcSinh[x^(5/2)]))/(20*x^(23/2))","A",1
367,1,42,50,0.0106765,"\int \frac{\sqrt{a x^{13}}}{\sqrt{1+x^5}} \, dx","Integrate[Sqrt[a*x^13]/Sqrt[1 + x^5],x]","\frac{\sqrt{a x^{13}} \left(x^{5/2} \sqrt{x^5+1}-\sinh ^{-1}\left(x^{5/2}\right)\right)}{5 x^{13/2}}","\frac{\sqrt{x^5+1} \sqrt{a x^{13}}}{5 x^4}-\frac{\sqrt{a x^{13}} \sinh ^{-1}\left(x^{5/2}\right)}{5 x^{13/2}}",1,"(Sqrt[a*x^13]*(x^(5/2)*Sqrt[1 + x^5] - ArcSinh[x^(5/2)]))/(5*x^(13/2))","A",1
368,1,24,24,0.0057305,"\int \frac{\sqrt{a x^3}}{\sqrt{1+x^5}} \, dx","Integrate[Sqrt[a*x^3]/Sqrt[1 + x^5],x]","\frac{2 \sqrt{a x^3} \sinh ^{-1}\left(x^{5/2}\right)}{5 x^{3/2}}","\frac{2 \sqrt{a x^3} \sinh ^{-1}\left(x^{5/2}\right)}{5 x^{3/2}}",1,"(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))","A",1
369,1,23,23,0.0041937,"\int \frac{\sqrt{\frac{a}{x^7}}}{\sqrt{1+x^5}} \, dx","Integrate[Sqrt[a/x^7]/Sqrt[1 + x^5],x]","-\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}}","-\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}}",1,"(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5","A",1
370,1,30,49,0.0064123,"\int \frac{\sqrt{\frac{a}{x^{17}}}}{\sqrt{1+x^5}} \, dx","Integrate[Sqrt[a/x^17]/Sqrt[1 + x^5],x]","-\frac{2}{15} x \left(1-2 x^5\right) \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}","\frac{4}{15} x^6 \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}-\frac{2}{15} x \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}",1,"(-2*Sqrt[a/x^17]*x*(1 - 2*x^5)*Sqrt[1 + x^5])/15","A",1
371,1,33,37,0.0137971,"\int \frac{\sqrt{a x^6}}{x \left(1-x^4\right)} \, dx","Integrate[Sqrt[a*x^6]/(x*(1 - x^4)),x]","-\frac{\sqrt{a x^6} \left(\log (1-x)-\log (x+1)+2 \tan ^{-1}(x)\right)}{4 x^3}","\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}",1,"-1/4*(Sqrt[a*x^6]*(2*ArcTan[x] + Log[1 - x] - Log[1 + x]))/x^3","A",1
372,1,33,37,0.0050841,"\int \frac{\sqrt{a x^6}}{x-x^5} \, dx","Integrate[Sqrt[a*x^6]/(x - x^5),x]","-\frac{\sqrt{a x^6} \left(\log (1-x)-\log (x+1)+2 \tan ^{-1}(x)\right)}{4 x^3}","\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}",1,"-1/4*(Sqrt[a*x^6]*(2*ArcTan[x] + Log[1 - x] - Log[1 + x]))/x^3","A",1
373,1,44,71,0.0151671,"\int \frac{\left(a x^6\right)^{3/2}}{x \left(1-x^4\right)} \, dx","Integrate[(a*x^6)^(3/2)/(x*(1 - x^4)),x]","-\frac{a \sqrt{a x^6} \left(4 x^5+20 x+5 \log (1-x)-5 \log (x+1)-10 \tan ^{-1}(x)\right)}{20 x^3}","\frac{a \sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{a \sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{1}{5} a x^2 \sqrt{a x^6}-\frac{a \sqrt{a x^6}}{x^2}",1,"-1/20*(a*Sqrt[a*x^6]*(20*x + 4*x^5 - 10*ArcTan[x] + 5*Log[1 - x] - 5*Log[1 + x]))/x^3","A",1
374,1,29,49,0.0610764,"\int \left(\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x \left(1-x^4\right)}\right) \, dx","Integrate[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x*(1 - x^4)),x]","\frac{1}{2} \left(\frac{\sqrt{a x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[a*x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
375,1,29,49,0.0164077,"\int \left(\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x-x^5}\right) \, dx","Integrate[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x - x^5),x]","\frac{1}{2} \left(\frac{\sqrt{a x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[a*x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
376,1,30,44,0.0096348,"\int \frac{\sqrt{a x^3}}{x-x^3} \, dx","Integrate[Sqrt[a*x^3]/(x - x^3),x]","\frac{\sqrt{a x^3} \left(\tanh ^{-1}\left(\sqrt{x}\right)-\tan ^{-1}\left(\sqrt{x}\right)\right)}{x^{3/2}}","\frac{\sqrt{a x^3} \tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}",1,"(Sqrt[a*x^3]*(-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]))/x^(3/2)","A",1
377,1,32,44,0.0090363,"\int \frac{\sqrt{a x^4}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a*x^4]/Sqrt[1 + x^2],x]","\frac{\sqrt{a x^4} \left(x \sqrt{x^2+1}-\sinh ^{-1}(x)\right)}{2 x^2}","\frac{\sqrt{x^2+1} \sqrt{a x^4}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2}",1,"(Sqrt[a*x^4]*(x*Sqrt[1 + x^2] - ArcSinh[x]))/(2*x^2)","A",1
378,1,43,83,0.0084625,"\int \frac{\sqrt{a x^3}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a*x^3]/Sqrt[1 + x^2],x]","\frac{2 \sqrt{a x^3} \left(\sqrt{x^2+1}-\, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^2\right)\right)}{3 x}","\frac{2 \sqrt{x^2+1} \sqrt{a x^3}}{3 x}-\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{a x^3} F\left(2 \tan ^{-1}\left(\sqrt{x}\right)|\frac{1}{2}\right)}{3 x^{3/2} \sqrt{x^2+1}}",1,"(2*Sqrt[a*x^3]*(Sqrt[1 + x^2] - Hypergeometric2F1[1/4, 1/2, 5/4, -x^2]))/(3*x)","C",1
379,1,22,22,0.0047533,"\int \frac{\sqrt{a x^2}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a*x^2]/Sqrt[1 + x^2],x]","\frac{\sqrt{x^2+1} \sqrt{a x^2}}{x}","\frac{\sqrt{x^2+1} \sqrt{a x^2}}{x}",1,"(Sqrt[a*x^2]*Sqrt[1 + x^2])/x","A",1
380,1,27,131,0.0055879,"\int \frac{\sqrt{a x}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a*x]/Sqrt[1 + x^2],x]","\frac{2}{3} x \sqrt{a x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-x^2\right)","\frac{2 \sqrt{x^2+1} \sqrt{a x}}{x+1}+\frac{\sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{a x}}{\sqrt{a}}\right)|\frac{1}{2}\right)}{\sqrt{x^2+1}}-\frac{2 \sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt{a x}}{\sqrt{a}}\right)|\frac{1}{2}\right)}{\sqrt{x^2+1}}",1,"(2*x*Sqrt[a*x]*Hypergeometric2F1[1/2, 3/4, 7/4, -x^2])/3","C",1
381,1,27,54,0.0050246,"\int \frac{\sqrt{\frac{a}{x}}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a/x]/Sqrt[1 + x^2],x]","2 x \sqrt{\frac{a}{x}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^2\right)","\frac{\sqrt{x} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{\frac{a}{x}} F\left(2 \tan ^{-1}\left(\sqrt{x}\right)|\frac{1}{2}\right)}{\sqrt{x^2+1}}",1,"2*Sqrt[a/x]*x*Hypergeometric2F1[1/4, 1/2, 5/4, -x^2]","C",1
382,1,22,22,0.0041607,"\int \frac{\sqrt{\frac{a}{x^2}}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a/x^2]/Sqrt[1 + x^2],x]","x \left(-\sqrt{\frac{a}{x^2}}\right) \tanh ^{-1}\left(\sqrt{x^2+1}\right)","x \left(-\sqrt{\frac{a}{x^2}}\right) \tanh ^{-1}\left(\sqrt{x^2+1}\right)",1,"-(Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^2]])","A",1
383,1,27,159,0.0062852,"\int \frac{\sqrt{\frac{a}{x^3}}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a/x^3]/Sqrt[1 + x^2],x]","-2 x \sqrt{\frac{a}{x^3}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-x^2\right)","\frac{2 \sqrt{x^2+1} x^2 \sqrt{\frac{a}{x^3}}}{x+1}-2 \sqrt{x^2+1} x \sqrt{\frac{a}{x^3}}+\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} F\left(2 \tan ^{-1}\left(\sqrt{x}\right)|\frac{1}{2}\right)}{\sqrt{x^2+1}}-\frac{2 (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} E\left(2 \tan ^{-1}\left(\sqrt{x}\right)|\frac{1}{2}\right)}{\sqrt{x^2+1}}",1,"-2*Sqrt[a/x^3]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -x^2]","C",1
384,1,21,21,0.0040978,"\int \frac{\sqrt{\frac{a}{x^4}}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[a/x^4]/Sqrt[1 + x^2],x]","x \sqrt{x^2+1} \left(-\sqrt{\frac{a}{x^4}}\right)","x \sqrt{x^2+1} \left(-\sqrt{\frac{a}{x^4}}\right)",1,"-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])","A",1
385,1,25,25,0.004416,"\int \frac{\sqrt{a x^4}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a*x^4]/Sqrt[1 + x^3],x]","\frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2}","\frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2}",1,"(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)","A",1
386,1,29,292,0.0053792,"\int \frac{\sqrt{a x^3}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a*x^3]/Sqrt[1 + x^3],x]","\frac{2}{5} x \sqrt{a x^3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};-x^3\right)","\frac{\left(1+\sqrt{3}\right) \sqrt{x^3+1} \sqrt{a x^3}}{x \left(\left(1+\sqrt{3}\right) x+1\right)}-\frac{\left(1-\sqrt{3}\right) (x+1) \sqrt{\frac{x^2-x+1}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{a x^3} F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) x+1}{\left(1+\sqrt{3}\right) x+1}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[4]{3} x \sqrt{\frac{x (x+1)}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{a x^3} E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) x+1}{\left(1+\sqrt{3}\right) x+1}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{x \sqrt{\frac{x (x+1)}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{x^3+1}}",1,"(2*x*Sqrt[a*x^3]*Hypergeometric2F1[1/2, 5/6, 11/6, -x^3])/5","C",1
387,1,29,260,0.0040134,"\int \frac{\sqrt{a x^2}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a*x^2]/Sqrt[1 + x^3],x]","\frac{1}{2} x \sqrt{a x^2} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right)","\frac{2 \sqrt{x^3+1} \sqrt{a x^2}}{x \left(x+\sqrt{3}+1\right)}+\frac{2 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{a x^2} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} x \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{a x^2} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{x \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"(x*Sqrt[a*x^2]*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3])/2","C",1
388,1,22,23,0.005078,"\int \frac{\sqrt{a x}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a*x]/Sqrt[1 + x^3],x]","\frac{2 \sqrt{a x} \sinh ^{-1}\left(x^{3/2}\right)}{3 \sqrt{x}}","\frac{2}{3} \sqrt{a} \sinh ^{-1}\left(\frac{(a x)^{3/2}}{a^{3/2}}\right)",1,"(2*Sqrt[a*x]*ArcSinh[x^(3/2)])/(3*Sqrt[x])","A",1
389,1,27,116,0.0052745,"\int \frac{\sqrt{\frac{a}{x}}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a/x]/Sqrt[1 + x^3],x]","2 x \sqrt{\frac{a}{x}} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};-x^3\right)","\frac{x (x+1) \sqrt{\frac{x^2-x+1}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{\frac{a}{x}} F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) x+1}{\left(1+\sqrt{3}\right) x+1}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[4]{3} \sqrt{\frac{x (x+1)}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{x^3+1}}",1,"2*Sqrt[a/x]*x*Hypergeometric2F1[1/6, 1/2, 7/6, -x^3]","C",1
390,1,24,24,0.0041228,"\int \frac{\sqrt{\frac{a}{x^2}}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a/x^2]/Sqrt[1 + x^3],x]","-\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left(\sqrt{x^3+1}\right)","-\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left(\sqrt{x^3+1}\right)",1,"(-2*Sqrt[a/x^2]*x*ArcTanh[Sqrt[1 + x^3]])/3","A",1
391,1,27,312,0.0055022,"\int \frac{\sqrt{\frac{a}{x^3}}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a/x^3]/Sqrt[1 + x^3],x]","-2 x \sqrt{\frac{a}{x^3}} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};-x^3\right)","-2 \sqrt{x^3+1} x \sqrt{\frac{a}{x^3}}+\frac{2 \left(1+\sqrt{3}\right) \sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^3}}}{\left(1+\sqrt{3}\right) x+1}-\frac{\left(1-\sqrt{3}\right) (x+1) \sqrt{\frac{x^2-x+1}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} x^2 \sqrt{\frac{a}{x^3}} F\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) x+1}{\left(1+\sqrt{3}\right) x+1}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[4]{3} \sqrt{\frac{x (x+1)}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{x^3+1}}-\frac{2 \sqrt[4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} x^2 \sqrt{\frac{a}{x^3}} E\left(\cos ^{-1}\left(\frac{\left(1-\sqrt{3}\right) x+1}{\left(1+\sqrt{3}\right) x+1}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt{\frac{x (x+1)}{\left(\left(1+\sqrt{3}\right) x+1\right)^2}} \sqrt{x^3+1}}",1,"-2*Sqrt[a/x^3]*x*Hypergeometric2F1[-1/6, 1/2, 5/6, -x^3]","C",1
392,1,27,281,0.0047605,"\int \frac{\sqrt{\frac{a}{x^4}}}{\sqrt{1+x^3}} \, dx","Integrate[Sqrt[a/x^4]/Sqrt[1 + x^3],x]","x \left(-\sqrt{\frac{a}{x^4}}\right) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};-x^3\right)","-\sqrt{x^3+1} x \sqrt{\frac{a}{x^4}}+\frac{\sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^4}}}{x+\sqrt{3}+1}+\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} x^2 \sqrt{\frac{a}{x^4}} F\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{\sqrt[4]{3} \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}-\frac{\sqrt[4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left(x+\sqrt{3}+1\right)^2}} x^2 \sqrt{\frac{a}{x^4}} E\left(\sin ^{-1}\left(\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 \sqrt{\frac{x+1}{\left(x+\sqrt{3}+1\right)^2}} \sqrt{x^3+1}}",1,"-(Sqrt[a/x^4]*x*Hypergeometric2F1[-1/3, 1/2, 2/3, -x^3])","C",1
393,1,37,37,0.0121229,"\int \frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}} \, dx","Integrate[Sqrt[a*x^(2*n)]/Sqrt[1 + x^n],x]","\frac{x \sqrt{a x^{2 n}} \, _2F_1\left(\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right)}{n+1}","\frac{x \sqrt{a x^{2 n}} \, _2F_1\left(\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right)}{n+1}",1,"(x*Sqrt[a*x^(2*n)]*Hypergeometric2F1[1/2, 1 + n^(-1), 2 + n^(-1), -x^n])/(1 + n)","A",1
394,1,40,48,0.0100379,"\int \frac{\sqrt{a x^n}}{\sqrt{1+x^n}} \, dx","Integrate[Sqrt[a*x^n]/Sqrt[1 + x^n],x]","\frac{2 x \sqrt{a x^n} \, _2F_1\left(\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-x^n\right)}{n+2}","\frac{2 x \sqrt{a x^n} \, _2F_1\left(\frac{1}{2},\frac{1}{2} \left(1+\frac{2}{n}\right);\frac{1}{2} \left(3+\frac{2}{n}\right);-x^n\right)}{n+2}",1,"(2*x*Sqrt[a*x^n]*Hypergeometric2F1[1/2, 1/2 + n^(-1), 3/2 + n^(-1), -x^n])/(2 + n)","A",1
395,1,44,52,0.0113517,"\int \frac{\sqrt{a x^{n/2}}}{\sqrt{1+x^n}} \, dx","Integrate[Sqrt[a*x^(n/2)]/Sqrt[1 + x^n],x]","\frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left(\frac{1}{2},\frac{1}{4}+\frac{1}{n};\frac{5}{4}+\frac{1}{n};-x^n\right)}{n+4}","\frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left(\frac{1}{2},\frac{1}{4} \left(1+\frac{4}{n}\right);\frac{1}{4} \left(5+\frac{4}{n}\right);-x^n\right)}{n+4}",1,"(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, 1/4 + n^(-1), 5/4 + n^(-1), -x^n])/(4 + n)","A",1
396,1,33,34,0.0285549,"\int \left(\frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}}+\frac{2 x^{-n} \sqrt{a x^{2 n}}}{(2+n) \sqrt{1+x^n}}\right) \, dx","Integrate[Sqrt[a*x^(2*n)]/Sqrt[1 + x^n] + (2*Sqrt[a*x^(2*n)])/((2 + n)*x^n*Sqrt[1 + x^n]),x]","\frac{2 a x^{n+1} \sqrt{x^n+1}}{(n+2) \sqrt{a x^{2 n}}}","\frac{2 x^{1-n} \sqrt{x^n+1} \sqrt{a x^{2 n}}}{n+2}",1,"(2*a*x^(1 + n)*Sqrt[1 + x^n])/((2 + n)*Sqrt[a*x^(2*n)])","A",1
397,1,106,114,0.2082428,"\int \frac{\sqrt{a x}}{\sqrt{d+e x} \sqrt{e+f x}} \, dx","Integrate[Sqrt[a*x]/(Sqrt[d + e*x]*Sqrt[e + f*x]),x]","-\frac{2 i e \sqrt{a x} \sqrt{\frac{f x}{e}+1} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{e x}{d}}\right)|\frac{d f}{e^2}\right)-F\left(i \sinh ^{-1}\left(\sqrt{\frac{e x}{d}}\right)|\frac{d f}{e^2}\right)\right)}{f \sqrt{\frac{e x}{d+e x}} \sqrt{d+e x} \sqrt{e+f x}}","\frac{2 \sqrt{a x} \sqrt{d f-e^2} \sqrt{\frac{e (e+f x)}{e^2-d f}} E\left(\sin ^{-1}\left(\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{d f-e^2}}\right)|1-\frac{e^2}{d f}\right)}{e \sqrt{f} \sqrt{-\frac{e x}{d}} \sqrt{e+f x}}",1,"((-2*I)*e*Sqrt[a*x]*Sqrt[1 + (f*x)/e]*(EllipticE[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)/e^2] - EllipticF[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)/e^2]))/(f*Sqrt[(e*x)/(d + e*x)]*Sqrt[d + e*x]*Sqrt[e + f*x])","C",1
398,1,16,16,0.0032966,"\int \left(a x^m\right)^r \, dx","Integrate[(a*x^m)^r,x]","\frac{x \left(a x^m\right)^r}{m r+1}","\frac{x \left(a x^m\right)^r}{m r+1}",1,"(x*(a*x^m)^r)/(1 + m*r)","A",1
399,1,26,26,0.006586,"\int \left(a x^m\right)^r \left(b x^n\right)^s \, dx","Integrate[(a*x^m)^r*(b*x^n)^s,x]","\frac{x \left(a x^m\right)^r \left(b x^n\right)^s}{m r+n s+1}","\frac{x \left(a x^m\right)^r \left(b x^n\right)^s}{m r+n s+1}",1,"(x*(a*x^m)^r*(b*x^n)^s)/(1 + m*r + n*s)","A",1
400,1,36,36,0.0102487,"\int \left(a x^m\right)^r \left(b x^n\right)^s \left(c x^p\right)^t \, dx","Integrate[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x]","\frac{x \left(a x^m\right)^r \left(b x^n\right)^s \left(c x^p\right)^t}{m r+n s+p t+1}","\frac{x \left(a x^m\right)^r \left(b x^n\right)^s \left(c x^p\right)^t}{m r+n s+p t+1}",1,"(x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t)/(1 + m*r + n*s + p*t)","A",1
401,1,140,147,0.1662064,"\int \frac{x^2}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x]),x]","\frac{2 \left(8 a^3 \sqrt{a+b x}-4 a^2 b x \sqrt{a+b x}+15 b^3 x^3 \left(\sqrt{a+b x}-\sqrt{b x+c}\right)+3 a b^2 x^2 \sqrt{a+b x}-3 b^2 c x^2 \sqrt{b x+c}-8 c^3 \sqrt{b x+c}+4 b c^2 x \sqrt{b x+c}\right)}{105 b^3 (a-c)}","\frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac{2 c^2 (b x+c)^{3/2}}{3 b^3 (a-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (a-c)}-\frac{2 (b x+c)^{7/2}}{7 b^3 (a-c)}+\frac{4 c (b x+c)^{5/2}}{5 b^3 (a-c)}",1,"(2*(8*a^3*Sqrt[a + b*x] - 4*a^2*b*x*Sqrt[a + b*x] + 3*a*b^2*x^2*Sqrt[a + b*x] - 8*c^3*Sqrt[c + b*x] + 4*b*c^2*x*Sqrt[c + b*x] - 3*b^2*c*x^2*Sqrt[c + b*x] + 15*b^3*x^3*(Sqrt[a + b*x] - Sqrt[c + b*x])))/(105*b^3*(a - c))","A",1
402,1,95,95,0.0978064,"\int \frac{x}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[c + b*x]),x]","\frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)}","\frac{2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac{2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac{2 c (b x+c)^{3/2}}{3 b^2 (a-c)}",1,"(-2*a*(a + b*x)^(3/2))/(3*b^2*(a - c)) + (2*(a + b*x)^(5/2))/(5*b^2*(a - c)) + (2*c*(c + b*x)^(3/2))/(3*b^2*(a - c)) - (2*(c + b*x)^(5/2))/(5*b^2*(a - c))","A",1
403,1,35,47,0.0489909,"\int \frac{1}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-1),x]","\frac{2 \left((a+b x)^{3/2}-(b x+c)^{3/2}\right)}{3 b (a-c)}","\frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)}",1,"(2*((a + b*x)^(3/2) - (c + b*x)^(3/2)))/(3*b*(a - c))","A",1
404,1,75,97,0.0737126,"\int \frac{1}{x \left(\sqrt{a+b x}+\sqrt{c+b x}\right)} \, dx","Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]","\frac{2 \left(\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)-\sqrt{b x+c}+\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)\right)}{a-c}","\frac{2 \sqrt{a+b x}}{a-c}-\frac{2 \sqrt{b x+c}}{a-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{a-c}+\frac{2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)}{a-c}",1,"(2*(Sqrt[a + b*x] - Sqrt[c + b*x] - Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]]))/(a - c)","A",1
405,1,99,103,0.26814,"\int \frac{1}{x^2 \left(\sqrt{a+b x}+\sqrt{c+b x}\right)} \, dx","Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]","\frac{\frac{b x \sqrt{\frac{b x}{c}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{c}+1}\right)+b x+c}{\sqrt{b x+c}}-\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{a}+1}\right)+a+b x}{\sqrt{a+b x}}}{x (a-c)}","-\frac{\sqrt{a+b x}}{x (a-c)}+\frac{\sqrt{b x+c}}{x (a-c)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a} (a-c)}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)}{\sqrt{c} (a-c)}",1,"(-((a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]])/Sqrt[a + b*x]) + (c + b*x + b*x*Sqrt[1 + (b*x)/c]*ArcTanh[Sqrt[1 + (b*x)/c]])/Sqrt[c + b*x])/((a - c)*x)","A",1
406,1,361,228,1.2912359,"\int \frac{x^2}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^2} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x])^2,x]","\frac{-15 a^3 \sqrt{a+b x} \sqrt{b x+c}+\frac{3 \sqrt{b} (c-a)^3 \left(5 a^2+6 a c+5 c^2\right) \sqrt{-\frac{b x+c}{a-c}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right)}{\sqrt{b (c-a)} \sqrt{b x+c}}+a^2 \sqrt{a+b x} \sqrt{b x+c} (10 b x+7 c)-16 b^3 x^3 \left(3 \sqrt{a+b x} \sqrt{b x+c}-2 c\right)-8 b^2 c x^2 \sqrt{a+b x} \sqrt{b x+c}-a \left(8 b^2 x^2 \sqrt{a+b x} \sqrt{b x+c}-7 c^2 \sqrt{a+b x} \sqrt{b x+c}+4 b c x \sqrt{a+b x} \sqrt{b x+c}-32 b^3 x^3\right)-15 c^3 \sqrt{a+b x} \sqrt{b x+c}+10 b c^2 x \sqrt{a+b x} \sqrt{b x+c}+48 b^4 x^4}{96 b^3 (a-c)^2}","\frac{5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac{\left(4 a c-5 (a+c)^2\right) (a+b x)^{3/2} \sqrt{b x+c}}{16 b^3 (a-c)^2}-\frac{\left(4 a c-5 (a+c)^2\right) \sqrt{a+b x} \sqrt{b x+c}}{32 b^3 (a-c)}-\frac{\left(4 a c-5 (a+c)^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right)}{32 b^3}-\frac{x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac{b x^4}{2 (a-c)^2}+\frac{x^3 (a+c)}{3 (a-c)^2}",1,"(48*b^4*x^4 - 15*a^3*Sqrt[a + b*x]*Sqrt[c + b*x] - 15*c^3*Sqrt[a + b*x]*Sqrt[c + b*x] + 10*b*c^2*x*Sqrt[a + b*x]*Sqrt[c + b*x] - 8*b^2*c*x^2*Sqrt[a + b*x]*Sqrt[c + b*x] + a^2*Sqrt[a + b*x]*Sqrt[c + b*x]*(7*c + 10*b*x) - 16*b^3*x^3*(-2*c + 3*Sqrt[a + b*x]*Sqrt[c + b*x]) - a*(-32*b^3*x^3 - 7*c^2*Sqrt[a + b*x]*Sqrt[c + b*x] + 4*b*c*x*Sqrt[a + b*x]*Sqrt[c + b*x] + 8*b^2*x^2*Sqrt[a + b*x]*Sqrt[c + b*x]) + (3*Sqrt[b]*(-a + c)^3*(5*a^2 + 6*a*c + 5*c^2)*Sqrt[-((c + b*x)/(a - c))]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b*(-a + c)]])/(Sqrt[b*(-a + c)]*Sqrt[c + b*x]))/(96*b^3*(a - c)^2)","A",1
407,1,229,165,0.8759315,"\int \frac{x}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^2} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^2,x]","\frac{3 a^2 \sqrt{a+b x} \sqrt{b x+c}-2 a \left(b x \sqrt{a+b x} \sqrt{b x+c}+c \sqrt{a+b x} \sqrt{b x+c}-3 b^2 x^2\right)+(4 b x+3 c) \left(-2 b x \sqrt{a+b x} \sqrt{b x+c}+c \sqrt{a+b x} \sqrt{b x+c}+2 b^2 x^2\right)}{12 b^2 (a-c)^2}-\frac{(a+c) \sqrt{b (c-a)} \sqrt{-\frac{b x+c}{a-c}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right)}{4 b^{5/2} \sqrt{b x+c}}","-\frac{2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2 (a-c)^2}+\frac{(a+c) (a+b x)^{3/2} \sqrt{b x+c}}{2 b^2 (a-c)^2}-\frac{(a+c) \sqrt{a+b x} \sqrt{b x+c}}{4 b^2 (a-c)}-\frac{(a+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right)}{4 b^2}+\frac{2 b x^3}{3 (a-c)^2}+\frac{x^2 (a+c)}{2 (a-c)^2}",1,"(3*a^2*Sqrt[a + b*x]*Sqrt[c + b*x] + (3*c + 4*b*x)*(2*b^2*x^2 + c*Sqrt[a + b*x]*Sqrt[c + b*x] - 2*b*x*Sqrt[a + b*x]*Sqrt[c + b*x]) - 2*a*(-3*b^2*x^2 + c*Sqrt[a + b*x]*Sqrt[c + b*x] + b*x*Sqrt[a + b*x]*Sqrt[c + b*x]))/(12*b^2*(a - c)^2) - (Sqrt[b*(-a + c)]*(a + c)*Sqrt[-((c + b*x)/(a - c))]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b*(-a + c)]])/(4*b^(5/2)*Sqrt[c + b*x])","A",1
408,1,179,63,0.5588089,"\int \frac{1}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^2} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-2),x]","\frac{2 b x \left(b x-\sqrt{a+b x} \sqrt{b x+c}\right)+a \left(2 b x-\sqrt{a+b x} \sqrt{b x+c}\right)+c \left(2 b x-\sqrt{a+b x} \sqrt{b x+c}\right)+\frac{\sqrt{b} (c-a)^3 \sqrt{\frac{b x+c}{c-a}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right)}{\sqrt{b (c-a)} \sqrt{b x+c}}+2 c^2}{2 b (a-c)^2}","\frac{(a-c)^2}{8 b \left(\sqrt{a+b x}+\sqrt{b x+c}\right)^4}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right)}{2 b}",1,"(2*c^2 + 2*b*x*(b*x - Sqrt[a + b*x]*Sqrt[c + b*x]) + a*(2*b*x - Sqrt[a + b*x]*Sqrt[c + b*x]) + c*(2*b*x - Sqrt[a + b*x]*Sqrt[c + b*x]) + (Sqrt[b]*(-a + c)^3*Sqrt[(c + b*x)/(-a + c)]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b*(-a + c)]])/(Sqrt[b*(-a + c)]*Sqrt[c + b*x]))/(2*b*(a - c)^2)","B",1
409,1,195,133,1.0684065,"\int \frac{1}{x \left(\sqrt{a+b x}+\sqrt{c+b x}\right)^2} \, dx","Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^2),x]","\frac{\sqrt{b} \left(-2 \left(c \sqrt{a+b x}+b x \left(\sqrt{a+b x}-\sqrt{b x+c}\right)\right)+(a+c) \log (x) \sqrt{b x+c}+4 \sqrt{a} \sqrt{c} \sqrt{b x+c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right)\right)-2 (a+c) \sqrt{b (c-a)} \sqrt{-\frac{b x+c}{a-c}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right)}{\sqrt{b} (a-c)^2 \sqrt{b x+c}}","\frac{2 b x}{(a-c)^2}-\frac{2 \sqrt{a+b x} \sqrt{b x+c}}{(a-c)^2}-\frac{2 (a+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right)}{(a-c)^2}+\frac{4 \sqrt{a} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right)}{(a-c)^2}+\frac{(a+c) \log (x)}{(a-c)^2}",1,"(-2*Sqrt[b*(-a + c)]*(a + c)*Sqrt[-((c + b*x)/(a - c))]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b*(-a + c)]] + Sqrt[b]*(-2*(c*Sqrt[a + b*x] + b*x*(Sqrt[a + b*x] - Sqrt[c + b*x])) + 4*Sqrt[a]*Sqrt[c]*Sqrt[c + b*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + b*x])] + (a + c)*Sqrt[c + b*x]*Log[x]))/(Sqrt[b]*(a - c)^2*Sqrt[c + b*x])","A",1
410,1,205,141,0.9595978,"\int \frac{1}{x^2 \left(\sqrt{a+b x}+\sqrt{c+b x}\right)^2} \, dx","Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])^2),x]","\frac{\frac{a \left(-\sqrt{b x+c}\right)+2 c \sqrt{a+b x}+2 b x \sqrt{a+b x}-c \sqrt{b x+c}+2 b x \log (x) \sqrt{b x+c}}{x}-4 \sqrt{b} \sqrt{b (c-a)} \sqrt{-\frac{b x+c}{a-c}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right)+\frac{2 b (a+c) \sqrt{b x+c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right)}{\sqrt{a} \sqrt{c}}}{(a-c)^2 \sqrt{b x+c}}","\frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x (a-c)^2}+\frac{2 b \log (x)}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right)}{\sqrt{a} \sqrt{c} (a-c)^2}-\frac{4 b \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right)}{(a-c)^2}-\frac{a+c}{x (a-c)^2}",1,"(-4*Sqrt[b]*Sqrt[b*(-a + c)]*Sqrt[-((c + b*x)/(a - c))]*ArcSinh[(Sqrt[b]*Sqrt[a + b*x])/Sqrt[b*(-a + c)]] + (2*b*(a + c)*Sqrt[c + b*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + b*x])])/(Sqrt[a]*Sqrt[c]) + (2*c*Sqrt[a + b*x] + 2*b*x*Sqrt[a + b*x] - a*Sqrt[c + b*x] - c*Sqrt[c + b*x] + 2*b*x*Sqrt[c + b*x]*Log[x])/x)/((a - c)^2*Sqrt[c + b*x])","A",1
411,1,282,375,0.4048953,"\int \frac{x^2}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^3} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]","\frac{2 \left(-40 a^4 \sqrt{a+b x}+4 a^3 \sqrt{a+b x} (5 b x+18 c)-3 a^2 b x \sqrt{a+b x} (5 b x+12 c)+a \left(5 b^3 x^3 \left(13 \sqrt{a+b x}-27 \sqrt{b x+c}\right)+27 b^2 c x^2 \left(\sqrt{a+b x}-\sqrt{b x+c}\right)-72 c^3 \sqrt{b x+c}+36 b c^2 x \sqrt{b x+c}\right)+5 \left(28 b^4 x^4 \left(\sqrt{a+b x}-\sqrt{b x+c}\right)+b^3 c x^3 \left(27 \sqrt{a+b x}-13 \sqrt{b x+c}\right)+3 b^2 c^2 x^2 \sqrt{b x+c}+8 c^4 \sqrt{b x+c}-4 b c^3 x \sqrt{b x+c}\right)\right)}{315 b^3 (a-c)^3}","-\frac{8 a^3 (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{24 a^2 (a+b x)^{5/2}}{5 b^3 (a-c)^3}+\frac{2 a^2 (a+3 c) (a+b x)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 c^3 (b x+c)^{3/2}}{3 b^3 (a-c)^3}-\frac{24 c^2 (b x+c)^{5/2}}{5 b^3 (a-c)^3}-\frac{2 c^2 (3 a+c) (b x+c)^{3/2}}{3 b^3 (a-c)^3}+\frac{8 (a+b x)^{9/2}}{9 b^3 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{24 a (a+b x)^{7/2}}{7 b^3 (a-c)^3}-\frac{4 a (a+3 c) (a+b x)^{5/2}}{5 b^3 (a-c)^3}-\frac{8 (b x+c)^{9/2}}{9 b^3 (a-c)^3}+\frac{24 c (b x+c)^{7/2}}{7 b^3 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{7/2}}{7 b^3 (a-c)^3}+\frac{4 c (3 a+c) (b x+c)^{5/2}}{5 b^3 (a-c)^3}",1,"(2*(-40*a^4*Sqrt[a + b*x] - 3*a^2*b*x*Sqrt[a + b*x]*(12*c + 5*b*x) + 4*a^3*Sqrt[a + b*x]*(18*c + 5*b*x) + a*(-72*c^3*Sqrt[c + b*x] + 36*b*c^2*x*Sqrt[c + b*x] + 5*b^3*x^3*(13*Sqrt[a + b*x] - 27*Sqrt[c + b*x]) + 27*b^2*c*x^2*(Sqrt[a + b*x] - Sqrt[c + b*x])) + 5*(8*c^4*Sqrt[c + b*x] - 4*b*c^3*x*Sqrt[c + b*x] + 3*b^2*c^2*x^2*Sqrt[c + b*x] + b^3*c*x^3*(27*Sqrt[a + b*x] - 13*Sqrt[c + b*x]) + 28*b^4*x^4*(Sqrt[a + b*x] - Sqrt[c + b*x]))))/(315*b^3*(a - c)^3)","A",1
412,1,214,261,0.2602246,"\int \frac{x}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^3} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]","\frac{2 \left(6 a^3 \sqrt{a+b x}-a^2 \sqrt{a+b x} (3 b x+14 c)+20 b^3 x^3 \left(\sqrt{a+b x}-\sqrt{b x+c}\right)+a \left(b^2 x^2 \left(11 \sqrt{a+b x}-21 \sqrt{b x+c}\right)+7 b c x \left(\sqrt{a+b x}-\sqrt{b x+c}\right)+14 c^2 \sqrt{b x+c}\right)+b^2 c x^2 \left(21 \sqrt{a+b x}-11 \sqrt{b x+c}\right)-6 c^3 \sqrt{b x+c}+3 b c^2 x \sqrt{b x+c}\right)}{35 b^2 (a-c)^3}","\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 c^2 (b x+c)^{3/2}}{3 b^2 (a-c)^3}+\frac{8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 (b x+c)^{7/2}}{7 b^2 (a-c)^3}+\frac{16 c (b x+c)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{5/2}}{5 b^2 (a-c)^3}+\frac{2 c (3 a+c) (b x+c)^{3/2}}{3 b^2 (a-c)^3}",1,"(2*(6*a^3*Sqrt[a + b*x] - 6*c^3*Sqrt[c + b*x] + 3*b*c^2*x*Sqrt[c + b*x] - a^2*Sqrt[a + b*x]*(14*c + 3*b*x) + b^2*c*x^2*(21*Sqrt[a + b*x] - 11*Sqrt[c + b*x]) + 20*b^3*x^3*(Sqrt[a + b*x] - Sqrt[c + b*x]) + a*(14*c^2*Sqrt[c + b*x] + b^2*x^2*(11*Sqrt[a + b*x] - 21*Sqrt[c + b*x]) + 7*b*c*x*(Sqrt[a + b*x] - Sqrt[c + b*x]))))/(35*b^2*(a - c)^3)","A",1
413,1,151,64,0.1507123,"\int \frac{1}{\left(\sqrt{a+b x}+\sqrt{c+b x}\right)^3} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[c + b*x])^(-3),x]","\frac{8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac{8 a (a+b x)^{3/2}}{3 b (a-c)^3}-\frac{8 (b x+c)^{5/2}}{5 b (a-c)^3}+\frac{8 c (b x+c)^{3/2}}{3 b (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{3/2}}{3 b (a-c)^3}","\frac{(a-c)^2}{10 b \left(\sqrt{a+b x}+\sqrt{b x+c}\right)^5}-\frac{1}{2 b \left(\sqrt{a+b x}+\sqrt{b x+c}\right)}",1,"(-8*a*(a + b*x)^(3/2))/(3*b*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(3/2))/(3*b*(a - c)^3) + (8*(a + b*x)^(5/2))/(5*b*(a - c)^3) + (8*c*(c + b*x)^(3/2))/(3*b*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(3/2))/(3*b*(a - c)^3) - (8*(c + b*x)^(5/2))/(5*b*(a - c)^3)","B",1
414,1,142,157,0.1944613,"\int \frac{1}{x \left(\sqrt{a+b x}+\sqrt{c+b x}\right)^3} \, dx","Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]","\frac{2 \left(-9 a \sqrt{b x+c}+9 c \sqrt{a+b x}-3 \sqrt{a} (a+3 c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)+3 \sqrt{c} (3 a+c) \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)+7 a \sqrt{a+b x}+4 b x \sqrt{a+b x}-7 c \sqrt{b x+c}-4 b x \sqrt{b x+c}\right)}{3 (a-c)^3}","\frac{8 (a+b x)^{3/2}}{3 (a-c)^3}+\frac{2 (a+3 c) \sqrt{a+b x}}{(a-c)^3}-\frac{8 (b x+c)^{3/2}}{3 (a-c)^3}-\frac{2 (3 a+c) \sqrt{b x+c}}{(a-c)^3}-\frac{2 \sqrt{a} (a+3 c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{(a-c)^3}+\frac{2 \sqrt{c} (3 a+c) \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)}{(a-c)^3}",1,"(2*(7*a*Sqrt[a + b*x] + 9*c*Sqrt[a + b*x] + 4*b*x*Sqrt[a + b*x] - 9*a*Sqrt[c + b*x] - 7*c*Sqrt[c + b*x] - 4*b*x*Sqrt[c + b*x] - 3*Sqrt[a]*(a + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + 3*Sqrt[c]*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]]))/(3*(a - c)^3)","A",1
415,1,187,162,0.5910912,"\int \frac{1}{x^2 \left(\sqrt{a+b x}+\sqrt{c+b x}\right)^3} \, dx","Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]","\frac{b \left(-\frac{(a+3 c) \left(b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{a}+1}\right)+a+b x\right)}{b x \sqrt{a+b x}}+\frac{(3 a+c) \left(b x \sqrt{\frac{b x}{c}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{c}+1}\right)+b x+c\right)}{b x \sqrt{b x+c}}+8 \sqrt{a+b x}-8 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)-8 \sqrt{b x+c}+8 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)\right)}{(a-c)^3}","\frac{8 b \sqrt{a+b x}}{(a-c)^3}-\frac{8 b \sqrt{b x+c}}{(a-c)^3}-\frac{(a+3 c) \sqrt{a+b x}}{x (a-c)^3}+\frac{(3 a+c) \sqrt{b x+c}}{x (a-c)^3}-\frac{3 b (3 a+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a} (a-c)^3}-\frac{3 b (a+3 c) \tanh ^{-1}\left(\frac{\sqrt{b x+c}}{\sqrt{c}}\right)}{\sqrt{c} (c-a)^3}",1,"(b*(8*Sqrt[a + b*x] - 8*Sqrt[c + b*x] - 8*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + 8*Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]] - ((a + 3*c)*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]]))/(b*x*Sqrt[a + b*x]) + ((3*a + c)*(c + b*x + b*x*Sqrt[1 + (b*x)/c]*ArcTanh[Sqrt[1 + (b*x)/c]]))/(b*x*Sqrt[c + b*x])))/(a - c)^3","A",1
416,1,21,21,0.0193365,"\int \frac{1}{\sqrt{x}+\sqrt{1+x}} \, dx","Integrate[(Sqrt[x] + Sqrt[1 + x])^(-1),x]","\frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3}","\frac{2}{3} (x+1)^{3/2}-\frac{2 x^{3/2}}{3}",1,"(-2*x^(3/2))/3 + (2*(1 + x)^(3/2))/3","A",1
417,1,21,21,0.01744,"\int \frac{1}{\sqrt{-1+x}+\sqrt{x}} \, dx","Integrate[(Sqrt[-1 + x] + Sqrt[x])^(-1),x]","\frac{2 x^{3/2}}{3}-\frac{2}{3} (x-1)^{3/2}","\frac{2 x^{3/2}}{3}-\frac{2}{3} (x-1)^{3/2}",1,"(-2*(-1 + x)^(3/2))/3 + (2*x^(3/2))/3","A",1
418,1,23,23,0.0199728,"\int \frac{1}{\sqrt{-1+x}+\sqrt{1+x}} \, dx","Integrate[(Sqrt[-1 + x] + Sqrt[1 + x])^(-1),x]","\frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2}","\frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2}",1,"-1/3*(-1 + x)^(3/2) + (1 + x)^(3/2)/3","A",1
419,1,38,38,0.0527469,"\int x^3 \left(\sqrt{1-x}+\sqrt{1+x}\right)^2 \, dx","Integrate[x^3*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]","\frac{x^4}{2}+\frac{2}{5} \left(1-x^2\right)^{5/2}-\frac{2}{3} \left(1-x^2\right)^{3/2}","\frac{x^4}{2}+\frac{2}{5} \left(1-x^2\right)^{5/2}-\frac{2}{3} \left(1-x^2\right)^{3/2}",1,"x^4/2 - (2*(1 - x^2)^(3/2))/3 + (2*(1 - x^2)^(5/2))/5","A",1
420,1,42,48,0.0421101,"\int x^2 \left(\sqrt{1-x}+\sqrt{1+x}\right)^2 \, dx","Integrate[x^2*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]","\frac{1}{12} \left(-3 \sqrt{1-x^2} x+\left(6 \sqrt{1-x^2}+8\right) x^3+3 \sin ^{-1}(x)\right)","\frac{2 x^3}{3}-\frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} \sqrt{1-x^2} x^3+\frac{1}{4} \sin ^{-1}(x)",1,"(-3*x*Sqrt[1 - x^2] + x^3*(8 + 6*Sqrt[1 - x^2]) + 3*ArcSin[x])/12","A",1
421,1,19,19,0.0223242,"\int x \left(\sqrt{1-x}+\sqrt{1+x}\right)^2 \, dx","Integrate[x*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]","x^2-\frac{2}{3} \left(1-x^2\right)^{3/2}","x^2-\frac{2}{3} \left(1-x^2\right)^{3/2}",1,"x^2 - (2*(1 - x^2)^(3/2))/3","A",1
422,1,18,19,0.0143435,"\int \left(\sqrt{1-x}+\sqrt{1+x}\right)^2 \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^2,x]","x \left(\sqrt{1-x^2}+2\right)+\sin ^{-1}(x)","\sqrt{1-x^2} x+2 x+\sin ^{-1}(x)",1,"x*(2 + Sqrt[1 - x^2]) + ArcSin[x]","A",1
423,1,32,32,0.0398785,"\int \frac{\left(\sqrt{1-x}+\sqrt{1+x}\right)^2}{x} \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^2/x,x]","2 \sqrt{1-x^2}-2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)+2 \log (x)","2 \sqrt{1-x^2}-2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)+2 \log (x)",1,"2*Sqrt[1 - x^2] - 2*ArcTanh[Sqrt[1 - x^2]] + 2*Log[x]","A",1
424,1,22,26,0.027788,"\int \frac{\left(\sqrt{1-x}+\sqrt{1+x}\right)^2}{x^2} \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^2/x^2,x]","-\frac{2 \left(\sqrt{1-x^2}+x \sin ^{-1}(x)+1\right)}{x}","-\frac{2 \sqrt{1-x^2}}{x}-\frac{2}{x}-2 \sin ^{-1}(x)",1,"(-2*(1 + Sqrt[1 - x^2] + x*ArcSin[x]))/x","A",1
425,1,45,34,0.041578,"\int \frac{\left(\sqrt{1-x}+\sqrt{1+x}\right)^2}{x^3} \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^2/x^3,x]","-\frac{1}{x^2 \sqrt{1-x^2}}+\frac{1}{\sqrt{1-x^2}}-\frac{1}{x^2}+\tanh ^{-1}\left(\sqrt{1-x^2}\right)","-\frac{\sqrt{1-x^2}}{x^2}-\frac{1}{x^2}+\tanh ^{-1}\left(\sqrt{1-x^2}\right)",1,"-x^(-2) + 1/Sqrt[1 - x^2] - 1/(x^2*Sqrt[1 - x^2]) + ArcTanh[Sqrt[1 - x^2]]","A",1
426,1,147,147,0.2230139,"\int \frac{x^3}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx","Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]","\frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)}","\frac{2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac{2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac{2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac{4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac{2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac{4 a (a+c x)^{5/2}}{5 c^3 (b-c)}",1,"(2*a^2*(a + b*x)^(3/2))/(3*b^3*(b - c)) - (4*a*(a + b*x)^(5/2))/(5*b^3*(b - c)) + (2*(a + b*x)^(7/2))/(7*b^3*(b - c)) - (2*a^2*(a + c*x)^(3/2))/(3*(b - c)*c^3) + (4*a*(a + c*x)^(5/2))/(5*(b - c)*c^3) - (2*(a + c*x)^(7/2))/(7*(b - c)*c^3)","A",1
427,1,70,95,0.1941255,"\int \frac{x^2}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]","\frac{2 \left(\frac{3 (a+b x)^{5/2}}{b^2}-\frac{5 a (a+b x)^{3/2}}{b^2}-\frac{3 (a+c x)^{5/2}}{c^2}+\frac{5 a (a+c x)^{3/2}}{c^2}\right)}{15 (b-c)}","\frac{2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac{2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac{2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac{2 a (a+c x)^{3/2}}{3 c^2 (b-c)}",1,"(2*((-5*a*(a + b*x)^(3/2))/b^2 + (3*(a + b*x)^(5/2))/b^2 + (5*a*(a + c*x)^(3/2))/c^2 - (3*(a + c*x)^(5/2))/c^2))/(15*(b - c))","A",1
428,1,39,47,0.0874425,"\int \frac{x}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]","\frac{2 \left(\frac{(a+b x)^{3/2}}{b}-\frac{(a+c x)^{3/2}}{c}\right)}{3 (b-c)}","\frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)}",1,"(2*((a + b*x)^(3/2)/b - (a + c*x)^(3/2)/c))/(3*(b - c))","A",1
429,1,75,97,0.047351,"\int \frac{1}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-1),x]","\frac{2 \left(\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)-\sqrt{a+c x}+\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)\right)}{b-c}","\frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{b-c}",1,"(2*(Sqrt[a + b*x] - Sqrt[a + c*x] - Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + Sqrt[a]*ArcTanh[Sqrt[a + c*x]/Sqrt[a]]))/(b - c)","A",1
430,1,135,103,0.2106083,"\int \frac{1}{x \left(\sqrt{a+b x}+\sqrt{a+c x}\right)} \, dx","Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])),x]","\frac{-\frac{a}{\sqrt{a+b x}}-\frac{b x}{\sqrt{a+b x}}-\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{a}+1}\right)}{\sqrt{a+b x}}+\frac{a}{\sqrt{a+c x}}+\frac{c x}{\sqrt{a+c x}}+\frac{c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x}{a}+1}\right)}{\sqrt{a+c x}}}{b x-c x}","-\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{\sqrt{a} (b-c)}",1,"(-(a/Sqrt[a + b*x]) - (b*x)/Sqrt[a + b*x] + a/Sqrt[a + c*x] + (c*x)/Sqrt[a + c*x] - (b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]])/Sqrt[a + b*x] + (c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]])/Sqrt[a + c*x])/(b*x - c*x)","A",1
431,1,75,171,0.0904406,"\int \frac{1}{x^2 \left(\sqrt{a+b x}+\sqrt{a+c x}\right)} \, dx","Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[a + c*x])),x]","\frac{2 c^2 (a+c x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x}{a}+1\right)-2 b^2 (a+b x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x}{a}+1\right)}{3 a^3 (b-c)}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{4 a^{3/2} (b-c)}-\frac{c^2 \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{4 a^{3/2} (b-c)}-\frac{\sqrt{a+b x}}{2 x^2 (b-c)}+\frac{\sqrt{a+c x}}{2 x^2 (b-c)}-\frac{b \sqrt{a+b x}}{4 a x (b-c)}+\frac{c \sqrt{a+c x}}{4 a x (b-c)}",1,"(-2*b^2*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x)/a] + 2*c^2*(a + c*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x)/a])/(3*a^3*(b - c))","C",1
432,1,238,195,0.6658146,"\int \frac{x^3}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^2,x]","\frac{\frac{3 a^4 (c-b)^3 (b+c) \sqrt{\frac{b (a+c x)}{a (b-c)}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a (b-c)}}\right)}{\sqrt{a (b-c)} \sqrt{a+c x}}+b \sqrt{c} \left(a^2 \left(3 b^2-2 b c+3 c^2\right) \sqrt{a+b x} \sqrt{a+c x}+4 b^2 c^2 x^2 \left(-2 \sqrt{a+b x} \sqrt{a+c x}+b x+c x\right)-2 a b c x \left(b \sqrt{a+b x} \sqrt{a+c x}+c \sqrt{a+b x} \sqrt{a+c x}-6 b c x\right)\right)}{12 b^3 c^{5/2} (b-c)^2}","-\frac{a^3 (b+c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right)}{4 b^{5/2} c^{5/2}}+\frac{a^2 (b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 b^2 c^2 (b-c)}+\frac{a (b+c) (a+b x)^{3/2} \sqrt{a+c x}}{2 b^2 c (b-c)^2}+\frac{a x^2}{(b-c)^2}-\frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 b c (b-c)^2}+\frac{x^3 (b+c)}{3 (b-c)^2}",1,"(b*Sqrt[c]*(a^2*(3*b^2 - 2*b*c + 3*c^2)*Sqrt[a + b*x]*Sqrt[a + c*x] + 4*b^2*c^2*x^2*(b*x + c*x - 2*Sqrt[a + b*x]*Sqrt[a + c*x]) - 2*a*b*c*x*(-6*b*c*x + b*Sqrt[a + b*x]*Sqrt[a + c*x] + c*Sqrt[a + b*x]*Sqrt[a + c*x])) + (3*a^4*(-b + c)^3*(b + c)*Sqrt[(b*(a + c*x))/(a*(b - c))]*ArcSinh[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*(b - c)]])/(Sqrt[a*(b - c)]*Sqrt[a + c*x]))/(12*b^3*(b - c)^2*c^(5/2))","A",1
433,1,177,142,0.561109,"\int \frac{x^2}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x])^2,x]","\frac{b \sqrt{c} \left(b c x \left(-2 \sqrt{a+b x} \sqrt{a+c x}+b x+c x\right)-a \left(b \sqrt{a+b x} \sqrt{a+c x}+c \sqrt{a+b x} \sqrt{a+c x}-4 b c x\right)\right)+\frac{(a (b-c))^{5/2} \sqrt{\frac{b (a+c x)}{a (b-c)}} \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a (b-c)}}\right)}{\sqrt{a+c x}}}{2 b^2 c^{3/2} (b-c)^2}","\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right)}{2 b^{3/2} c^{3/2}}+\frac{2 a x}{(b-c)^2}-\frac{a \sqrt{a+b x} \sqrt{a+c x}}{2 b c (b-c)}-\frac{(a+b x)^{3/2} \sqrt{a+c x}}{b (b-c)^2}+\frac{x^2 (b+c)}{2 (b-c)^2}",1,"(b*Sqrt[c]*(b*c*x*(b*x + c*x - 2*Sqrt[a + b*x]*Sqrt[a + c*x]) - a*(-4*b*c*x + b*Sqrt[a + b*x]*Sqrt[a + c*x] + c*Sqrt[a + b*x]*Sqrt[a + c*x])) + ((a*(b - c))^(5/2)*Sqrt[(b*(a + c*x))/(a*(b - c))]*ArcSinh[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*(b - c)]])/Sqrt[a + c*x])/(2*b^2*(b - c)^2*c^(3/2))","A",1
434,1,195,135,0.8587131,"\int \frac{x}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^2,x]","\frac{\frac{2 (b+c) \sqrt{a (b-c)} (a+c x) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a (b-c)}}\right)}{\sqrt{c} \sqrt{\frac{b (a+c x)}{a (b-c)}}}-(b-c) \left(-2 c x \sqrt{a+b x}+b x \sqrt{a+c x}+4 a \sqrt{a+c x} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)-2 a \sqrt{a+b x}+c x \sqrt{a+c x}+2 a \log (x) \sqrt{a+c x}\right)}{(c-b)^3 \sqrt{a+c x}}","-\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{(b-c)^2}+\frac{2 a \log (x)}{(b-c)^2}+\frac{4 a \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)}{(b-c)^2}-\frac{2 a (b+c) \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right)}{\sqrt{b} \sqrt{c} (b-c)^2}+\frac{x (b+c)}{(b-c)^2}",1,"((2*Sqrt[a*(b - c)]*(b + c)*(a + c*x)*ArcSinh[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*(b - c)]])/(Sqrt[c]*Sqrt[(b*(a + c*x))/(a*(b - c))]) - (b - c)*(-2*a*Sqrt[a + b*x] - 2*c*x*Sqrt[a + b*x] + b*x*Sqrt[a + c*x] + c*x*Sqrt[a + c*x] + 4*a*Sqrt[a + c*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]] + 2*a*Sqrt[a + c*x]*Log[x]))/((-b + c)^3*Sqrt[a + c*x])","A",1
435,1,178,138,0.7130158,"\int \frac{1}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-2),x]","\frac{2 c \sqrt{a+b x}+\frac{2 a \left(\sqrt{a+b x}-\sqrt{a+c x}\right)}{x}+(b+c) \log (x) \sqrt{a+c x}-\frac{4 b \sqrt{c} (a+c x) \sinh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a (b-c)}}\right)}{\sqrt{a (b-c)} \sqrt{\frac{b (a+c x)}{a (b-c)}}}+2 (b+c) \sqrt{a+c x} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)}{(b-c)^2 \sqrt{a+c x}}","-\frac{2 a}{x (b-c)^2}+\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{x (b-c)^2}+\frac{2 (b+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)}{(b-c)^2}-\frac{4 \sqrt{b} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right)}{(b-c)^2}+\frac{(b+c) \log (x)}{(b-c)^2}",1,"(2*c*Sqrt[a + b*x] + (2*a*(Sqrt[a + b*x] - Sqrt[a + c*x]))/x - (4*b*Sqrt[c]*(a + c*x)*ArcSinh[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*(b - c)]])/(Sqrt[a*(b - c)]*Sqrt[(b*(a + c*x))/(a*(b - c))]) + 2*(b + c)*Sqrt[a + c*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]] + (b + c)*Sqrt[a + c*x]*Log[x])/((b - c)^2*Sqrt[a + c*x])","A",1
436,1,109,123,0.1452792,"\int \frac{1}{x \left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]","\frac{-2 a^2-x^2 (b-c)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)+2 a \left(\sqrt{a+b x} \sqrt{a+c x}-b x-c x\right)+x (b+c) \sqrt{a+b x} \sqrt{a+c x}}{2 a x^2 (b-c)^2}","\frac{\sqrt{a+b x} (a+c x)^{3/2}}{a x^2 (b-c)^2}-\frac{a}{x^2 (b-c)^2}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a x (b-c)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)}{2 a}-\frac{b+c}{x (b-c)^2}",1,"(-2*a^2 + (b + c)*x*Sqrt[a + b*x]*Sqrt[a + c*x] + 2*a*(-(b*x) - c*x + Sqrt[a + b*x]*Sqrt[a + c*x]) - (b - c)^2*x^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]])/(2*a*(b - c)^2*x^2)","A",1
437,1,153,174,0.1810641,"\int \frac{1}{x^2 \left(\sqrt{a+b x}+\sqrt{a+c x}\right)^2} \, dx","Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[a + c*x])^2),x]","\frac{-8 a^3+a^2 \left(8 \sqrt{a+b x} \sqrt{a+c x}-6 b x-6 c x\right)+x^2 \left(-3 b^2+2 b c-3 c^2\right) \sqrt{a+b x} \sqrt{a+c x}+3 x^3 (b-c)^2 (b+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)+2 a x (b+c) \sqrt{a+b x} \sqrt{a+c x}}{12 a^2 x^3 (b-c)^2}","\frac{2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac{(b+c) \sqrt{a+b x} \sqrt{a+c x}}{4 a^2 x (b-c)}+\frac{(b+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right)}{4 a^2}-\frac{2 a}{3 x^3 (b-c)^2}-\frac{b+c}{2 x^2 (b-c)^2}",1,"(-8*a^3 + 2*a*(b + c)*x*Sqrt[a + b*x]*Sqrt[a + c*x] + (-3*b^2 + 2*b*c - 3*c^2)*x^2*Sqrt[a + b*x]*Sqrt[a + c*x] + a^2*(-6*b*x - 6*c*x + 8*Sqrt[a + b*x]*Sqrt[a + c*x]) + 3*(b - c)^2*(b + c)*x^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a + c*x]])/(12*a^2*(b - c)^2*x^3)","A",1
438,1,271,277,0.3927132,"\int \frac{x^4}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^3} \, dx","Integrate[x^4/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]","\frac{2 \left(8 a^3 \left(b^4 \left(-\sqrt{a+c x}\right)+2 b^3 c \sqrt{a+c x}+c^4 \sqrt{a+b x}-2 b c^3 \sqrt{a+b x}\right)+4 a^2 b c x \left(b^3 \sqrt{a+c x}-2 b^2 c \sqrt{a+c x}-c^3 \sqrt{a+b x}+2 b c^2 \sqrt{a+b x}\right)+5 b^3 c^3 x^3 \left(-3 b \sqrt{a+c x}+3 c \sqrt{a+b x}+b \sqrt{a+b x}-c \sqrt{a+c x}\right)+a b^2 c^2 x^2 \left(-3 b^2 \sqrt{a+c x}+3 c^2 \sqrt{a+b x}+29 b c \left(\sqrt{a+b x}-\sqrt{a+c x}\right)\right)\right)}{35 b^3 c^3 (b-c)^3}","\frac{2 a^2 (b+3 c) (a+b x)^{3/2}}{3 b^3 (b-c)^3}-\frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 a^2 (3 b+c) (a+c x)^{3/2}}{3 c^3 (b-c)^3}+\frac{8 a^2 (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{2 (b+3 c) (a+b x)^{7/2}}{7 b^3 (b-c)^3}-\frac{4 a (b+3 c) (a+b x)^{5/2}}{5 b^3 (b-c)^3}+\frac{8 a (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{7/2}}{7 c^3 (b-c)^3}+\frac{4 a (3 b+c) (a+c x)^{5/2}}{5 c^3 (b-c)^3}-\frac{8 a (a+c x)^{5/2}}{5 c^2 (b-c)^3}",1,"(2*(5*b^3*c^3*x^3*(b*Sqrt[a + b*x] + 3*c*Sqrt[a + b*x] - 3*b*Sqrt[a + c*x] - c*Sqrt[a + c*x]) + 4*a^2*b*c*x*(2*b*c^2*Sqrt[a + b*x] - c^3*Sqrt[a + b*x] + b^3*Sqrt[a + c*x] - 2*b^2*c*Sqrt[a + c*x]) + 8*a^3*(-2*b*c^3*Sqrt[a + b*x] + c^4*Sqrt[a + b*x] - b^4*Sqrt[a + c*x] + 2*b^3*c*Sqrt[a + c*x]) + a*b^2*c^2*x^2*(3*c^2*Sqrt[a + b*x] - 3*b^2*Sqrt[a + c*x] + 29*b*c*(Sqrt[a + b*x] - Sqrt[a + c*x]))))/(35*b^3*(b - c)^3*c^3)","A",1
439,1,120,163,0.442706,"\int \frac{x^3}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^3} \, dx","Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]","\frac{2 \left(\frac{3 (b+3 c) (a+b x)^{5/2}}{b^2}-\frac{5 a (b+3 c) (a+b x)^{3/2}}{b^2}-\frac{3 (3 b+c) (a+c x)^{5/2}}{c^2}+\frac{5 a (3 b+c) (a+c x)^{3/2}}{c^2}+\frac{20 a (a+b x)^{3/2}}{b}-\frac{20 a (a+c x)^{3/2}}{c}\right)}{15 (b-c)^3}","\frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3}",1,"(2*((20*a*(a + b*x)^(3/2))/b - (5*a*(b + 3*c)*(a + b*x)^(3/2))/b^2 + (3*(b + 3*c)*(a + b*x)^(5/2))/b^2 - (20*a*(a + c*x)^(3/2))/c + (5*a*(3*b + c)*(a + c*x)^(3/2))/c^2 - (3*(3*b + c)*(a + c*x)^(5/2))/c^2))/(15*(b - c)^3)","A",1
440,1,119,155,0.2709143,"\int \frac{x^2}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^3} \, dx","Integrate[x^2/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]","\frac{2 \left(-12 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)+12 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)+\frac{(b+3 c) (a+b x)^{3/2}}{b}-\frac{(3 b+c) (a+c x)^{3/2}}{c}+12 a \sqrt{a+b x}-12 a \sqrt{a+c x}\right)}{3 (b-c)^3}","-\frac{8 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{(b-c)^3}+\frac{8 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{(b-c)^3}+\frac{8 a \sqrt{a+b x}}{(b-c)^3}-\frac{8 a \sqrt{a+c x}}{(b-c)^3}+\frac{2 (b+3 c) (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{3/2}}{3 c (b-c)^3}",1,"(2*(12*a*Sqrt[a + b*x] + ((b + 3*c)*(a + b*x)^(3/2))/b - 12*a*Sqrt[a + c*x] - ((3*b + c)*(a + c*x)^(3/2))/c - 12*a^(3/2)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + 12*a^(3/2)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]]))/(3*(b - c)^3)","A",1
441,1,192,157,0.8826197,"\int \frac{x}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^3} \, dx","Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]","\frac{2 \left(-(3 b+c) \sqrt{a+c x}+(b+3 c) \sqrt{a+b x}+\sqrt{a} (3 b+c) \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)-\sqrt{a} (b+3 c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)-\frac{2 a \left(b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{a}+1}\right)+a+b x\right)}{x \sqrt{a+b x}}+\frac{2 a \left(c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x}{a}+1}\right)+a+c x\right)}{x \sqrt{a+c x}}\right)}{(b-c)^3}","-\frac{4 a \sqrt{a+b x}}{x (b-c)^3}+\frac{4 a \sqrt{a+c x}}{x (b-c)^3}+\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}-\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{(b-c)^3}+\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{(b-c)^3}",1,"(2*((b + 3*c)*Sqrt[a + b*x] - (3*b + c)*Sqrt[a + c*x] - Sqrt[a]*(b + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] - (2*a*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]]))/(x*Sqrt[a + b*x]) + Sqrt[a]*(3*b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]] + (2*a*(a + c*x + c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]]))/(x*Sqrt[a + c*x])))/(b - c)^3","A",1
442,1,182,164,0.2664677,"\int \frac{1}{\left(\sqrt{a+b x}+\sqrt{a+c x}\right)^3} \, dx","Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-3),x]","\frac{-\frac{8 b^2 (a+b x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{b x}{a}+1\right)}{a^2}+\frac{8 c^2 (a+c x)^{3/2} \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};\frac{c x}{a}+1\right)}{a^2}-\frac{3 (b+3 c) \left(b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{b x}{a}+1}\right)+a+b x\right)}{x \sqrt{a+b x}}+\frac{3 (3 b+c) \left(c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left(\sqrt{\frac{c x}{a}+1}\right)+a+c x\right)}{x \sqrt{a+c x}}}{3 (b-c)^3}","-\frac{2 a \sqrt{a+b x}}{x^2 (b-c)^3}+\frac{2 a \sqrt{a+c x}}{x^2 (b-c)^3}-\frac{(2 b+3 c) \sqrt{a+b x}}{x (b-c)^3}+\frac{(3 b+2 c) \sqrt{a+c x}}{x (b-c)^3}-\frac{3 b c \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a} (b-c)^3}+\frac{3 b c \tanh ^{-1}\left(\frac{\sqrt{a+c x}}{\sqrt{a}}\right)}{\sqrt{a} (b-c)^3}",1,"((-3*(b + 3*c)*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]]))/(x*Sqrt[a + b*x]) + (3*(3*b + c)*(a + c*x + c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]]))/(x*Sqrt[a + c*x]) - (8*b^2*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x)/a])/a^2 + (8*c^2*(a + c*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (c*x)/a])/a^2)/(3*(b - c)^3)","C",1
443,1,31,31,0.0148102,"\int \sqrt{1-x} \left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx","Integrate[Sqrt[1 - x]*(Sqrt[1 - x] + Sqrt[1 + x]),x]","-\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x)","-\frac{x^2}{2}+\frac{1}{2} \sqrt{1-x^2} x+x+\frac{1}{2} \sin ^{-1}(x)",1,"x - x^2/2 + (x*Sqrt[1 - x^2])/2 + ArcSin[x]/2","A",1
444,1,38,38,0.0396102,"\int x^3 \left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx","Integrate[x^3*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]","-\frac{x^4}{2}-\frac{2}{5} \left(1-x^2\right)^{5/2}+\frac{2}{3} \left(1-x^2\right)^{3/2}","-\frac{x^4}{2}-\frac{2}{5} \left(1-x^2\right)^{5/2}+\frac{2}{3} \left(1-x^2\right)^{3/2}",1,"-1/2*x^4 + (2*(1 - x^2)^(3/2))/3 - (2*(1 - x^2)^(5/2))/5","A",1
445,1,43,48,0.0374543,"\int x^2 \left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx","Integrate[x^2*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]","\frac{1}{12} \left(3 \sqrt{1-x^2} x-\left(\left(6 \sqrt{1-x^2}+8\right) x^3\right)-3 \sin ^{-1}(x)\right)","-\frac{2 x^3}{3}+\frac{1}{4} \sqrt{1-x^2} x-\frac{1}{2} \sqrt{1-x^2} x^3-\frac{1}{4} \sin ^{-1}(x)",1,"(3*x*Sqrt[1 - x^2] - x^3*(8 + 6*Sqrt[1 - x^2]) - 3*ArcSin[x])/12","A",1
446,1,21,21,0.0204798,"\int x \left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx","Integrate[x*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]","\frac{2}{3} \left(1-x^2\right)^{3/2}-x^2","\frac{2}{3} \left(1-x^2\right)^{3/2}-x^2",1,"-x^2 + (2*(1 - x^2)^(3/2))/3","A",1
447,1,21,22,0.0144206,"\int \left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right) \, dx","Integrate[(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]","-x \left(\sqrt{1-x^2}+2\right)-\sin ^{-1}(x)","-\sqrt{1-x^2} x-2 x-\sin ^{-1}(x)",1,"-(x*(2 + Sqrt[1 - x^2])) - ArcSin[x]","A",1
448,1,32,32,0.0286087,"\int \frac{\left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right)}{x} \, dx","Integrate[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x,x]","-2 \sqrt{1-x^2}+2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)-2 \log (x)","-2 \sqrt{1-x^2}+2 \tanh ^{-1}\left(\sqrt{1-x^2}\right)-2 \log (x)",1,"-2*Sqrt[1 - x^2] + 2*ArcTanh[Sqrt[1 - x^2]] - 2*Log[x]","A",1
449,1,22,26,0.0255669,"\int \frac{\left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right)}{x^2} \, dx","Integrate[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x^2,x]","\frac{2 \left(\sqrt{1-x^2}+x \sin ^{-1}(x)+1\right)}{x}","\frac{2 \sqrt{1-x^2}}{x}+\frac{2}{x}+2 \sin ^{-1}(x)",1,"(2*(1 + Sqrt[1 - x^2] + x*ArcSin[x]))/x","A",1
450,1,46,33,0.0390143,"\int \frac{\left(-\sqrt{1-x}-\sqrt{1+x}\right) \left(\sqrt{1-x}+\sqrt{1+x}\right)}{x^3} \, dx","Integrate[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x^3,x]","\frac{1}{x^2 \sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}+\frac{1}{x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)","\frac{\sqrt{1-x^2}}{x^2}+\frac{1}{x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)",1,"x^(-2) - 1/Sqrt[1 - x^2] + 1/(x^2*Sqrt[1 - x^2]) - ArcTanh[Sqrt[1 - x^2]]","A",1
451,1,48,28,0.156432,"\int \frac{\sqrt{1-x}+\sqrt{1+x}}{-\sqrt{1-x}+\sqrt{1+x}} \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])/(-Sqrt[1 - x] + Sqrt[1 + x]),x]","\sqrt{1-x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)+\log (x)+2 \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)+\sin ^{-1}(x)","\sqrt{1-x^2}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)+\log (x)",1,"Sqrt[1 - x^2] + 2*ArcSin[Sqrt[1 - x]/Sqrt[2]] + ArcSin[x] - ArcTanh[Sqrt[1 - x^2]] + Log[x]","A",0
452,1,58,33,0.161671,"\int \frac{-\sqrt{-1+x}+\sqrt{1+x}}{\sqrt{-1+x}+\sqrt{1+x}} \, dx","Integrate[(-Sqrt[-1 + x] + Sqrt[1 + x])/(Sqrt[-1 + x] + Sqrt[1 + x]),x]","\frac{1}{2} \left(x^2-\sqrt{x-1} \sqrt{x+1} x+\frac{2 (x-1) \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)}{\sqrt{-(x-1)^2}}+1\right)","\frac{x^2}{2}-\frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x)",1,"(1 + x^2 - Sqrt[-1 + x]*x*Sqrt[1 + x] + (2*(-1 + x)*ArcSin[Sqrt[1 - x]/Sqrt[2]])/Sqrt[-(-1 + x)^2])/2","A",0
453,1,86,121,0.1160398,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1} \left(a f^2 \, _2F_1\left(2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right)+d^2\right)}{2 d^2 e (n+1)}","\frac{a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right)}{2 d^2 e (n+1)}+\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{2 e (n+1)}",1,"((d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(1 + n)*(d^2 + a*f^2*Hypergeometric2F1[2, 1 + n, 2 + n, (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])/d]))/(2*d^2*e*(1 + n))","A",1
454,1,158,175,0.3195389,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^3 \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3,x]","\frac{-\frac{4 a d^3 f^2}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}+12 a d^2 f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)+\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^4+2 a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^2+8 a d f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{8 e}","-\frac{a d^3 f^2}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{3 a d^2 f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{2 e}+\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^4}{8 e}+\frac{a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^2}{4 e}+\frac{a d f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{e}",1,"((-4*a*d^3*f^2)/(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + 8*a*d*f^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + 2*a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2 + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^4 + 12*a*d^2*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(8*e)","A",1
455,1,128,136,0.1976895,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^2 \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2,x]","\frac{\frac{a d^2 f^2}{f \left(-\sqrt{a+\frac{e^2 x^2}{f^2}}\right)-e x}+\frac{1}{3} \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^3+2 a d f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)+a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{2 e}","-\frac{a d^2 f^2}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^3}{6 e}+\frac{a d f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{e}+\frac{a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{2 e}",1,"((a*d^2*f^2)/(-(e*x) - f*Sqrt[a + (e^2*x^2)/f^2]) + a*f^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3/3 + 2*a*d*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e)","A",1
456,1,81,68,0.0465177,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right) \, dx","Integrate[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2],x]","\frac{1}{2} f x \sqrt{\frac{a f^2+e^2 x^2}{f^2}}+\frac{a f^2 \log \left(e f \sqrt{\frac{a f^2+e^2 x^2}{f^2}}+e^2 x\right)}{2 e}+d x+\frac{e x^2}{2}","\frac{1}{2} f x \sqrt{a+\frac{e^2 x^2}{f^2}}+\frac{a f^2 \tanh ^{-1}\left(\frac{e x}{f \sqrt{a+\frac{e^2 x^2}{f^2}}}\right)}{2 e}+d x+\frac{e x^2}{2}",1,"d*x + (e*x^2)/2 + (f*x*Sqrt[(a*f^2 + e^2*x^2)/f^2])/2 + (a*f^2*Log[e^2*x + e*f*Sqrt[(a*f^2 + e^2*x^2)/f^2]])/(2*e)","A",1
457,1,109,117,0.1449984,"\int \frac{1}{d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-1),x]","\frac{-\frac{a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{d^2}+\left(\frac{a f^2}{d^2}+1\right) \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)+\frac{a f^2}{d \left(f \left(-\sqrt{a+\frac{e^2 x^2}{f^2}}\right)-e x\right)}}{2 e}","-\frac{a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{2 d^2 e}+\frac{\left(\frac{a f^2}{d^2}+1\right) \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}{2 e}-\frac{a f^2}{2 d e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}",1,"((a*f^2)/(d*(-(e*x) - f*Sqrt[a + (e^2*x^2)/f^2])) - (a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/d^2 + (1 + (a*f^2)/d^2)*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e)","A",1
458,1,141,151,0.2899782,"\int \frac{1}{\left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^2} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-2),x]","-\frac{\frac{2 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{d^3}-\frac{2 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}{d^3}+\frac{a f^2}{d^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{\frac{a f^2}{d^2}+1}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e}","-\frac{a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{d^3 e}+\frac{a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}{d^3 e}-\frac{a f^2}{2 d^2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}-\frac{\frac{a f^2}{d^2}+1}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}",1,"-1/2*((a*f^2)/(d^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (1 + (a*f^2)/d^2)/(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + (2*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/d^3 - (2*a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/d^3)/e","A",1
459,1,180,193,0.5747774,"\int \frac{1}{\left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^3} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-3),x]","\frac{-\frac{3 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{d^4}+\frac{3 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}{d^4}+\frac{a f^2}{d^3 \left(f \left(-\sqrt{a+\frac{e^2 x^2}{f^2}}\right)-e x\right)}-\frac{2 a f^2}{d^3 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}-\frac{\frac{a f^2}{d^2}+1}{2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^2}}{2 e}","-\frac{3 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}{2 d^4 e}+\frac{3 a f^2 \log \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}{2 d^4 e}-\frac{a f^2}{2 d^3 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}-\frac{a f^2}{d^3 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)}-\frac{\frac{a f^2}{d^2}+1}{4 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^2}",1,"((a*f^2)/(d^3*(-(e*x) - f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2) - (2*a*f^2)/(d^3*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (3*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/d^4 + (3*a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/d^4)/(2*e)","A",1
460,1,213,225,0.3383058,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^{5/2} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2),x]","\frac{-5 a d^{3/2} f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)-\frac{a d^2 f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}+\frac{2}{7} \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{7/2}+\frac{2}{3} a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}+4 a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e}","-\frac{5 a d^{3/2} f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 e}-\frac{a d^2 f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{7/2}}{7 e}+\frac{a f^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}{3 e}+\frac{2 a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e}",1,"(4*a*d*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]] - (a*d^2*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + (2*a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2))/3 + (2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(7/2))/7 - 5*a*d^(3/2)*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/(2*e)","A",1
461,1,175,183,0.2352151,"\int \left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^{3/2} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2),x]","\frac{\frac{2}{5} \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{5/2}+2 a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}-\frac{a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}-3 a \sqrt{d} f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 e}","\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{5/2}}{5 e}+\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e}-\frac{a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}-\frac{3 a \sqrt{d} f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 e}",1,"(2*a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]] - (a*d*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) + (2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2))/5 - 3*a*Sqrt[d]*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/(2*e)","A",1
462,1,139,147,0.3341176,"\int \sqrt{d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}} \, dx","Integrate[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]],x]","-\frac{\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}-\frac{2}{3} \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}+\frac{a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{2 e}","-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{\left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}{3 e}-\frac{a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 \sqrt{d} e}",1,"-1/2*((a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) - (2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2))/3 + (a*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/Sqrt[d])/e","A",1
463,1,143,147,0.2495999,"\int \frac{1}{\sqrt{d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}}} \, dx","Integrate[1/Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]],x]","\frac{\frac{a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 d^{3/2}}+\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d \left(f \left(-\sqrt{a+\frac{e^2 x^2}{f^2}}\right)-e x\right)}+\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e}","\frac{a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 d^{3/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}+\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{e}",1,"(Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]] + (a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d*(-(e*x) - f*Sqrt[a + (e^2*x^2)/f^2])) + (a*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/(2*d^(3/2)))/e","A",1
464,1,167,158,0.4251016,"\int \frac{1}{\left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^{3/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-3/2),x]","\frac{\frac{3 a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{d^{5/2}}+\frac{-2 d^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)-a f^2 \left(3 f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+3 e x\right)}{d^2 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right) \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}}{2 e}","\frac{3 a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 d^{5/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^2 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}-\frac{\frac{a f^2}{d^2}+1}{e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}",1,"((-2*d^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) - a*f^2*(d + 3*e*x + 3*f*Sqrt[a + (e^2*x^2)/f^2]))/(d^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]) + (3*a*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/d^(5/2))/(2*e)","A",1
465,1,186,199,0.6022611,"\int \frac{1}{\left(d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right)^{5/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-5/2),x]","-\frac{\frac{2 d \left(a f^2+d^2\right)}{3 \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}+\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x}+\frac{4 a f^2}{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{5 a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{\sqrt{d}}}{2 d^3 e}","\frac{5 a f^2 \tanh ^{-1}\left(\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right)}{2 d^{7/2} e}-\frac{a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 d^3 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right)}-\frac{2 a f^2}{d^3 e \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{\frac{a f^2}{d^2}+1}{3 e \left(f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}",1,"-1/2*((2*d*(d^2 + a*f^2))/(3*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(3/2)) + (4*a*f^2)/Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]] + (a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(e*x + f*Sqrt[a + (e^2*x^2)/f^2]) - (5*a*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]]/Sqrt[d]])/Sqrt[d])/(d^3*e)","A",1
466,1,40,41,0.0146925,"\int \sqrt{x-\sqrt{-4+x^2}} \, dx","Integrate[Sqrt[x - Sqrt[-4 + x^2]],x]","\frac{2 x^2-2 \sqrt{x^2-4} x+8}{3 \sqrt{x-\sqrt{x^2-4}}}","\frac{1}{3} \left(x-\sqrt{x^2-4}\right)^{3/2}+\frac{4}{\sqrt{x-\sqrt{x^2-4}}}",1,"(8 + 2*x^2 - 2*x*Sqrt[-4 + x^2])/(3*Sqrt[x - Sqrt[-4 + x^2]])","A",1
467,1,67,69,0.0584728,"\int \sqrt{a x+b \sqrt{c+\frac{a^2 x^2}{b^2}}} \, dx","Integrate[Sqrt[a*x + b*Sqrt[c + (a^2*x^2)/b^2]],x]","\frac{2 \left(a b x \sqrt{\frac{a^2 x^2}{b^2}+c}+a^2 x^2+b^2 (-c)\right)}{3 a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}}","\frac{\left(b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x\right)^{3/2}}{3 a}-\frac{b^2 c}{a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}}",1,"(2*(-(b^2*c) + a^2*x^2 + a*b*x*Sqrt[c + (a^2*x^2)/b^2]))/(3*a*Sqrt[a*x + b*Sqrt[c + (a^2*x^2)/b^2]])","A",1
468,1,35,45,0.1016121,"\int \sqrt{1+\sqrt{1-x^2}} \, dx","Integrate[Sqrt[1 + Sqrt[1 - x^2]],x]","\frac{2 x \left(\sqrt{1-x^2}+2\right)}{3 \sqrt{\sqrt{1-x^2}+1}}","\frac{2 x}{\sqrt{\sqrt{1-x^2}+1}}-\frac{2 x^3}{3 \left(\sqrt{1-x^2}+1\right)^{3/2}}",1,"(2*x*(2 + Sqrt[1 - x^2]))/(3*Sqrt[1 + Sqrt[1 - x^2]])","A",1
469,1,44,41,0.0595764,"\int \sqrt{1+\sqrt{1+x^2}} \, dx","Integrate[Sqrt[1 + Sqrt[1 + x^2]],x]","\frac{2 \left(\sqrt{x^2+1}-1\right) \sqrt{\sqrt{x^2+1}+1} \left(\sqrt{x^2+1}+2\right)}{3 x}","\frac{2 x}{\sqrt{\sqrt{x^2+1}+1}}+\frac{2 x^3}{3 \left(\sqrt{x^2+1}+1\right)^{3/2}}",1,"(2*(-1 + Sqrt[1 + x^2])*Sqrt[1 + Sqrt[1 + x^2]]*(2 + Sqrt[1 + x^2]))/(3*x)","A",1
470,1,44,41,0.0616517,"\int \sqrt{5+\sqrt{25+x^2}} \, dx","Integrate[Sqrt[5 + Sqrt[25 + x^2]],x]","\frac{2 \left(\sqrt{x^2+25}-5\right) \sqrt{\sqrt{x^2+25}+5} \left(\sqrt{x^2+25}+10\right)}{3 x}","\frac{10 x}{\sqrt{\sqrt{x^2+25}+5}}+\frac{2 x^3}{3 \left(\sqrt{x^2+25}+5\right)^{3/2}}",1,"(2*(-5 + Sqrt[25 + x^2])*Sqrt[5 + Sqrt[25 + x^2]]*(10 + Sqrt[25 + x^2]))/(3*x)","A",1
471,1,55,66,0.2259893,"\int \sqrt{a+b \sqrt{\frac{a^2}{b^2}+c x^2}} \, dx","Integrate[Sqrt[a + b*Sqrt[a^2/b^2 + c*x^2]],x]","\frac{2 b x \sqrt{\frac{a^2}{b^2}+c x^2}+4 a x}{3 \sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}}","\frac{2 a x}{\sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}}+\frac{2 b^2 c x^3}{3 \left(b \sqrt{\frac{a^2}{b^2}+c x^2}+a\right)^{3/2}}",1,"(4*a*x + 2*b*x*Sqrt[a^2/b^2 + c*x^2])/(3*Sqrt[a + b*Sqrt[a^2/b^2 + c*x^2]])","A",1
472,1,134,166,0.3163997,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{n+1} \left(\left(4 a e^2 f^2-b^2 f^4\right) \, _2F_1\left(2,n+1;n+2;\frac{2 e \left(d+e x+f \sqrt{a+x \left(\frac{x e^2}{f^2}+b\right)}\right)}{2 d e-b f^2}\right)+\left(b f^2-2 d e\right)^2\right)}{2 e (n+1) \left(b f^2-2 d e\right)^2}","\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{2 e \left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+b x+a}\right)}{2 d e-b f^2}\right)}{2 e (n+1) \left(2 d e-b f^2\right)^2}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{2 e (n+1)}",1,"((d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(1 + n)*((-2*d*e + b*f^2)^2 + (4*a*e^2*f^2 - b^2*f^4)*Hypergeometric2F1[2, 1 + n, 2 + n, (2*e*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]))/(2*d*e - b*f^2)]))/(2*e*(-2*d*e + b*f^2)^2*(1 + n))","A",1
473,1,276,303,0.5520175,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^3 \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^3,x]","\frac{2 e^2 f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^2+4 e f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-\frac{f^2 \left(b^2 f^2-4 a e^2\right) \left(b f^2-2 d e\right)^3}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}-3 \left(b^2 f^2-4 a e^2\right) \left(b f^3-2 d e f\right)^2 \log \left(-2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-b f^2\right)+4 e^4 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^4}{32 e^5}","-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^3}{32 e^5 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{3 f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^2 \log \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}{32 e^5}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right)}{8 e^4}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^2}{16 e^3}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^4}{8 e}",1,"(4*e*f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]) + 2*e^2*f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^2 + 4*e^4*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^4 - (f^2*(-2*d*e + b*f^2)^3*(-4*a*e^2 + b^2*f^2))/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) - 3*(-4*a*e^2 + b^2*f^2)*(-2*d*e*f + b*f^3)^2*Log[-(b*f^2) - 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/(32*e^5)","A",1
474,1,213,237,0.3181476,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^2 \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^2,x]","\frac{6 f^2 \left(b^2 f^2-4 a e^2\right) \left(b f^2-2 d e\right) \log \left(-2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-b f^2\right)+\frac{3 \left(b^2 f^2-4 a e^2\right) \left(b f^3-2 d e f\right)^2}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+6 e f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+8 e^3 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^3}{48 e^4}","-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^2}{16 e^4 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \log \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}{8 e^4}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right)}{8 e^3}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^3}{6 e}",1,"(6*e*f^2*(4*a*e^2 - b^2*f^2)*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]) + 8*e^3*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^3 + (3*(-4*a*e^2 + b^2*f^2)*(-2*d*e*f + b*f^3)^2)/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) + 6*f^2*(-2*d*e + b*f^2)*(-4*a*e^2 + b^2*f^2)*Log[-(b*f^2) - 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/(48*e^4)","A",1
475,1,120,118,0.2068499,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right) \, dx","Integrate[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2],x]","\frac{1}{8} \left(\frac{\left(4 a e^2 f^2-b^2 f^4\right) \log \left(2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2\right)}{e^3}+4 f x \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+\frac{2 b f^3 \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}}{e^2}+8 d x+4 e x^2\right)","\frac{f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{b f^2+2 e^2 x}{2 e f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}\right)}{8 e^3}+\frac{f \left(b f^2+2 e^2 x\right) \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}{4 e^2}+d x+\frac{e x^2}{2}",1,"(8*d*x + 4*e*x^2 + (2*b*f^3*Sqrt[a + x*(b + (e^2*x)/f^2)])/e^2 + 4*f*x*Sqrt[a + x*(b + (e^2*x)/f^2)] + ((4*a*e^2*f^2 - b^2*f^4)*Log[b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/e^3)/8","A",1
476,1,187,215,0.2045288,"\int \frac{1}{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-1),x]","\frac{-\frac{f^2 \left(b^2 f^2-4 a e^2\right) \left(b f^2-2 d e\right)}{e \left(2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2\right)}+\frac{f^2 \left(b^2 f^2-4 a e^2\right) \log \left(-2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-b f^2\right)}{e}+4 \left(a e f^2-b d f^2+d^2 e\right) \log \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)}{2 \left(b f^2-2 d e\right)^2}","-\frac{f^2 \left(4 a e^2-b^2 f^2\right)}{2 e \left(2 d e-b f^2\right) \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \log \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}{2 e \left(2 d e-b f^2\right)^2}+\frac{2 \left(a e f^2-b d f^2+d^2 e\right) \log \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)}{\left(2 d e-b f^2\right)^2}",1,"(-((f^2*(-2*d*e + b*f^2)*(-4*a*e^2 + b^2*f^2))/(e*(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])))) + 4*(d^2*e - b*d*f^2 + a*e*f^2)*Log[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] + (f^2*(-4*a*e^2 + b^2*f^2)*Log[-(b*f^2) - 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/e)/(2*(-2*d*e + b*f^2)^2)","A",1
477,1,237,266,0.3208132,"\int \frac{1}{\left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^2} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-2),x]","-\frac{\frac{f^2 \left(b^2 f^2-4 a e^2\right) \left(b f^2-2 d e\right)}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+2 f^2 \left(b^2 f^2-4 a e^2\right) \log \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)-2 f^2 \left(b^2 f^2-4 a e^2\right) \log \left(-2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-b f^2\right)+\frac{2 \left(2 d e-b f^2\right) \left(a e f^2-b d f^2+d^2 e\right)}{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\left(2 d e-b f^2\right)^3}","\frac{2 f^2 \left(4 a e^2-b^2 f^2\right) \log \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)}{\left(2 d e-b f^2\right)^3}-\frac{f^2 \left(4 a e^2-b^2 f^2\right)}{\left(2 d e-b f^2\right)^2 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}-\frac{2 f^2 \left(4 a e^2-b^2 f^2\right) \log \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}{\left(2 d e-b f^2\right)^3}-\frac{2 \left(a e f^2-b d f^2+d^2 e\right)}{\left(2 d e-b f^2\right)^2 \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)}",1,"-(((2*(2*d*e - b*f^2)*(d^2*e - b*d*f^2 + a*e*f^2))/(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]) + (f^2*(-2*d*e + b*f^2)*(-4*a*e^2 + b^2*f^2))/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) + 2*f^2*(-4*a*e^2 + b^2*f^2)*Log[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] - 2*f^2*(-4*a*e^2 + b^2*f^2)*Log[-(b*f^2) - 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/(2*d*e - b*f^2)^3)","A",1
478,1,300,330,0.6886092,"\int \frac{1}{\left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^3} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-3),x]","-\frac{\frac{2 f^2 \left(b^2 f^2-4 a e^2\right) \left(b f^2-2 d e\right)}{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}+\frac{2 e f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}-6 e f^2 \left(4 a e^2-b^2 f^2\right) \log \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)+6 e f^2 \left(4 a e^2-b^2 f^2\right) \log \left(-2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)-b f^2\right)+\frac{\left(b f^2-2 d e\right)^2 \left(a e f^2-b d f^2+d^2 e\right)}{\left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^2}}{\left(b f^2-2 d e\right)^4}","-\frac{2 f^2 \left(4 a e^2-b^2 f^2\right)}{\left(2 d e-b f^2\right)^3 \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)}+\frac{6 e f^2 \left(4 a e^2-b^2 f^2\right) \log \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)}{\left(2 d e-b f^2\right)^4}-\frac{2 e f^2 \left(4 a e^2-b^2 f^2\right)}{\left(2 d e-b f^2\right)^3 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}-\frac{6 e f^2 \left(4 a e^2-b^2 f^2\right) \log \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}{\left(2 d e-b f^2\right)^4}-\frac{a e f^2-b d f^2+d^2 e}{\left(2 d e-b f^2\right)^2 \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^2}",1,"-((((-2*d*e + b*f^2)^2*(d^2*e - b*d*f^2 + a*e*f^2))/(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^2 + (2*f^2*(-2*d*e + b*f^2)*(-4*a*e^2 + b^2*f^2))/(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]) + (2*e*f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2))/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) - 6*e*f^2*(4*a*e^2 - b^2*f^2)*Log[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] + 6*e*f^2*(4*a*e^2 - b^2*f^2)*Log[-(b*f^2) - 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/(-2*d*e + b*f^2)^4)","A",1
479,1,357,370,1.060579,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^{5/2} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(5/2),x]","\frac{\frac{4}{3} e^2 f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{3/2}+4 e f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}-\frac{5 \sqrt{e} f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{2}}-\frac{\left(4 a e^3 f^2-b^2 e f^4\right) \left(b f^2-2 d e\right)^2 \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+\frac{16}{7} e^4 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{7/2}}{16 e^5}","-\frac{5 f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{16 \sqrt{2} e^{9/2}}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 e^4}-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right)^2 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{16 e^4 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}{12 e^3}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{7/2}}{7 e}",1,"(4*e*f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] + (4*e^2*f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(3/2))/3 + (16*e^4*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(7/2))/7 - ((-2*d*e + b*f^2)^2*(4*a*e^3*f^2 - b^2*e*f^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) - (5*Sqrt[e]*f^2*(2*d*e - b*f^2)^(3/2)*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/Sqrt[2])/(16*e^5)","A",1
480,1,291,302,0.5682422,"\int \left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^{3/2} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(3/2),x]","\frac{2 e f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}-\frac{3 \sqrt{e} f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{2 d e-b f^2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{2}}-\frac{\left(4 a e^3 f^2-b^2 e f^4\right) \left(2 d e-b f^2\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+\frac{8}{5} e^3 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{5/2}}{8 e^4}","-\frac{3 f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{2 d e-b f^2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{8 \sqrt{2} e^{7/2}}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 e^3}-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \left(2 d e-b f^2\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{8 e^3 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{5/2}}{5 e}",1,"(2*e*f^2*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] + (8*e^3*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(5/2))/5 - ((2*d*e - b*f^2)*(4*a*e^3*f^2 - b^2*e*f^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) - (3*Sqrt[e]*f^2*Sqrt[2*d*e - b*f^2]*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/Sqrt[2])/(8*e^4)","A",1
481,1,223,233,0.3926032,"\int \sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}} \, dx","Integrate[Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]],x]","\frac{-\frac{\sqrt{e} f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{4 d e-2 b f^2}}+\frac{\left(b^2 e f^4-4 a e^3 f^2\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+\frac{4}{3} e^2 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{3/2}}{4 e^3}","-\frac{f^2 \left(4 a-\frac{b^2 f^2}{e^2}\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{4 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{4 \sqrt{2} e^{5/2} \sqrt{2 d e-b f^2}}+\frac{\left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}{3 e}",1,"((4*e^2*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(3/2))/3 + ((-4*a*e^3*f^2 + b^2*e*f^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) - (Sqrt[e]*f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/Sqrt[4*d*e - 2*b*f^2])/(4*e^3)","A",1
482,1,238,244,0.5724824,"\int \frac{1}{\sqrt{d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}}} \, dx","Integrate[1/Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]],x]","\frac{f^2 \left(b^2 f^2-4 a e^2\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(2 d e-b f^2\right) \left(2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2\right)}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{2 \sqrt{2} e^{3/2} \left(2 d e-b f^2\right)^{3/2}}+\frac{\sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{e}","-\frac{f^2 \left(4 a e-\frac{b^2 f^2}{e}\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{2 \left(2 d e-b f^2\right) \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{2 \sqrt{2} e^{3/2} \left(2 d e-b f^2\right)^{3/2}}+\frac{\sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{e}",1,"Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]]/e + (f^2*(-4*a*e^2 + b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(2*e*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]))) + (f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/(2*Sqrt[2]*e^(3/2)*(2*d*e - b*f^2)^(3/2))","A",1
483,1,257,269,0.486184,"\int \frac{1}{\left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^{3/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-3/2),x]","\frac{-\frac{2 e^2 \left(4 a e^2 f^2-b^2 f^4\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}+\frac{3 e^{3/2} f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{d e-\frac{b f^2}{2}}}-\frac{8 e^2 \left(a e f^2-b d f^2+d^2 e\right)}{\sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}}{2 e^2 \left(b f^2-2 d e\right)^2}","-\frac{f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left(2 d e-b f^2\right)^2 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{3 f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{2} \sqrt{e} \left(2 d e-b f^2\right)^{5/2}}-\frac{4 \left(a e f^2-b d f^2+d^2 e\right)}{\left(2 d e-b f^2\right)^2 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}",1,"((-8*e^2*(d^2*e - b*d*f^2 + a*e*f^2))/Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] - (2*e^2*(4*a*e^2*f^2 - b^2*f^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) + (3*e^(3/2)*f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/Sqrt[d*e - (b*f^2)/2])/(2*e^2*(-2*d*e + b*f^2)^2)","A",1
484,1,315,335,0.8609722,"\int \frac{1}{\left(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right)^{5/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-5/2),x]","\frac{\frac{8 f^2 \left(b^2 f^2-4 a e^2\right)}{\sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}+\frac{10 \sqrt{2} \sqrt{e} f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\sqrt{2 d e-b f^2}}-\frac{4 \left(4 a e^3 f^2-b^2 e f^4\right) \sqrt{f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x}}{2 e \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+e x\right)+b f^2}-\frac{8 \left(2 d e-b f^2\right) \left(a e f^2-b d f^2+d^2 e\right)}{3 \left(f \sqrt{a+x \left(b+\frac{e^2 x}{f^2}\right)}+d+e x\right)^{3/2}}}{2 \left(2 d e-b f^2\right)^3}","-\frac{4 f^2 \left(4 a e^2-b^2 f^2\right)}{\left(2 d e-b f^2\right)^3 \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}-\frac{2 e f^2 \left(4 a e^2-b^2 f^2\right) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\left(2 d e-b f^2\right)^3 \left(2 e \left(f \sqrt{a+\frac{x \left(b f^2+e^2 x\right)}{f^2}}+e x\right)+b f^2\right)}+\frac{5 \sqrt{2} \sqrt{e} f^2 \left(4 a e^2-b^2 f^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right)}{\left(2 d e-b f^2\right)^{7/2}}-\frac{4 \left(a e f^2-b d f^2+d^2 e\right)}{3 \left(2 d e-b f^2\right)^2 \left(f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right)^{3/2}}",1,"((-8*(2*d*e - b*f^2)*(d^2*e - b*d*f^2 + a*e*f^2))/(3*(d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])^(3/2)) + (8*f^2*(-4*a*e^2 + b^2*f^2))/Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]] - (4*(4*a*e^3*f^2 - b^2*e*f^4)*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])) + (10*Sqrt[2]*Sqrt[e]*f^2*(4*a*e^2 - b^2*f^2)*ArcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)]])/Sqrt[2*d*e - b*f^2]])/Sqrt[2*d*e - b*f^2])/(2*(2*d*e - b*f^2)^3)","A",1
485,1,138,164,0.383106,"\int \left(a+x^2\right)^2 \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^2*(x + Sqrt[a + x^2])^n,x]","\frac{1}{32} \left(\sqrt{a+x^2}+x\right)^{n-5} \left(\frac{a^5}{n-5}+\frac{5 a^4 \left(\sqrt{a+x^2}+x\right)^2}{n-3}+\frac{10 a^3 \left(\sqrt{a+x^2}+x\right)^4}{n-1}+\frac{10 a^2 \left(\sqrt{a+x^2}+x\right)^6}{n+1}+\frac{\left(\sqrt{a+x^2}+x\right)^{10}}{n+5}+\frac{5 a \left(\sqrt{a+x^2}+x\right)^8}{n+3}\right)","-\frac{a^5 \left(\sqrt{a+x^2}+x\right)^{n-5}}{32 (5-n)}-\frac{5 a^4 \left(\sqrt{a+x^2}+x\right)^{n-3}}{32 (3-n)}-\frac{5 a^3 \left(\sqrt{a+x^2}+x\right)^{n-1}}{16 (1-n)}+\frac{5 a^2 \left(\sqrt{a+x^2}+x\right)^{n+1}}{16 (n+1)}+\frac{5 a \left(\sqrt{a+x^2}+x\right)^{n+3}}{32 (n+3)}+\frac{\left(\sqrt{a+x^2}+x\right)^{n+5}}{32 (n+5)}",1,"((x + Sqrt[a + x^2])^(-5 + n)*(a^5/(-5 + n) + (5*a^4*(x + Sqrt[a + x^2])^2)/(-3 + n) + (10*a^3*(x + Sqrt[a + x^2])^4)/(-1 + n) + (10*a^2*(x + Sqrt[a + x^2])^6)/(1 + n) + (5*a*(x + Sqrt[a + x^2])^8)/(3 + n) + (x + Sqrt[a + x^2])^10/(5 + n)))/32","A",1
486,1,92,108,0.1396974,"\int \left(a+x^2\right) \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)*(x + Sqrt[a + x^2])^n,x]","\frac{1}{8} \left(\sqrt{a+x^2}+x\right)^{n-3} \left(\frac{a^3}{n-3}+\frac{3 a^2 \left(\sqrt{a+x^2}+x\right)^2}{n-1}+\frac{\left(\sqrt{a+x^2}+x\right)^6}{n+3}+\frac{3 a \left(\sqrt{a+x^2}+x\right)^4}{n+1}\right)","-\frac{a^3 \left(\sqrt{a+x^2}+x\right)^{n-3}}{8 (3-n)}-\frac{3 a^2 \left(\sqrt{a+x^2}+x\right)^{n-1}}{8 (1-n)}+\frac{3 a \left(\sqrt{a+x^2}+x\right)^{n+1}}{8 (n+1)}+\frac{\left(\sqrt{a+x^2}+x\right)^{n+3}}{8 (n+3)}",1,"((x + Sqrt[a + x^2])^(-3 + n)*(a^3/(-3 + n) + (3*a^2*(x + Sqrt[a + x^2])^2)/(-1 + n) + (3*a*(x + Sqrt[a + x^2])^4)/(1 + n) + (x + Sqrt[a + x^2])^6/(3 + n)))/8","A",1
487,1,43,52,0.0342837,"\int \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[(x + Sqrt[a + x^2])^n,x]","\frac{\left(\sqrt{a+x^2}+x\right)^{n-1} \left((n-1) x \left(\sqrt{a+x^2}+x\right)+a n\right)}{n^2-1}","\frac{\left(\sqrt{a+x^2}+x\right)^{n+1}}{2 (n+1)}-\frac{a \left(\sqrt{a+x^2}+x\right)^{n-1}}{2 (1-n)}",1,"((x + Sqrt[a + x^2])^(-1 + n)*(a*n + (-1 + n)*x*(x + Sqrt[a + x^2])))/(-1 + n^2)","A",1
488,1,61,59,0.0252009,"\int \frac{\left(x+\sqrt{a+x^2}\right)^n}{a+x^2} \, dx","Integrate[(x + Sqrt[a + x^2])^n/(a + x^2),x]","\frac{2 \left(\sqrt{a+x^2}+x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a (n+1)}","\frac{2 \left(\sqrt{a+x^2}+x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a (n+1)}",1,"(2*(x + Sqrt[a + x^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -((x + Sqrt[a + x^2])^2/a)])/(a*(1 + n))","A",1
489,1,61,59,0.0294281,"\int \frac{\left(x+\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^2} \, dx","Integrate[(x + Sqrt[a + x^2])^n/(a + x^2)^2,x]","\frac{8 \left(\sqrt{a+x^2}+x\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+3}{2}+1;-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^3 (n+3)}","\frac{8 \left(\sqrt{a+x^2}+x\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+5}{2};-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^3 (n+3)}",1,"(8*(x + Sqrt[a + x^2])^(3 + n)*Hypergeometric2F1[3, (3 + n)/2, 1 + (3 + n)/2, -((x + Sqrt[a + x^2])^2/a)])/(a^3*(3 + n))","A",1
490,1,150,176,0.3921344,"\int \left(a+x^2\right)^2 \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^2*(x - Sqrt[a + x^2])^n,x]","\frac{1}{32} \left(x-\sqrt{a+x^2}\right)^{n-5} \left(\frac{a^5}{n-5}+\frac{5 a^4 \left(x-\sqrt{a+x^2}\right)^2}{n-3}+\frac{10 a^3 \left(x-\sqrt{a+x^2}\right)^4}{n-1}+\frac{10 a^2 \left(x-\sqrt{a+x^2}\right)^6}{n+1}+\frac{\left(x-\sqrt{a+x^2}\right)^{10}}{n+5}+\frac{5 a \left(x-\sqrt{a+x^2}\right)^8}{n+3}\right)","-\frac{a^5 \left(x-\sqrt{a+x^2}\right)^{n-5}}{32 (5-n)}-\frac{5 a^4 \left(x-\sqrt{a+x^2}\right)^{n-3}}{32 (3-n)}-\frac{5 a^3 \left(x-\sqrt{a+x^2}\right)^{n-1}}{16 (1-n)}+\frac{5 a^2 \left(x-\sqrt{a+x^2}\right)^{n+1}}{16 (n+1)}+\frac{5 a \left(x-\sqrt{a+x^2}\right)^{n+3}}{32 (n+3)}+\frac{\left(x-\sqrt{a+x^2}\right)^{n+5}}{32 (n+5)}",1,"((x - Sqrt[a + x^2])^(-5 + n)*(a^5/(-5 + n) + (5*a^4*(x - Sqrt[a + x^2])^2)/(-3 + n) + (10*a^3*(x - Sqrt[a + x^2])^4)/(-1 + n) + (10*a^2*(x - Sqrt[a + x^2])^6)/(1 + n) + (5*a*(x - Sqrt[a + x^2])^8)/(3 + n) + (x - Sqrt[a + x^2])^10/(5 + n)))/32","A",1
491,1,100,116,0.1346082,"\int \left(a+x^2\right) \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)*(x - Sqrt[a + x^2])^n,x]","\frac{1}{8} \left(x-\sqrt{a+x^2}\right)^{n-3} \left(\frac{a^3}{n-3}+\frac{3 a^2 \left(x-\sqrt{a+x^2}\right)^2}{n-1}+\frac{\left(x-\sqrt{a+x^2}\right)^6}{n+3}+\frac{3 a \left(x-\sqrt{a+x^2}\right)^4}{n+1}\right)","-\frac{a^3 \left(x-\sqrt{a+x^2}\right)^{n-3}}{8 (3-n)}-\frac{3 a^2 \left(x-\sqrt{a+x^2}\right)^{n-1}}{8 (1-n)}+\frac{3 a \left(x-\sqrt{a+x^2}\right)^{n+1}}{8 (n+1)}+\frac{\left(x-\sqrt{a+x^2}\right)^{n+3}}{8 (n+3)}",1,"((x - Sqrt[a + x^2])^(-3 + n)*(a^3/(-3 + n) + (3*a^2*(x - Sqrt[a + x^2])^2)/(-1 + n) + (3*a*(x - Sqrt[a + x^2])^4)/(1 + n) + (x - Sqrt[a + x^2])^6/(3 + n)))/8","A",1
492,1,50,56,0.0609141,"\int \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[(x - Sqrt[a + x^2])^n,x]","\frac{1}{2} \left(x-\sqrt{a+x^2}\right)^{n-1} \left(\frac{\left(x-\sqrt{a+x^2}\right)^2}{n+1}+\frac{a}{n-1}\right)","\frac{\left(x-\sqrt{a+x^2}\right)^{n+1}}{2 (n+1)}-\frac{a \left(x-\sqrt{a+x^2}\right)^{n-1}}{2 (1-n)}",1,"((x - Sqrt[a + x^2])^(-1 + n)*(a/(-1 + n) + (x - Sqrt[a + x^2])^2/(1 + n)))/2","A",1
493,1,65,63,0.0274153,"\int \frac{\left(x-\sqrt{a+x^2}\right)^n}{a+x^2} \, dx","Integrate[(x - Sqrt[a + x^2])^n/(a + x^2),x]","\frac{2 \left(x-\sqrt{a+x^2}\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a (n+1)}","\frac{2 \left(x-\sqrt{a+x^2}\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a (n+1)}",1,"(2*(x - Sqrt[a + x^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a*(1 + n))","A",1
494,1,65,63,0.0265064,"\int \frac{\left(x-\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^2} \, dx","Integrate[(x - Sqrt[a + x^2])^n/(a + x^2)^2,x]","\frac{8 \left(x-\sqrt{a+x^2}\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+3}{2}+1;-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^3 (n+3)}","\frac{8 \left(x-\sqrt{a+x^2}\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+5}{2};-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^3 (n+3)}",1,"(8*(x - Sqrt[a + x^2])^(3 + n)*Hypergeometric2F1[3, (3 + n)/2, 1 + (3 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a^3*(3 + n))","A",1
495,1,157,187,0.3794932,"\int \left(a+x^2\right)^{5/2} \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^(5/2)*(x + Sqrt[a + x^2])^n,x]","\frac{1}{64} \left(\sqrt{a+x^2}+x\right)^n \left(\frac{a^6}{(n-6) \left(\sqrt{a+x^2}+x\right)^6}+\frac{6 a^5}{(n-4) \left(\sqrt{a+x^2}+x\right)^4}+\frac{15 a^4}{(n-2) \left(\sqrt{a+x^2}+x\right)^2}+\frac{20 a^3}{n}+\frac{15 a^2 \left(\sqrt{a+x^2}+x\right)^2}{n+2}+\frac{6 a \left(\sqrt{a+x^2}+x\right)^4}{n+4}+\frac{\left(\sqrt{a+x^2}+x\right)^6}{n+6}\right)","-\frac{a^6 \left(\sqrt{a+x^2}+x\right)^{n-6}}{64 (6-n)}-\frac{3 a^5 \left(\sqrt{a+x^2}+x\right)^{n-4}}{32 (4-n)}-\frac{15 a^4 \left(\sqrt{a+x^2}+x\right)^{n-2}}{64 (2-n)}+\frac{5 a^3 \left(\sqrt{a+x^2}+x\right)^n}{16 n}+\frac{15 a^2 \left(\sqrt{a+x^2}+x\right)^{n+2}}{64 (n+2)}+\frac{3 a \left(\sqrt{a+x^2}+x\right)^{n+4}}{32 (n+4)}+\frac{\left(\sqrt{a+x^2}+x\right)^{n+6}}{64 (n+6)}",1,"((x + Sqrt[a + x^2])^n*((20*a^3)/n + a^6/((-6 + n)*(x + Sqrt[a + x^2])^6) + (6*a^5)/((-4 + n)*(x + Sqrt[a + x^2])^4) + (15*a^4)/((-2 + n)*(x + Sqrt[a + x^2])^2) + (15*a^2*(x + Sqrt[a + x^2])^2)/(2 + n) + (6*a*(x + Sqrt[a + x^2])^4)/(4 + n) + (x + Sqrt[a + x^2])^6/(6 + n)))/64","A",1
496,1,111,131,0.2308518,"\int \left(a+x^2\right)^{3/2} \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^(3/2)*(x + Sqrt[a + x^2])^n,x]","\frac{1}{16} \left(\sqrt{a+x^2}+x\right)^n \left(\frac{a^4}{(n-4) \left(\sqrt{a+x^2}+x\right)^4}+\frac{4 a^3}{(n-2) \left(\sqrt{a+x^2}+x\right)^2}+\frac{6 a^2}{n}+\frac{4 a \left(\sqrt{a+x^2}+x\right)^2}{n+2}+\frac{\left(\sqrt{a+x^2}+x\right)^4}{n+4}\right)","-\frac{a^4 \left(\sqrt{a+x^2}+x\right)^{n-4}}{16 (4-n)}-\frac{a^3 \left(\sqrt{a+x^2}+x\right)^{n-2}}{4 (2-n)}+\frac{3 a^2 \left(\sqrt{a+x^2}+x\right)^n}{8 n}+\frac{a \left(\sqrt{a+x^2}+x\right)^{n+2}}{4 (n+2)}+\frac{\left(\sqrt{a+x^2}+x\right)^{n+4}}{16 (n+4)}",1,"((x + Sqrt[a + x^2])^n*((6*a^2)/n + a^4/((-4 + n)*(x + Sqrt[a + x^2])^4) + (4*a^3)/((-2 + n)*(x + Sqrt[a + x^2])^2) + (4*a*(x + Sqrt[a + x^2])^2)/(2 + n) + (x + Sqrt[a + x^2])^4/(4 + n)))/16","A",1
497,1,65,75,0.0732274,"\int \sqrt{a+x^2} \left(x+\sqrt{a+x^2}\right)^n \, dx","Integrate[Sqrt[a + x^2]*(x + Sqrt[a + x^2])^n,x]","\frac{1}{4} \left(\sqrt{a+x^2}+x\right)^n \left(\frac{a^2}{(n-2) \left(\sqrt{a+x^2}+x\right)^2}+\frac{\left(\sqrt{a+x^2}+x\right)^2}{n+2}+\frac{2 a}{n}\right)","-\frac{a^2 \left(\sqrt{a+x^2}+x\right)^{n-2}}{4 (2-n)}+\frac{a \left(\sqrt{a+x^2}+x\right)^n}{2 n}+\frac{\left(\sqrt{a+x^2}+x\right)^{n+2}}{4 (n+2)}",1,"((x + Sqrt[a + x^2])^n*((2*a)/n + a^2/((-2 + n)*(x + Sqrt[a + x^2])^2) + (x + Sqrt[a + x^2])^2/(2 + n)))/4","A",1
498,1,17,17,0.0069604,"\int \frac{\left(x+\sqrt{a+x^2}\right)^n}{\sqrt{a+x^2}} \, dx","Integrate[(x + Sqrt[a + x^2])^n/Sqrt[a + x^2],x]","\frac{\left(\sqrt{a+x^2}+x\right)^n}{n}","\frac{\left(\sqrt{a+x^2}+x\right)^n}{n}",1,"(x + Sqrt[a + x^2])^n/n","A",1
499,1,61,59,0.0280395,"\int \frac{\left(x+\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^{3/2}} \, dx","Integrate[(x + Sqrt[a + x^2])^n/(a + x^2)^(3/2),x]","\frac{4 \left(\sqrt{a+x^2}+x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+2}{2}+1;-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^2 (n+2)}","\frac{4 \left(\sqrt{a+x^2}+x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^2 (n+2)}",1,"(4*(x + Sqrt[a + x^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, 1 + (2 + n)/2, -((x + Sqrt[a + x^2])^2/a)])/(a^2*(2 + n))","A",1
500,1,61,59,0.0328742,"\int \frac{\left(x+\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^{5/2}} \, dx","Integrate[(x + Sqrt[a + x^2])^n/(a + x^2)^(5/2),x]","\frac{16 \left(\sqrt{a+x^2}+x\right)^{n+4} \, _2F_1\left(4,\frac{n+4}{2};\frac{n+4}{2}+1;-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^4 (n+4)}","\frac{16 \left(\sqrt{a+x^2}+x\right)^{n+4} \, _2F_1\left(4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left(x+\sqrt{x^2+a}\right)^2}{a}\right)}{a^4 (n+4)}",1,"(16*(x + Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, 1 + (4 + n)/2, -((x + Sqrt[a + x^2])^2/a)])/(a^4*(4 + n))","A",1
501,1,173,201,0.4856202,"\int \left(a+x^2\right)^{5/2} \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^(5/2)*(x - Sqrt[a + x^2])^n,x]","\frac{1}{64} \left(x-\sqrt{a+x^2}\right)^n \left(-\frac{a^6}{(n-6) \left(x-\sqrt{a+x^2}\right)^6}-\frac{6 a^5}{(n-4) \left(x-\sqrt{a+x^2}\right)^4}-\frac{15 a^4}{(n-2) \left(x-\sqrt{a+x^2}\right)^2}-\frac{20 a^3}{n}-\frac{15 a^2 \left(x-\sqrt{a+x^2}\right)^2}{n+2}-\frac{6 a \left(x-\sqrt{a+x^2}\right)^4}{n+4}-\frac{\left(x-\sqrt{a+x^2}\right)^6}{n+6}\right)","\frac{a^6 \left(x-\sqrt{a+x^2}\right)^{n-6}}{64 (6-n)}+\frac{3 a^5 \left(x-\sqrt{a+x^2}\right)^{n-4}}{32 (4-n)}+\frac{15 a^4 \left(x-\sqrt{a+x^2}\right)^{n-2}}{64 (2-n)}-\frac{5 a^3 \left(x-\sqrt{a+x^2}\right)^n}{16 n}-\frac{15 a^2 \left(x-\sqrt{a+x^2}\right)^{n+2}}{64 (n+2)}-\frac{3 a \left(x-\sqrt{a+x^2}\right)^{n+4}}{32 (n+4)}-\frac{\left(x-\sqrt{a+x^2}\right)^{n+6}}{64 (n+6)}",1,"((x - Sqrt[a + x^2])^n*((-20*a^3)/n - a^6/((-6 + n)*(x - Sqrt[a + x^2])^6) - (6*a^5)/((-4 + n)*(x - Sqrt[a + x^2])^4) - (15*a^4)/((-2 + n)*(x - Sqrt[a + x^2])^2) - (15*a^2*(x - Sqrt[a + x^2])^2)/(2 + n) - (6*a*(x - Sqrt[a + x^2])^4)/(4 + n) - (x - Sqrt[a + x^2])^6/(6 + n)))/64","A",1
502,1,123,141,0.2531885,"\int \left(a+x^2\right)^{3/2} \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[(a + x^2)^(3/2)*(x - Sqrt[a + x^2])^n,x]","\frac{1}{16} \left(x-\sqrt{a+x^2}\right)^n \left(-\frac{a^4}{(n-4) \left(x-\sqrt{a+x^2}\right)^4}-\frac{4 a^3}{(n-2) \left(x-\sqrt{a+x^2}\right)^2}-\frac{6 a^2}{n}-\frac{4 a \left(x-\sqrt{a+x^2}\right)^2}{n+2}-\frac{\left(x-\sqrt{a+x^2}\right)^4}{n+4}\right)","\frac{a^4 \left(x-\sqrt{a+x^2}\right)^{n-4}}{16 (4-n)}+\frac{a^3 \left(x-\sqrt{a+x^2}\right)^{n-2}}{4 (2-n)}-\frac{3 a^2 \left(x-\sqrt{a+x^2}\right)^n}{8 n}-\frac{a \left(x-\sqrt{a+x^2}\right)^{n+2}}{4 (n+2)}-\frac{\left(x-\sqrt{a+x^2}\right)^{n+4}}{16 (n+4)}",1,"((x - Sqrt[a + x^2])^n*((-6*a^2)/n - a^4/((-4 + n)*(x - Sqrt[a + x^2])^4) - (4*a^3)/((-2 + n)*(x - Sqrt[a + x^2])^2) - (4*a*(x - Sqrt[a + x^2])^2)/(2 + n) - (x - Sqrt[a + x^2])^4/(4 + n)))/16","A",1
503,1,73,81,0.0728645,"\int \sqrt{a+x^2} \left(x-\sqrt{a+x^2}\right)^n \, dx","Integrate[Sqrt[a + x^2]*(x - Sqrt[a + x^2])^n,x]","\frac{1}{4} \left(x-\sqrt{a+x^2}\right)^n \left(-\frac{a^2}{(n-2) \left(x-\sqrt{a+x^2}\right)^2}-\frac{\left(x-\sqrt{a+x^2}\right)^2}{n+2}-\frac{2 a}{n}\right)","\frac{a^2 \left(x-\sqrt{a+x^2}\right)^{n-2}}{4 (2-n)}-\frac{a \left(x-\sqrt{a+x^2}\right)^n}{2 n}-\frac{\left(x-\sqrt{a+x^2}\right)^{n+2}}{4 (n+2)}",1,"((x - Sqrt[a + x^2])^n*((-2*a)/n - a^2/((-2 + n)*(x - Sqrt[a + x^2])^2) - (x - Sqrt[a + x^2])^2/(2 + n)))/4","A",1
504,1,20,20,0.0076722,"\int \frac{\left(x-\sqrt{a+x^2}\right)^n}{\sqrt{a+x^2}} \, dx","Integrate[(x - Sqrt[a + x^2])^n/Sqrt[a + x^2],x]","-\frac{\left(x-\sqrt{a+x^2}\right)^n}{n}","-\frac{\left(x-\sqrt{a+x^2}\right)^n}{n}",1,"-((x - Sqrt[a + x^2])^n/n)","A",1
505,1,65,63,0.0245792,"\int \frac{\left(x-\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^{3/2}} \, dx","Integrate[(x - Sqrt[a + x^2])^n/(a + x^2)^(3/2),x]","-\frac{4 \left(x-\sqrt{a+x^2}\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+2}{2}+1;-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^2 (n+2)}","-\frac{4 \left(x-\sqrt{a+x^2}\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^2 (n+2)}",1,"(-4*(x - Sqrt[a + x^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, 1 + (2 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a^2*(2 + n))","A",1
506,1,65,63,0.0294584,"\int \frac{\left(x-\sqrt{a+x^2}\right)^n}{\left(a+x^2\right)^{5/2}} \, dx","Integrate[(x - Sqrt[a + x^2])^n/(a + x^2)^(5/2),x]","-\frac{16 \left(x-\sqrt{a+x^2}\right)^{n+4} \, _2F_1\left(4,\frac{n+4}{2};\frac{n+4}{2}+1;-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^4 (n+4)}","-\frac{16 \left(x-\sqrt{a+x^2}\right)^{n+4} \, _2F_1\left(4,\frac{n+4}{2};\frac{n+6}{2};-\frac{\left(x-\sqrt{x^2+a}\right)^2}{a}\right)}{a^4 (n+4)}",1,"(-16*(x - Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, 1 + (4 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a^4*(4 + n))","A",1
507,1,280,365,2.9126715,"\int \left(a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right)^2 \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n-5} \left(-\frac{5 \left(d^2-a f^2\right) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^8}{n+3}+\frac{10 \left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^6}{n+1}-\frac{10 \left(d^2-a f^2\right)^3 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^4}{n-1}+\frac{5 \left(d^2-a f^2\right)^4 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n-3}-\frac{\left(d^2-a f^2\right)^5}{n-5}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{10}}{n+5}\right)}{32 e f^4}","\frac{\left(d^2-a f^2\right)^5 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-5}}{32 e f^4 (5-n)}-\frac{5 \left(d^2-a f^2\right)^4 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-3}}{32 e f^4 (3-n)}+\frac{5 \left(d^2-a f^2\right)^3 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-1}}{16 e f^4 (1-n)}+\frac{5 \left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{16 e f^4 (n+1)}-\frac{5 \left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+3}}{32 e f^4 (n+3)}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+5}}{32 e f^4 (n+5)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-5 + n)*(-((d^2 - a*f^2)^5/(-5 + n)) + (5*(d^2 - a*f^2)^4*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2)/(-3 + n) - (10*(d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^4)/(-1 + n) + (10*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^6)/(1 + n) - (5*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^8)/(3 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^10/(5 + n)))/(32*e*f^4)","A",1
508,1,186,239,0.6123117,"\int \left(a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right) \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n-3} \left(-\frac{3 \left(d^2-a f^2\right) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^4}{n+1}+\frac{3 \left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n-1}-\frac{\left(d^2-a f^2\right)^3}{n-3}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^6}{n+3}\right)}{8 e f^2}","\frac{\left(d^2-a f^2\right)^3 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-3}}{8 e f^2 (3-n)}-\frac{3 \left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-1}}{8 e f^2 (1-n)}-\frac{3 \left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{8 e f^2 (n+1)}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+3}}{8 e f^2 (n+3)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-3 + n)*(-((d^2 - a*f^2)^3/(-3 + n)) + (3*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2)/(-1 + n) - (3*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^4)/(1 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^6/(3 + n)))/(8*e*f^2)","A",1
509,1,89,107,0.3919991,"\int \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n-1} \left(\frac{a f^2-d^2}{n-1}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n+1}\right)}{2 e}","\frac{\left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-1}}{2 e (1-n)}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{2 e (n+1)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-1 + n)*((-d^2 + a*f^2)/(-1 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(1 + n)))/(2*e)","A",1
510,1,112,122,0.1433337,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2),x]","-\frac{2 f^2 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left(d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right)^2}{d^2-a f^2}\right)}{e (n+1) \left(d^2-a f^2\right)}","-\frac{2 f^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right)^2}{d^2-a f^2}\right)}{e (n+1) \left(d^2-a f^2\right)}",1,"(-2*f^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))","A",1
511,1,112,122,0.1917741,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{\left(a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right)^2} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2,x]","-\frac{8 f^4 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+5}{2};\frac{\left(d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right)^2}{d^2-a f^2}\right)}{e (n+3) \left(d^2-a f^2\right)^3}","-\frac{8 f^4 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+3} \, _2F_1\left(3,\frac{n+3}{2};\frac{n+5}{2};\frac{\left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right)^2}{d^2-a f^2}\right)}{e (n+3) \left(d^2-a f^2\right)^3}",1,"(-8*f^4*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(3 + n)*Hypergeometric2F1[3, (3 + n)/2, (5 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)^3*(3 + n))","A",1
512,1,89,107,0.0407003,"\int \left(d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right)^n \, dx","Integrate[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n-1} \left(\frac{a f^2-d^2}{n-1}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n+1}\right)}{2 e}","\frac{\left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-1}}{2 e (1-n)}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1}}{2 e (n+1)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-1 + n)*((-d^2 + a*f^2)/(-1 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(1 + n)))/(2*e)","A",1
513,1,112,122,0.0352547,"\int \frac{\left(d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right)^n}{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \, dx","Integrate[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n/(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2),x]","-\frac{2 f^2 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left(d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right)^2}{d^2-a f^2}\right)}{e (n+1) \left(d^2-a f^2\right)}","-\frac{2 f^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\frac{\left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right)^2}{d^2-a f^2}\right)}{e (n+1) \left(d^2-a f^2\right)}",1,"(-2*f^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))","A",1
514,1,228,297,1.1995268,"\int \left(a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right)^{3/2} \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^(3/2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n \left(\frac{\left(d^2-a f^2\right)^4}{(n-4) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^4}-\frac{4 \left(d^2-a f^2\right)^3}{(n-2) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}-\frac{4 \left(d^2-a f^2\right) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n+2}+\frac{6 \left(d^2-a f^2\right)^2}{n}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^4}{n+4}\right)}{16 e f^3}","-\frac{\left(d^2-a f^2\right)^4 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-4}}{16 e f^3 (4-n)}+\frac{\left(d^2-a f^2\right)^3 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-2}}{4 e f^3 (2-n)}+\frac{3 \left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{8 e f^3 n}-\frac{\left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+2}}{4 e f^3 (n+2)}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+4}}{16 e f^3 (n+4)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n*((6*(d^2 - a*f^2)^2)/n + (d^2 - a*f^2)^4/((-4 + n)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^4) - (4*(d^2 - a*f^2)^3)/((-2 + n)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2) - (4*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2)/(2 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^4/(4 + n)))/(16*e*f^3)","A",1
515,1,135,171,0.3507533,"\int \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n \left(\frac{\left(d^2-a f^2\right)^2}{(n-2) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}+\frac{2 \left(a f^2-d^2\right)}{n}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n+2}\right)}{4 e f}","-\frac{\left(d^2-a f^2\right)^2 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-2}}{4 e f (2-n)}-\frac{\left(d^2-a f^2\right) \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{2 e f n}+\frac{\left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+2}}{4 e f (n+2)}",1,"((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n*((2*(-d^2 + a*f^2))/n + (d^2 - a*f^2)^2/((-2 + n)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(2 + n)))/(4*e*f)","A",1
516,1,36,41,0.0753224,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{\sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2],x]","\frac{f \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n}{e n}","\frac{f \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{e n}",1,"(f*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n)/(e*n)","A",1
517,1,112,122,0.1501767,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{\left(a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right)^{3/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^(3/2),x]","\frac{4 f^3 \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\frac{\left(d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right)^2}{d^2-a f^2}\right)}{e (n+2) \left(d^2-a f^2\right)^2}","\frac{4 f^3 \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\frac{\left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right)^2}{d^2-a f^2}\right)}{e (n+2) \left(d^2-a f^2\right)^2}",1,"(4*f^3*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)^2*(2 + n))","A",1
518,1,36,41,0.0336675,"\int \frac{\left(d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right)^n}{\sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n/Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2],x]","\frac{f \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n}{e n}","\frac{f \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{e n}",1,"(f*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n)/(e*n)","A",1
519,1,175,327,0.2344731,"\int \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n \, dx","Integrate[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]","\frac{\sqrt{g \left(a+\frac{e x (2 d+e x)}{f^2}\right)} \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n \left(\frac{\left(d^2-a f^2\right)^2}{(n-2) \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}+\frac{2 \left(a f^2-d^2\right)}{n}+\frac{\left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^2}{n+2}\right)}{4 e f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}}","-\frac{\left(d^2-a f^2\right)^2 \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n-2}}{4 e f (2-n) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}-\frac{\left(d^2-a f^2\right) \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{2 e f n \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}+\frac{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+2}}{4 e f (n+2) \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}}",1,"(Sqrt[g*(a + (e*x*(2*d + e*x))/f^2)]*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n*((2*(-d^2 + a*f^2))/n + (d^2 - a*f^2)^2/((-2 + n)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(2 + n)))/(4*e*f*Sqrt[a + (e*x*(2*d + e*x))/f^2])","A",1
520,1,76,93,0.1001603,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2],x]","\frac{f \sqrt{a+\frac{e x (2 d+e x)}{f^2}} \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n}{e n \sqrt{g \left(a+\frac{e x (2 d+e x)}{f^2}\right)}}","\frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}",1,"(f*Sqrt[a + (e*x*(2*d + e*x))/f^2]*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n)/(e*n*Sqrt[g*(a + (e*x*(2*d + e*x))/f^2)])","A",1
521,1,152,177,0.1905815,"\int \frac{\left(d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right)^n}{\left(a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}\right)^{3/2}} \, dx","Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2)^(3/2),x]","\frac{4 f^3 \left(a+\frac{e x (2 d+e x)}{f^2}\right)^{3/2} \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\frac{\left(d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right)^2}{d^2-a f^2}\right)}{e (n+2) \left(d^2-a f^2\right)^2 \left(g \left(a+\frac{e x (2 d+e x)}{f^2}\right)\right)^{3/2}}","\frac{4 f^3 \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^{n+2} \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\frac{\left(d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right)^2}{d^2-a f^2}\right)}{e g (n+2) \left(d^2-a f^2\right)^2 \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}",1,"(4*f^3*(a + (e*x*(2*d + e*x))/f^2)^(3/2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)^2*(2 + n)*(g*(a + (e*x*(2*d + e*x))/f^2))^(3/2))","A",1
522,1,76,93,0.0381861,"\int \frac{\left(d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right)^n}{\sqrt{\frac{a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx","Integrate[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n/Sqrt[(a*f^2*g + e*g*x*(2*d + e*x))/f^2],x]","\frac{f \sqrt{a+\frac{e x (2 d+e x)}{f^2}} \left(f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right)^n}{e n \sqrt{g \left(a+\frac{e x (2 d+e x)}{f^2}\right)}}","\frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left(f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right)^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}",1,"(f*Sqrt[a + (e*x*(2*d + e*x))/f^2]*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^n)/(e*n*Sqrt[g*(a + (e*x*(2*d + e*x))/f^2)])","A",1
523,1,772,191,1.6190893,"\int \frac{1}{(a+b x) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx","Integrate[1/((a + b*x)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]","\frac{2 \sqrt{d} \left(\sqrt{c}+i \sqrt{d} x\right) \left(\sqrt{e}+i \sqrt{f} x\right) \sqrt{\frac{\left(\sqrt{d} x+i \sqrt{c}\right) \left(\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right)}{\left(\sqrt{d} x-i \sqrt{c}\right) \left(\sqrt{c} \sqrt{f}+\sqrt{d} \sqrt{e}\right)}} \sqrt{\frac{\sqrt{c} \sqrt{d} \left(\sqrt{f} x+i \sqrt{e}\right)}{\left(\sqrt{d} x-i \sqrt{c}\right) \left(\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right)}} \left(\left(b \sqrt{c}+i a \sqrt{d}\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right) \left(\sqrt{d} x+i \sqrt{c}\right)}{\left(\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right) \left(\sqrt{d} x-i \sqrt{c}\right)}}\right)|\frac{\left(\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right)^2}{\left(\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right)^2}\right)-2 b \sqrt{c} \Pi \left(\frac{\left(b \sqrt{c}-i a \sqrt{d}\right) \left(\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right)}{\left(i \sqrt{d} a+b \sqrt{c}\right) \left(\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right)};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right) \left(\sqrt{d} x+i \sqrt{c}\right)}{\left(\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right) \left(\sqrt{d} x-i \sqrt{c}\right)}}\right)|\frac{\left(\sqrt{d} \sqrt{e}+\sqrt{c} \sqrt{f}\right)^2}{\left(\sqrt{d} \sqrt{e}-\sqrt{c} \sqrt{f}\right)^2}\right)\right)}{\sqrt{c+d x^2} \sqrt{e+f x^2} \left(b \sqrt{c}-i a \sqrt{d}\right) \left(b \sqrt{c}+i a \sqrt{d}\right) \left(\sqrt{c} \sqrt{f}-\sqrt{d} \sqrt{e}\right) \sqrt{\frac{\sqrt{c} \sqrt{d} \left(\sqrt{e}+i \sqrt{f} x\right)}{\left(\sqrt{c}+i \sqrt{d} x\right) \left(\sqrt{c} \sqrt{f}+\sqrt{d} \sqrt{e}\right)}}}","\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left(-\frac{b^2 c}{a^2 d};\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{-c}}\right)|\frac{c f}{d e}\right)}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c+d x^2} \sqrt{a^2 f+b^2 e}}{\sqrt{e+f x^2} \sqrt{a^2 d+b^2 c}}\right)}{\sqrt{a^2 d+b^2 c} \sqrt{a^2 f+b^2 e}}",1,"(2*Sqrt[d]*(Sqrt[c] + I*Sqrt[d]*x)*Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]*(Sqrt[e] + I*Sqrt[f]*x)*Sqrt[(Sqrt[c]*Sqrt[d]*(I*Sqrt[e] + Sqrt[f]*x))/((-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]*((b*Sqrt[c] + I*a*Sqrt[d])*EllipticF[ArcSin[Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2] - 2*b*Sqrt[c]*EllipticPi[((b*Sqrt[c] - I*a*Sqrt[d])*(Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f]))/((b*Sqrt[c] + I*a*Sqrt[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])), ArcSin[Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2]))/((b*Sqrt[c] - I*a*Sqrt[d])*(b*Sqrt[c] + I*a*Sqrt[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*Sqrt[(Sqrt[c]*Sqrt[d]*(Sqrt[e] + I*Sqrt[f]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*(Sqrt[c] + I*Sqrt[d]*x))]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])","C",1
524,1,191,81,0.1166144,"\int \frac{e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx","Integrate[(e - 2*f*x^2)/(e^2 + 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]","\frac{-\frac{\left(\sqrt{d} \sqrt{d+2 e}-d-2 e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}\right)}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}-\frac{\left(\sqrt{d} \sqrt{d+2 e}+d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}\right)}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{d+2 e}}","\frac{\log \left(2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right)}{4 \sqrt{-d} \sqrt{f}}-\frac{\log \left(-2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right)}{4 \sqrt{-d} \sqrt{f}}",1,"(-(((-d - 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e - Sqrt[d]*Sqrt[d + 2*e]]])/Sqrt[d + e - Sqrt[d]*Sqrt[d + 2*e]]) - ((d + 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]]])/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[d + 2*e]*Sqrt[f])","B",1
525,1,233,73,0.135982,"\int \frac{e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx","Integrate[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]","\frac{-\frac{\left(\sqrt{d} \sqrt{2 e-d}-i d+2 i e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}\right)}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}-\frac{\left(\sqrt{d} \sqrt{2 e-d}+i d-2 i e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{f} x}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}\right)}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{2 e-d}}","\frac{\log \left(2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right)}{4 \sqrt{d} \sqrt{f}}-\frac{\log \left(-2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right)}{4 \sqrt{d} \sqrt{f}}",1,"(-((((-I)*d + (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2*e]]])/Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2*e]]) - ((I*d - (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[-d + e + I*Sqrt[d]*Sqrt[-d + 2*e]]])/Sqrt[-d + e + I*Sqrt[d]*Sqrt[-d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[-d + 2*e]*Sqrt[f])","C",1
526,1,87,38,0.0600854,"\int \frac{e-4 f x^3}{e^2+4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx","Integrate[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]","-\frac{\text{RootSum}\left[4 \text{$\#$1}^6 f^2+4 \text{$\#$1}^3 e f+4 \text{$\#$1}^2 d f+e^2\&,\frac{4 \text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^5 f+3 \text{$\#$1}^2 e+2 \text{$\#$1} d}\&\right]}{4 f}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"-1/4*RootSum[e^2 + 4*d*f*#1^2 + 4*e*f*#1^3 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + 4*f*Log[x - #1]*#1^3)/(2*d*#1 + 3*e*#1^2 + 6*f*#1^5) & ]/f","C",1
527,1,87,38,0.0592985,"\int \frac{e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx","Integrate[(e - 4*f*x^3)/(e^2 - 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]","-\frac{\text{RootSum}\left[4 \text{$\#$1}^6 f^2+4 \text{$\#$1}^3 e f-4 \text{$\#$1}^2 d f+e^2\&,\frac{4 \text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^5 f+3 \text{$\#$1}^2 e-2 \text{$\#$1} d}\&\right]}{4 f}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"-1/4*RootSum[e^2 - 4*d*f*#1^2 + 4*e*f*#1^3 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + 4*f*Log[x - #1]*#1^3)/(-2*d*#1 + 3*e*#1^2 + 6*f*#1^5) & ]/f","C",1
528,0,0,38,0.2671783,"\int \frac{e-2 f (-1+n) x^n}{e^2+4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx","Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 + 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)),x]","\int \frac{e-2 f (-1+n) x^n}{e^2+4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^n}\right)}{2 \sqrt{d} \sqrt{f}}",1,"Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 + 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x]","F",-1
529,0,0,38,0.2705977,"\int \frac{e-2 f (-1+n) x^n}{e^2-4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx","Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)),x]","\int \frac{e-2 f (-1+n) x^n}{e^2-4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^n}\right)}{2 \sqrt{d} \sqrt{f}}",1,"Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x]","F",-1
530,1,42,42,0.0204296,"\int \frac{x}{e^2+4 e f x^2+4 d f x^4+4 f^2 x^4} \, dx","Integrate[x/(e^2 + 4*e*f*x^2 + 4*d*f*x^4 + 4*f^2*x^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} \left(2 x^2 (d+f)+e\right)}{\sqrt{d} e}\right)}{4 \sqrt{d} e \sqrt{f}}","\frac{\tan ^{-1}\left(\frac{\sqrt{f} \left(2 x^2 (d+f)+e\right)}{\sqrt{d} e}\right)}{4 \sqrt{d} e \sqrt{f}}",1,"ArcTan[(Sqrt[f]*(e + 2*(d + f)*x^2))/(Sqrt[d]*e)]/(4*Sqrt[d]*e*Sqrt[f])","A",1
531,1,46,44,0.0224189,"\int \frac{x}{e^2+4 e f x^2-4 d f x^4+4 f^2 x^4} \, dx","Integrate[x/(e^2 + 4*e*f*x^2 - 4*d*f*x^4 + 4*f^2*x^4),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \left(-2 d x^2+e+2 f x^2\right)}{\sqrt{d} e}\right)}{4 \sqrt{d} e \sqrt{f}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \left(e-2 x^2 (d-f)\right)}{\sqrt{d} e}\right)}{4 \sqrt{d} e \sqrt{f}}",1,"-1/4*ArcTanh[(Sqrt[f]*(e - 2*d*x^2 + 2*f*x^2))/(Sqrt[d]*e)]/(Sqrt[d]*e*Sqrt[f])","A",1
532,1,85,40,0.0491326,"\int \frac{x^2 \left(3 e+2 f x^2\right)}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx","Integrate[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^6),x]","\frac{\text{RootSum}\left[4 \text{$\#$1}^6 d f+4 \text{$\#$1}^4 f^2+4 \text{$\#$1}^2 e f+e^2\&,\frac{2 \text{$\#$1}^3 f \log (x-\text{$\#$1})+3 \text{$\#$1} e \log (x-\text{$\#$1})}{3 \text{$\#$1}^4 d+2 \text{$\#$1}^2 f+e}\&\right]}{8 f}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}",1,"RootSum[e^2 + 4*e*f*#1^2 + 4*f^2*#1^4 + 4*d*f*#1^6 & , (3*e*Log[x - #1]*#1 + 2*f*Log[x - #1]*#1^3)/(e + 2*f*#1^2 + 3*d*#1^4) & ]/(8*f)","C",1
533,1,85,40,0.0515284,"\int \frac{x^2 \left(3 e+2 f x^2\right)}{e^2+4 e f x^2+4 f^2 x^4-4 d f x^6} \, dx","Integrate[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6),x]","\frac{\text{RootSum}\left[-4 \text{$\#$1}^6 d f+4 \text{$\#$1}^4 f^2+4 \text{$\#$1}^2 e f+e^2\&,\frac{2 \text{$\#$1}^3 f \log (x-\text{$\#$1})+3 \text{$\#$1} e \log (x-\text{$\#$1})}{-3 \text{$\#$1}^4 d+2 \text{$\#$1}^2 f+e}\&\right]}{8 f}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}",1,"RootSum[e^2 + 4*e*f*#1^2 + 4*f^2*#1^4 - 4*d*f*#1^6 & , (3*e*Log[x - #1]*#1 + 2*f*Log[x - #1]*#1^3)/(e + 2*f*#1^2 - 3*d*#1^4) & ]/(8*f)","C",1
534,1,42,42,0.3150188,"\int \frac{x^m \left(e (1+m)+2 f (-1+m) x^2\right)}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^{2+2 m}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^(2 + 2*m)),x]","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}",1,"ArcTan[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])","A",1
535,1,42,42,0.0420755,"\int \frac{x^m \left(e (1+m)+2 f (-1+m) x^2\right)}{e^2+4 e f x^2+4 f^2 x^4-4 d f x^{2+2 m}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^(2 + 2*m)),x]","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right)}{2 \sqrt{d} \sqrt{f}}",1,"ArcTanh[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])","A",1
536,1,86,40,0.0491521,"\int \frac{x \left(2 e-2 f x^3\right)}{e^2+4 e f x^3+4 d f x^4+4 f^2 x^6} \, dx","Integrate[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 + 4*d*f*x^4 + 4*f^2*x^6),x]","-\frac{\text{RootSum}\left[4 \text{$\#$1}^6 f^2+4 \text{$\#$1}^4 d f+4 \text{$\#$1}^3 e f+e^2\&,\frac{\text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^4 f+4 \text{$\#$1}^2 d+3 \text{$\#$1} e}\&\right]}{2 f}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"-1/2*RootSum[e^2 + 4*e*f*#1^3 + 4*d*f*#1^4 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + f*Log[x - #1]*#1^3)/(3*e*#1 + 4*d*#1^2 + 6*f*#1^4) & ]/f","C",1
537,1,86,40,0.0474782,"\int \frac{x \left(2 e-2 f x^3\right)}{e^2+4 e f x^3-4 d f x^4+4 f^2 x^6} \, dx","Integrate[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 - 4*d*f*x^4 + 4*f^2*x^6),x]","-\frac{\text{RootSum}\left[4 \text{$\#$1}^6 f^2-4 \text{$\#$1}^4 d f+4 \text{$\#$1}^3 e f+e^2\&,\frac{\text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^4 f-4 \text{$\#$1}^2 d+3 \text{$\#$1} e}\&\right]}{2 f}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"-1/2*RootSum[e^2 + 4*e*f*#1^3 - 4*d*f*#1^4 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + f*Log[x - #1]*#1^3)/(3*e*#1 - 4*d*#1^2 + 6*f*#1^4) & ]/f","C",1
538,1,42,42,0.019637,"\int \frac{x^2}{e^2+4 e f x^3+4 d f x^6+4 f^2 x^6} \, dx","Integrate[x^2/(e^2 + 4*e*f*x^3 + 4*d*f*x^6 + 4*f^2*x^6),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{f} \left(2 x^3 (d+f)+e\right)}{\sqrt{d} e}\right)}{6 \sqrt{d} e \sqrt{f}}","\frac{\tan ^{-1}\left(\frac{\sqrt{f} \left(2 x^3 (d+f)+e\right)}{\sqrt{d} e}\right)}{6 \sqrt{d} e \sqrt{f}}",1,"ArcTan[(Sqrt[f]*(e + 2*(d + f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])","A",1
539,1,46,44,0.0221008,"\int \frac{x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx","Integrate[x^2/(e^2 + 4*e*f*x^3 - 4*d*f*x^6 + 4*f^2*x^6),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \left(-2 d x^3+e+2 f x^3\right)}{\sqrt{d} e}\right)}{6 \sqrt{d} e \sqrt{f}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{f} \left(e-2 x^3 (d-f)\right)}{\sqrt{d} e}\right)}{6 \sqrt{d} e \sqrt{f}}",1,"-1/6*ArcTanh[(Sqrt[f]*(e - 2*d*x^3 + 2*f*x^3))/(Sqrt[d]*e)]/(Sqrt[d]*e*Sqrt[f])","A",1
540,1,42,42,0.3174424,"\int \frac{x^m \left(e (1+m)+2 f (-2+m) x^3\right)}{e^2+4 e f x^3+4 f^2 x^6+4 d f x^{2+2 m}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 + 4*d*f*x^(2 + 2*m)),x]","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"ArcTan[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^3)]/(2*Sqrt[d]*Sqrt[f])","A",1
541,1,42,42,0.0449554,"\int \frac{x^m \left(e (1+m)+2 f (-2+m) x^3\right)}{e^2+4 e f x^3+4 f^2 x^6-4 d f x^{2+2 m}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 - 4*d*f*x^(2 + 2*m)),x]","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^3}\right)}{2 \sqrt{d} \sqrt{f}}",1,"ArcTanh[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^3)]/(2*Sqrt[d]*Sqrt[f])","A",1
542,0,0,42,0.4737136,"\int \frac{x^m \left(e (1+m)+2 f (1+m-n) x^n\right)}{e^2+4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)),x]","\int \frac{x^m \left(e (1+m)+2 f (1+m-n) x^n\right)}{e^2+4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx","\frac{\tan ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^n}\right)}{2 \sqrt{d} \sqrt{f}}",1,"Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x]","F",-1
543,0,0,42,0.5028403,"\int \frac{x^m \left(e (1+m)+2 f (1+m-n) x^n\right)}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx","Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)),x]","\int \frac{x^m \left(e (1+m)+2 f (1+m-n) x^n\right)}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx","\frac{\tanh ^{-1}\left(\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^n}\right)}{2 \sqrt{d} \sqrt{f}}",1,"Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x]","F",-1
544,1,126,134,0.2154793,"\int \frac{x^5}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx","Integrate[x^5/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]","\frac{c \left(a \left(20 c^2 d \sqrt{a+b x^2}-6 b c^3 x^2\right)+2 b c d x^2 \left(3 d-2 c \sqrt{a+b x^2}\right)-12 d^3 \sqrt{a+b x^2}+3 b^2 c^3 x^4\right)+12 \left(d^2-a c^2\right)^2 \log \left(c \sqrt{a+b x^2}+d\right)}{12 b^3 c^5}","-\frac{d \left(a+b x^2\right)^{3/2}}{3 b^3 c^2}+\frac{\left(a c^2-d^2\right)^2 \log \left(c \sqrt{a+b x^2}+d\right)}{b^3 c^5}+\frac{d \sqrt{a+b x^2} \left(2 a c^2-d^2\right)}{b^3 c^4}+\frac{\left(a+b x^2\right)^2}{4 b^3 c}-\frac{x^2 \left(2 a c^2-d^2\right)}{2 b^2 c^3}",1,"(c*(3*b^2*c^3*x^4 - 12*d^3*Sqrt[a + b*x^2] + 2*b*c*d*x^2*(3*d - 2*c*Sqrt[a + b*x^2]) + a*(-6*b*c^3*x^2 + 20*c^2*d*Sqrt[a + b*x^2])) + 12*(-(a*c^2) + d^2)^2*Log[d + c*Sqrt[a + b*x^2]])/(12*b^3*c^5)","A",1
545,1,65,69,0.0907641,"\int \frac{x^3}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx","Integrate[x^3/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]","\frac{-\frac{d \sqrt{a+b x^2}}{c^2}-\frac{\left(a c^2-d^2\right) \log \left(c \sqrt{a+b x^2}+d\right)}{c^3}+\frac{b x^2}{2 c}}{b^2}","-\frac{d \sqrt{a+b x^2}}{b^2 c^2}-\frac{\left(a c^2-d^2\right) \log \left(c \sqrt{a+b x^2}+d\right)}{b^2 c^3}+\frac{x^2}{2 b c}",1,"((b*x^2)/(2*c) - (d*Sqrt[a + b*x^2])/c^2 - ((a*c^2 - d^2)*Log[d + c*Sqrt[a + b*x^2]])/c^3)/b^2","A",1
546,1,23,23,0.0243232,"\int \frac{x}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx","Integrate[x/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]","\frac{\log \left(c \sqrt{a+b x^2}+d\right)}{b c}","\frac{\log \left(c \sqrt{a+b x^2}+d\right)}{b c}",1,"Log[d + c*Sqrt[a + b*x^2]]/(b*c)","A",1
547,1,107,88,0.1289532,"\int \frac{1}{x \left(a c+b c x^2+d \sqrt{a+b x^2}\right)} \, dx","Integrate[1/(x*(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])),x]","\frac{\left(\sqrt{a} c-d\right) \log \left(\sqrt{a}-\sqrt{a+b x^2}\right)+\left(\sqrt{a} c+d\right) \log \left(\sqrt{a+b x^2}+\sqrt{a}\right)-2 \sqrt{a} c \log \left(c \sqrt{a+b x^2}+d\right)}{2 \sqrt{a} \left(a c^2-d^2\right)}","-\frac{c \log \left(c \sqrt{a+b x^2}+d\right)}{a c^2-d^2}+\frac{d \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{\sqrt{a} \left(a c^2-d^2\right)}+\frac{c \log (x)}{a c^2-d^2}",1,"((Sqrt[a]*c - d)*Log[Sqrt[a] - Sqrt[a + b*x^2]] + (Sqrt[a]*c + d)*Log[Sqrt[a] + Sqrt[a + b*x^2]] - 2*Sqrt[a]*c*Log[d + c*Sqrt[a + b*x^2]])/(2*Sqrt[a]*(a*c^2 - d^2))","A",1
548,1,291,151,1.0816097,"\int \frac{1}{x^3 \left(a c+b c x^2+d \sqrt{a+b x^2}\right)} \, dx","Integrate[1/(x^3*(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])),x]","\frac{\frac{\sqrt{a} \left(-a^2 c^3 \sqrt{a+b x^2}+a^2 c^2 d+2 a b c^3 x^2 \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{c \sqrt{a+b x^2}}{d}\right)-2 a b c^3 x^2 \log (x) \sqrt{a+b x^2}+b d x^2 \sqrt{\frac{b x^2}{a}+1} \left(a c^2-d^2\right) \tanh ^{-1}\left(\sqrt{\frac{b x^2}{a}+1}\right)+a b c^2 d x^2+a b c^3 x^2 \sqrt{a+b x^2} \log \left(a c^2+b c^2 x^2-d^2\right)+a c d^2 \sqrt{a+b x^2}-a d^3-b d^3 x^2\right)}{x^2 \sqrt{a+b x^2}}+2 b d \left(d^2-2 a c^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2} \left(d^2-a c^2\right)^2}","-\frac{b d \left(3 a c^2-d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right)}{2 a^{3/2} \left(a c^2-d^2\right)^2}-\frac{a c-d \sqrt{a+b x^2}}{2 a x^2 \left(a c^2-d^2\right)}+\frac{b c^3 \log \left(c \sqrt{a+b x^2}+d\right)}{\left(a c^2-d^2\right)^2}-\frac{b c^3 \log (x)}{\left(a c^2-d^2\right)^2}",1,"(2*b*d*(-2*a*c^2 + d^2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]] + (Sqrt[a]*(a^2*c^2*d - a*d^3 + a*b*c^2*d*x^2 - b*d^3*x^2 - a^2*c^3*Sqrt[a + b*x^2] + a*c*d^2*Sqrt[a + b*x^2] + 2*a*b*c^3*x^2*Sqrt[a + b*x^2]*ArcTanh[(c*Sqrt[a + b*x^2])/d] + b*d*(a*c^2 - d^2)*x^2*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]] - 2*a*b*c^3*x^2*Sqrt[a + b*x^2]*Log[x] + a*b*c^3*x^2*Sqrt[a + b*x^2]*Log[a*c^2 - d^2 + b*c^2*x^2]))/(x^2*Sqrt[a + b*x^2]))/(2*a^(3/2)*(-(a*c^2) + d^2)^2)","A",1
549,1,157,147,0.2655035,"\int \frac{x^2}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx","Integrate[x^2/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]","\frac{\sqrt{a c^2-d^2} \left(\sqrt{b} c x-d \log \left(\sqrt{b} \sqrt{a+b x^2}+b x\right)\right)+\left(a c^2-d^2\right) \tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)+\left(d^2-a c^2\right) \tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)}{b^{3/2} c^2 \sqrt{a c^2-d^2}}","\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)}{b^{3/2} c^2}-\frac{\sqrt{a c^2-d^2} \tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)}{b^{3/2} c^2}-\frac{d \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right)}{b^{3/2} c^2}+\frac{x}{b c}",1,"((-(a*c^2) + d^2)*ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]] + (a*c^2 - d^2)*ArcTan[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])] + Sqrt[a*c^2 - d^2]*(Sqrt[b]*c*x - d*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]))/(b^(3/2)*c^2*Sqrt[a*c^2 - d^2])","A",1
550,1,83,103,0.1273738,"\int \frac{1}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx","Integrate[(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)-\tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)}{\sqrt{b} \sqrt{a c^2-d^2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)}{\sqrt{b} \sqrt{a c^2-d^2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)}{\sqrt{b} \sqrt{a c^2-d^2}}",1,"(ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]] - ArcTan[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(Sqrt[b]*Sqrt[a*c^2 - d^2])","A",1
551,1,139,160,0.3382801,"\int \frac{1}{x^2 \left(a c+b c x^2+d \sqrt{a+b x^2}\right)} \, dx","Integrate[1/(x^2*(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])),x]","\frac{\sqrt{a c^2-d^2} \left(d \sqrt{a+b x^2}-a c\right)+a \sqrt{b} c^2 x \tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)-a \sqrt{b} c^2 x \tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)}{a x \left(a c^2-d^2\right)^{3/2}}","\frac{d \sqrt{a+b x^2}}{a x \left(a c^2-d^2\right)}+\frac{\sqrt{b} c^2 \tan ^{-1}\left(\frac{\sqrt{b} d x}{\sqrt{a+b x^2} \sqrt{a c^2-d^2}}\right)}{\left(a c^2-d^2\right)^{3/2}}-\frac{\sqrt{b} c^2 \tan ^{-1}\left(\frac{\sqrt{b} c x}{\sqrt{a c^2-d^2}}\right)}{\left(a c^2-d^2\right)^{3/2}}-\frac{c}{x \left(a c^2-d^2\right)}",1,"(Sqrt[a*c^2 - d^2]*(-(a*c) + d*Sqrt[a + b*x^2]) - a*Sqrt[b]*c^2*x*ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]] + a*Sqrt[b]*c^2*x*ArcTan[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])])/(a*(a*c^2 - d^2)^(3/2)*x)","A",1
552,1,126,140,0.1977654,"\int \frac{x^8}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[x^8/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]","\frac{c \left(a \left(20 c^2 d \sqrt{a+b x^3}-6 b c^3 x^3\right)+2 b c d x^3 \left(3 d-2 c \sqrt{a+b x^3}\right)-12 d^3 \sqrt{a+b x^3}+3 b^2 c^3 x^6\right)+12 \left(d^2-a c^2\right)^2 \log \left(c \sqrt{a+b x^3}+d\right)}{18 b^3 c^5}","-\frac{2 d \left(a+b x^3\right)^{3/2}}{9 b^3 c^2}+\frac{2 \left(a c^2-d^2\right)^2 \log \left(c \sqrt{a+b x^3}+d\right)}{3 b^3 c^5}+\frac{2 d \sqrt{a+b x^3} \left(2 a c^2-d^2\right)}{3 b^3 c^4}+\frac{\left(a+b x^3\right)^2}{6 b^3 c}-\frac{x^3 \left(2 a c^2-d^2\right)}{3 b^2 c^3}",1,"(c*(3*b^2*c^3*x^6 - 12*d^3*Sqrt[a + b*x^3] + 2*b*c*d*x^3*(3*d - 2*c*Sqrt[a + b*x^3]) + a*(-6*b*c^3*x^3 + 20*c^2*d*Sqrt[a + b*x^3])) + 12*(-(a*c^2) + d^2)^2*Log[d + c*Sqrt[a + b*x^3]])/(18*b^3*c^5)","A",1
553,1,63,73,0.0702878,"\int \frac{x^5}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]","\frac{\left(2 d^2-2 a c^2\right) \log \left(c \sqrt{a+b x^3}+d\right)+c \left(b c x^3-2 d \sqrt{a+b x^3}\right)}{3 b^2 c^3}","-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left(a c^2-d^2\right) \log \left(c \sqrt{a+b x^3}+d\right)}{3 b^2 c^3}+\frac{x^3}{3 b c}",1,"(c*(b*c*x^3 - 2*d*Sqrt[a + b*x^3]) + (-2*a*c^2 + 2*d^2)*Log[d + c*Sqrt[a + b*x^3]])/(3*b^2*c^3)","A",1
554,1,26,26,0.0286546,"\int \frac{x^2}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[x^2/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]","\frac{2 \log \left(c \sqrt{a+b x^3}+d\right)}{3 b c}","\frac{2 \log \left(c \sqrt{a+b x^3}+d\right)}{3 b c}",1,"(2*Log[d + c*Sqrt[a + b*x^3]])/(3*b*c)","A",1
555,1,107,93,0.1109155,"\int \frac{1}{x \left(a c+b c x^3+d \sqrt{a+b x^3}\right)} \, dx","Integrate[1/(x*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]","\frac{\left(\sqrt{a} c-d\right) \log \left(\sqrt{a}-\sqrt{a+b x^3}\right)+\left(\sqrt{a} c+d\right) \log \left(\sqrt{a+b x^3}+\sqrt{a}\right)-2 \sqrt{a} c \log \left(c \sqrt{a+b x^3}+d\right)}{3 \sqrt{a} \left(a c^2-d^2\right)}","-\frac{2 c \log \left(c \sqrt{a+b x^3}+d\right)}{3 \left(a c^2-d^2\right)}+\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 \sqrt{a} \left(a c^2-d^2\right)}+\frac{c \log (x)}{a c^2-d^2}",1,"((Sqrt[a]*c - d)*Log[Sqrt[a] - Sqrt[a + b*x^3]] + (Sqrt[a]*c + d)*Log[Sqrt[a] + Sqrt[a + b*x^3]] - 2*Sqrt[a]*c*Log[d + c*Sqrt[a + b*x^3]])/(3*Sqrt[a]*(a*c^2 - d^2))","A",1
556,1,307,154,0.67306,"\int \frac{1}{x^4 \left(a c+b c x^3+d \sqrt{a+b x^3}\right)} \, dx","Integrate[1/(x^4*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]","\frac{\sqrt{a} \left(-a^2 c^3 \sqrt{a+b x^3}+a^2 c^2 d+2 a b c^3 x^3 \sqrt{a+b x^3} \tanh ^{-1}\left(\frac{c \sqrt{a+b x^3}}{d}\right)-3 a b c^3 x^3 \log (x) \sqrt{a+b x^3}+b d x^3 \sqrt{\frac{b x^3}{a}+1} \left(a c^2-d^2\right) \tanh ^{-1}\left(\sqrt{\frac{b x^3}{a}+1}\right)+a b c^2 d x^3+a b c^3 x^3 \sqrt{a+b x^3} \log \left(a c^2+b c^2 x^3-d^2\right)+a c d^2 \sqrt{a+b x^3}-a d^3-b d^3 x^3\right)-2 b d x^3 \sqrt{a+b x^3} \left(2 a c^2-d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2} x^3 \sqrt{a+b x^3} \left(d^2-a c^2\right)^2}","-\frac{b d \left(3 a c^2-d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right)}{3 a^{3/2} \left(a c^2-d^2\right)^2}-\frac{a c-d \sqrt{a+b x^3}}{3 a x^3 \left(a c^2-d^2\right)}+\frac{2 b c^3 \log \left(c \sqrt{a+b x^3}+d\right)}{3 \left(a c^2-d^2\right)^2}-\frac{b c^3 \log (x)}{\left(a c^2-d^2\right)^2}",1,"(-2*b*d*(2*a*c^2 - d^2)*x^3*Sqrt[a + b*x^3]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]] + Sqrt[a]*(a^2*c^2*d - a*d^3 + a*b*c^2*d*x^3 - b*d^3*x^3 - a^2*c^3*Sqrt[a + b*x^3] + a*c*d^2*Sqrt[a + b*x^3] + 2*a*b*c^3*x^3*Sqrt[a + b*x^3]*ArcTanh[(c*Sqrt[a + b*x^3])/d] + b*d*(a*c^2 - d^2)*x^3*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]] - 3*a*b*c^3*x^3*Sqrt[a + b*x^3]*Log[x] + a*b*c^3*x^3*Sqrt[a + b*x^3]*Log[a*c^2 - d^2 + b*c^2*x^3]))/(3*a^(3/2)*(-(a*c^2) + d^2)^2*x^3*Sqrt[a + b*x^3])","A",1
557,1,296,311,0.5468581,"\int \frac{x^3}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[x^3/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]","\frac{\sqrt[3]{a c^2-d^2} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)-2 \sqrt[3]{a c^2-d^2} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)-2 \sqrt{3} \sqrt[3]{a c^2-d^2} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}-1}{\sqrt{3}}\right)+6 \sqrt[3]{b} c^{2/3} x}{6 b^{4/3} c^{5/3}}-\frac{d x^4 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{\sqrt{a+b x^3} \left(4 a c^2-4 d^2\right)}","-\frac{d x^4 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{4 \sqrt{a+b x^3} \left(a c^2-d^2\right)}+\frac{\sqrt[3]{a c^2-d^2} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)}{6 b^{4/3} c^{5/3}}-\frac{\sqrt[3]{a c^2-d^2} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)}{3 b^{4/3} c^{5/3}}+\frac{\sqrt[3]{a c^2-d^2} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}}{\sqrt{3}}\right)}{\sqrt{3} b^{4/3} c^{5/3}}+\frac{x}{b c}",1,"-((d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))])/((4*a*c^2 - 4*d^2)*Sqrt[a + b*x^3])) + (6*b^(1/3)*c^(2/3)*x - 2*Sqrt[3]*(a*c^2 - d^2)^(1/3)*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*(a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + (a*c^2 - d^2)^(1/3)*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*b^(4/3)*c^(5/3))","A",1
558,0,0,304,0.1604942,"\int \frac{x}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[x/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]","\int \frac{x}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","-\frac{d x^2 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{2 \sqrt{a+b x^3} \left(a c^2-d^2\right)}+\frac{\log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)}{6 b^{2/3} \sqrt[3]{c} \sqrt[3]{a c^2-d^2}}-\frac{\log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)}{3 b^{2/3} \sqrt[3]{c} \sqrt[3]{a c^2-d^2}}-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}}{\sqrt{3}}\right)}{\sqrt{3} b^{2/3} \sqrt[3]{c} \sqrt[3]{a c^2-d^2}}",1,"Integrate[x/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]), x]","F",-1
559,0,0,300,0.1403061,"\int \frac{1}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","Integrate[(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])^(-1),x]","\int \frac{1}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx","-\frac{d x \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{\sqrt{a+b x^3} \left(a c^2-d^2\right)}-\frac{\sqrt[3]{c} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)}{6 \sqrt[3]{b} \left(a c^2-d^2\right)^{2/3}}+\frac{\sqrt[3]{c} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)}{3 \sqrt[3]{b} \left(a c^2-d^2\right)^{2/3}}-\frac{\sqrt[3]{c} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}}{\sqrt{3}}\right)}{\sqrt{3} \sqrt[3]{b} \left(a c^2-d^2\right)^{2/3}}",1,"Integrate[(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])^(-1), x]","F",-1
560,1,496,319,0.6883154,"\int \frac{1}{x^2 \left(a c+b c x^3+d \sqrt{a+b x^3}\right)} \, dx","Integrate[1/(x^2*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]","\frac{-6 b^2 c^2 d x^6 \sqrt{\frac{b x^3}{a}+1} \sqrt[3]{a c^2-d^2} F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)+15 b d x^3 \sqrt{\frac{b x^3}{a}+1} \sqrt[3]{a c^2-d^2} \left(a c^2+d^2\right) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)-10 \left(a c^2-d^2\right) \left(a \sqrt[3]{b} c^{5/3} x \sqrt{a+b x^3} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)-6 b d x^3 \sqrt[3]{a c^2-d^2}+6 a c \sqrt{a+b x^3} \sqrt[3]{a c^2-d^2}-2 a \sqrt[3]{b} c^{5/3} x \sqrt{a+b x^3} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)+2 \sqrt{3} a \sqrt[3]{b} c^{5/3} x \sqrt{a+b x^3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}-1}{\sqrt{3}}\right)-6 a d \sqrt[3]{a c^2-d^2}\right)}{60 a x \sqrt{a+b x^3} \left(a c^2-d^2\right)^{7/3}}","\frac{d \sqrt{\frac{b x^3}{a}+1} F_1\left(-\frac{1}{3};\frac{1}{2},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{x \sqrt{a+b x^3} \left(a c^2-d^2\right)}-\frac{\sqrt[3]{b} c^{5/3} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)}{6 \left(a c^2-d^2\right)^{4/3}}+\frac{\sqrt[3]{b} c^{5/3} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)}{3 \left(a c^2-d^2\right)^{4/3}}+\frac{\sqrt[3]{b} c^{5/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}}{\sqrt{3}}\right)}{\sqrt{3} \left(a c^2-d^2\right)^{4/3}}-\frac{c}{x \left(a c^2-d^2\right)}",1,"(15*b*d*(a*c^2 - d^2)^(1/3)*(a*c^2 + d^2)*x^3*Sqrt[1 + (b*x^3)/a]*AppellF1[2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 6*b^2*c^2*d*(a*c^2 - d^2)^(1/3)*x^6*Sqrt[1 + (b*x^3)/a]*AppellF1[5/3, 1/2, 1, 8/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 10*(a*c^2 - d^2)*(-6*a*d*(a*c^2 - d^2)^(1/3) - 6*b*d*(a*c^2 - d^2)^(1/3)*x^3 + 6*a*c*(a*c^2 - d^2)^(1/3)*Sqrt[a + b*x^3] + 2*Sqrt[3]*a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2]))/(60*a*(a*c^2 - d^2)^(7/3)*x*Sqrt[a + b*x^3])","A",0
561,1,604,324,5.2900709,"\int \frac{1}{x^3 \left(a c+b c x^3+d \sqrt{a+b x^3}\right)} \, dx","Integrate[1/(x^3*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]","\frac{b^2 c^2 d x^4 \sqrt{\frac{b x^3}{a}+1} F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{16 a \sqrt{a+b x^3} \left(d^2-a c^2\right)^2}+\frac{2 b d x \left(d^2-5 a c^2\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{\sqrt{a+b x^3} \left(a c^2+b c^2 x^3-d^2\right) \left(3 b x^3 \left(2 a c^2 F_1\left(\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)+\left(a c^2-d^2\right) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)\right)+8 a \left(d^2-a c^2\right) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)\right)}+\frac{-2 a b^{2/3} c^{7/3} x^2 \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)+a b^{2/3} c^{7/3} x^2 \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)-2 \sqrt{3} a b^{2/3} c^{7/3} x^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}-1}{\sqrt{3}}\right)+3 d \sqrt{a+b x^3} \left(a c^2-d^2\right)^{2/3}-3 a c \left(a c^2-d^2\right)^{2/3}}{6 a x^2 \left(a c^2-d^2\right)^{5/3}}","\frac{d \sqrt{\frac{b x^3}{a}+1} F_1\left(-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right)}{2 x^2 \sqrt{a+b x^3} \left(a c^2-d^2\right)}+\frac{b^{2/3} c^{7/3} \log \left(-\sqrt[3]{b} c^{2/3} x \sqrt[3]{a c^2-d^2}+\left(a c^2-d^2\right)^{2/3}+b^{2/3} c^{4/3} x^2\right)}{6 \left(a c^2-d^2\right)^{5/3}}-\frac{b^{2/3} c^{7/3} \log \left(\sqrt[3]{a c^2-d^2}+\sqrt[3]{b} c^{2/3} x\right)}{3 \left(a c^2-d^2\right)^{5/3}}+\frac{b^{2/3} c^{7/3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{b} c^{2/3} x}{\sqrt[3]{a c^2-d^2}}}{\sqrt{3}}\right)}{\sqrt{3} \left(a c^2-d^2\right)^{5/3}}-\frac{c}{2 x^2 \left(a c^2-d^2\right)}",1,"(b^2*c^2*d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))])/(16*a*(-(a*c^2) + d^2)^2*Sqrt[a + b*x^3]) + (2*b*d*(-5*a*c^2 + d^2)*x*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))])/(Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(8*a*(-(a*c^2) + d^2)*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + 3*b*x^3*(2*a*c^2*AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + (a*c^2 - d^2)*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))]))) + (-3*a*c*(a*c^2 - d^2)^(2/3) + 3*d*(a*c^2 - d^2)^(2/3)*Sqrt[a + b*x^3] - 2*Sqrt[3]*a*b^(2/3)*c^(7/3)*x^2*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*a*b^(2/3)*c^(7/3)*x^2*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + a*b^(2/3)*c^(7/3)*x^2*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*a*(a*c^2 - d^2)^(5/3)*x^2)","A",0
562,1,320,135,0.6010187,"\int \frac{1}{a c+b c x^n+d \sqrt{a+b x^n}} \, dx","Integrate[(a*c + b*c*x^n + d*Sqrt[a + b*x^n])^(-1),x]","\frac{c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right)}{a c^2-d^2}-\frac{2 a d (n+1) x \left(a c^2-d^2\right) F_1\left(\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)}{\sqrt{a+b x^n} \left(a c^2+b c^2 x^n-d^2\right) \left(\left(a c^2-d^2\right) \left(2 a (n+1) F_1\left(\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)-b n x^n F_1\left(1+\frac{1}{n};\frac{3}{2},1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)\right)-2 a b c^2 n x^n F_1\left(1+\frac{1}{n};\frac{1}{2},2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)\right)}","\frac{c x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right)}{a c^2-d^2}-\frac{d x \sqrt{\frac{b x^n}{a}+1} F_1\left(\frac{1}{n};\frac{1}{2},1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)}{\left(a c^2-d^2\right) \sqrt{a+b x^n}}",1,"(-2*a*d*(a*c^2 - d^2)*(1 + n)*x*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))])/(Sqrt[a + b*x^n]*(a*c^2 - d^2 + b*c^2*x^n)*(-2*a*b*c^2*n*x^n*AppellF1[1 + n^(-1), 1/2, 2, 2 + n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))] + (a*c^2 - d^2)*(-(b*n*x^n*AppellF1[1 + n^(-1), 3/2, 1, 2 + n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]) + 2*a*(1 + n)*AppellF1[n^(-1), 1/2, 1, 1 + n^(-1), -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]))) + (c*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*c^2*x^n)/(a*c^2 - d^2))])/(a*c^2 - d^2)","B",0
563,1,156,167,0.2902435,"\int \frac{x^m}{a c+b c x^n+d \sqrt{a+b x^n}} \, dx","Integrate[x^m/(a*c + b*c*x^n + d*Sqrt[a + b*x^n]),x]","\frac{x^{m+1} \left(c \sqrt{a+b x^n} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right)-d \sqrt{\frac{b x^n}{a}+1} F_1\left(\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)\right)}{(m+1) \left(a c^2-d^2\right) \sqrt{a+b x^n}}","\frac{c x^{m+1} \, _2F_1\left(1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b c^2 x^n}{a c^2-d^2}\right)}{(m+1) \left(a c^2-d^2\right)}-\frac{d x^{m+1} \sqrt{\frac{b x^n}{a}+1} F_1\left(\frac{m+1}{n};\frac{1}{2},1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{b c^2 x^n}{a c^2-d^2}\right)}{(m+1) \left(a c^2-d^2\right) \sqrt{a+b x^n}}",1,"(x^(1 + m)*(-(d*Sqrt[1 + (b*x^n)/a]*AppellF1[(1 + m)/n, 1/2, 1, (1 + m + n)/n, -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))]) + c*Sqrt[a + b*x^n]*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*c^2*x^n)/(a*c^2 - d^2))]))/((a*c^2 - d^2)*(1 + m)*Sqrt[a + b*x^n])","A",1
564,1,27,27,0.0639875,"\int \frac{x^{-1+n}}{a c+b c x^n+d \sqrt{a+b x^n}} \, dx","Integrate[x^(-1 + n)/(a*c + b*c*x^n + d*Sqrt[a + b*x^n]),x]","\frac{2 \log \left(c \sqrt{a+b x^n}+d\right)}{b c n}","\frac{2 \log \left(c \sqrt{a+b x^n}+d\right)}{b c n}",1,"(2*Log[d + c*Sqrt[a + b*x^n]])/(b*c*n)","A",1
565,1,8,8,0.0028348,"\int \frac{1}{\sqrt{x}+4 x^{3/2}} \, dx","Integrate[(Sqrt[x] + 4*x^(3/2))^(-1),x]","\tan ^{-1}\left(2 \sqrt{x}\right)","\tan ^{-1}\left(2 \sqrt{x}\right)",1,"ArcTan[2*Sqrt[x]]","A",1
566,1,13,13,0.0039315,"\int \frac{1}{\sqrt{x}-x^{5/2}} \, dx","Integrate[(Sqrt[x] - x^(5/2))^(-1),x]","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)",1,"ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]","A",1
567,1,27,27,0.0099854,"\int \frac{1}{-\sqrt[4]{x}+\sqrt{x}} \, dx","Integrate[(-x^(1/4) + Sqrt[x])^(-1),x]","2 \sqrt{x}+4 \sqrt[4]{x}+4 \log \left(1-\sqrt[4]{x}\right)","2 \sqrt{x}+4 \sqrt[4]{x}+4 \log \left(1-\sqrt[4]{x}\right)",1,"4*x^(1/4) + 2*Sqrt[x] + 4*Log[1 - x^(1/4)]","A",1
568,1,32,32,0.0116219,"\int \frac{1}{\sqrt[3]{x}+\sqrt{x}} \, dx","Integrate[(x^(1/3) + Sqrt[x])^(-1),x]","2 \sqrt{x}-3 \sqrt[3]{x}+6 \sqrt[6]{x}-6 \log \left(\sqrt[6]{x}+1\right)","2 \sqrt{x}-3 \sqrt[3]{x}+6 \sqrt[6]{x}-6 \log \left(\sqrt[6]{x}+1\right)",1,"6*x^(1/6) - 3*x^(1/3) + 2*Sqrt[x] - 6*Log[1 + x^(1/6)]","A",1
569,1,25,25,0.008595,"\int \frac{1}{\sqrt[4]{x}+\sqrt{x}} \, dx","Integrate[(x^(1/4) + Sqrt[x])^(-1),x]","2 \sqrt{x}-4 \sqrt[4]{x}+4 \log \left(\sqrt[4]{x}+1\right)","2 \sqrt{x}-4 \sqrt[4]{x}+4 \log \left(\sqrt[4]{x}+1\right)",1,"-4*x^(1/4) + 2*Sqrt[x] + 4*Log[1 + x^(1/4)]","A",1
570,1,18,20,0.0063147,"\int \frac{1}{-\sqrt[3]{x}+x^{2/3}} \, dx","Integrate[(-x^(1/3) + x^(2/3))^(-1),x]","3 \left(\sqrt[3]{x}+\log \left(1-\sqrt[3]{x}\right)\right)","3 \sqrt[3]{x}+3 \log \left(1-\sqrt[3]{x}\right)",1,"3*(x^(1/3) + Log[1 - x^(1/3)])","A",1
571,1,24,62,0.0066552,"\int \frac{1}{\frac{1}{\sqrt[4]{x}}+\sqrt{x}} \, dx","Integrate[(x^(-1/4) + Sqrt[x])^(-1),x]","-2 \sqrt{x} \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-x^{3/4}\right)-1\right)","2 \sqrt{x}+\frac{4}{3} \log \left(\sqrt[4]{x}+1\right)-\frac{2}{3} \log \left(\sqrt{x}-\sqrt[4]{x}+1\right)+\frac{4 \tan ^{-1}\left(\frac{1-2 \sqrt[4]{x}}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-2*Sqrt[x]*(-1 + Hypergeometric2F1[2/3, 1, 5/3, -x^(3/4)])","C",1
572,1,73,73,0.0237684,"\int \frac{1}{\sqrt[4]{x}+\sqrt[3]{x}} \, dx","Integrate[(x^(1/4) + x^(1/3))^(-1),x]","\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt[3]{x}-4 \sqrt[4]{x}+6 \sqrt[6]{x}-12 \sqrt[12]{x}+12 \log \left(\sqrt[12]{x}+1\right)","\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt[3]{x}-4 \sqrt[4]{x}+6 \sqrt[6]{x}-12 \sqrt[12]{x}+12 \log \left(\sqrt[12]{x}+1\right)",1,"-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))/2 + 12*Log[1 + x^(1/12)]","A",1
573,1,130,130,0.0351863,"\int \frac{1}{\frac{1}{\sqrt[3]{x}}+\frac{1}{\sqrt[4]{x}}} \, dx","Integrate[(x^(-1/3) + x^(-1/4))^(-1),x]","\frac{4 x^{5/4}}{5}-\frac{6 x^{7/6}}{7}+\frac{12 x^{13/12}}{13}+\frac{12 x^{11/12}}{11}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}-\frac{3 x^{2/3}}{2}+\frac{12 x^{7/12}}{7}+\frac{12 x^{5/12}}{5}-x-2 \sqrt{x}-3 \sqrt[3]{x}+4 \sqrt[4]{x}-6 \sqrt[6]{x}+12 \sqrt[12]{x}-12 \log \left(\sqrt[12]{x}+1\right)","\frac{4 x^{5/4}}{5}-\frac{6 x^{7/6}}{7}+\frac{12 x^{13/12}}{13}+\frac{12 x^{11/12}}{11}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}-\frac{3 x^{2/3}}{2}+\frac{12 x^{7/12}}{7}+\frac{12 x^{5/12}}{5}-x-2 \sqrt{x}-3 \sqrt[3]{x}+4 \sqrt[4]{x}-6 \sqrt[6]{x}+12 \sqrt[12]{x}-12 \log \left(\sqrt[12]{x}+1\right)",1,"12*x^(1/12) - 6*x^(1/6) + 4*x^(1/4) - 3*x^(1/3) + (12*x^(5/12))/5 - 2*Sqrt[x] + (12*x^(7/12))/7 - (3*x^(2/3))/2 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(11/12))/11 - x + (12*x^(13/12))/13 - (6*x^(7/6))/7 + (4*x^(5/4))/5 - 12*Log[1 + x^(1/12)]","A",1
574,1,22,200,0.0074566,"\int \frac{1}{-\frac{1}{\sqrt[3]{x}}+\sqrt{x}} \, dx","Integrate[(-x^(-1/3) + Sqrt[x])^(-1),x]","-2 \sqrt{x} \left(\, _2F_1\left(\frac{3}{5},1;\frac{8}{5};x^{5/6}\right)-1\right)","2 \sqrt{x}+\frac{6}{5} \log \left(1-\sqrt[6]{x}\right)-\frac{3}{10} \left(1+\sqrt{5}\right) \log \left(2 \sqrt[3]{x}-\sqrt{5} \sqrt[6]{x}+\sqrt[6]{x}+2\right)-\frac{3}{10} \left(1-\sqrt{5}\right) \log \left(2 \sqrt[3]{x}+\sqrt{5} \sqrt[6]{x}+\sqrt[6]{x}+2\right)+\frac{3}{5} \sqrt{2 \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{4 \sqrt[6]{x}-\sqrt{5}+1}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)-\frac{3}{5} \sqrt{2 \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \left(4 \sqrt[6]{x}+\sqrt{5}+1\right)\right)",1,"-2*Sqrt[x]*(-1 + Hypergeometric2F1[3/5, 1, 8/5, x^(5/6)])","C",1
575,1,8,8,0.0018278,"\int \frac{\sqrt{x}}{x+x^2} \, dx","Integrate[Sqrt[x]/(x + x^2),x]","2 \tan ^{-1}\left(\sqrt{x}\right)","2 \tan ^{-1}\left(\sqrt{x}\right)",1,"2*ArcTan[Sqrt[x]]","A",1
576,1,19,19,0.0094012,"\int \frac{x}{4 \sqrt{x}+x} \, dx","Integrate[x/(4*Sqrt[x] + x),x]","x-8 \sqrt{x}+32 \log \left(\sqrt{x}+4\right)","x-8 \sqrt{x}+32 \log \left(\sqrt{x}+4\right)",1,"-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]","A",1
577,1,24,108,0.0070595,"\int \frac{\sqrt{x}}{\sqrt[3]{x}+x} \, dx","Integrate[Sqrt[x]/(x^(1/3) + x),x]","-2 \sqrt{x} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-x^{2/3}\right)-1\right)","2 \sqrt{x}-\frac{3 \log \left(\sqrt[3]{x}-\sqrt{2} \sqrt[6]{x}+1\right)}{2 \sqrt{2}}+\frac{3 \log \left(\sqrt[3]{x}+\sqrt{2} \sqrt[6]{x}+1\right)}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left(1-\sqrt{2} \sqrt[6]{x}\right)}{\sqrt{2}}-\frac{3 \tan ^{-1}\left(\sqrt{2} \sqrt[6]{x}+1\right)}{\sqrt{2}}",1,"-2*Sqrt[x]*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -x^(2/3)])","C",1
578,1,83,76,0.0175699,"\int \frac{\sqrt[3]{x}}{\sqrt[4]{x}+\sqrt{x}} \, dx","Integrate[x^(1/3)/(x^(1/4) + Sqrt[x]),x]","\frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt[3]{x}-12 \sqrt[12]{x}+4 \log \left(\sqrt[12]{x}+1\right)-2 \log \left(\sqrt[6]{x}-\sqrt[12]{x}+1\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[12]{x}-1}{\sqrt{3}}\right)","\frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt[3]{x}-12 \sqrt[12]{x}+6 \log \left(\sqrt[12]{x}+1\right)-2 \log \left(\sqrt[4]{x}+1\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \sqrt[12]{x}}{\sqrt{3}}\right)",1,"-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 + 4*Sqrt[3]*ArcTan[(-1 + 2*x^(1/12))/Sqrt[3]] + 4*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]","A",1
579,1,119,119,0.0320499,"\int \frac{\sqrt{x}}{\sqrt[4]{x}+\sqrt[3]{x}} \, dx","Integrate[Sqrt[x]/(x^(1/4) + x^(1/3)),x]","\frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt[3]{x}-4 \sqrt[4]{x}+6 \sqrt[6]{x}-12 \sqrt[12]{x}+12 \log \left(\sqrt[12]{x}+1\right)","\frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt[3]{x}-4 \sqrt[4]{x}+6 \sqrt[6]{x}-12 \sqrt[12]{x}+12 \log \left(\sqrt[12]{x}+1\right)",1,"-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^(1/12)]","A",1
580,1,29,201,0.0060689,"\int \frac{\sqrt{x}}{-\frac{1}{\sqrt[3]{x}}+\sqrt{x}} \, dx","Integrate[Sqrt[x]/(-x^(-1/3) + Sqrt[x]),x]","-6 \sqrt[6]{x} \, _2F_1\left(\frac{1}{5},1;\frac{6}{5};x^{5/6}\right)+x+6 \sqrt[6]{x}","x+6 \sqrt[6]{x}+\frac{6}{5} \log \left(1-\sqrt[6]{x}\right)-\frac{3}{10} \left(1-\sqrt{5}\right) \log \left(2 \sqrt[3]{x}-\sqrt{5} \sqrt[6]{x}+\sqrt[6]{x}+2\right)-\frac{3}{10} \left(1+\sqrt{5}\right) \log \left(2 \sqrt[3]{x}+\sqrt{5} \sqrt[6]{x}+\sqrt[6]{x}+2\right)-\frac{3}{5} \sqrt{2 \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\frac{4 \sqrt[6]{x}-\sqrt{5}+1}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)-\frac{3}{5} \sqrt{2 \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} \left(4 \sqrt[6]{x}+\sqrt{5}+1\right)\right)",1,"6*x^(1/6) + x - 6*x^(1/6)*Hypergeometric2F1[1/5, 1, 6/5, x^(5/6)]","C",1
581,1,35,36,0.0216727,"\int \frac{\sqrt{b-\frac{a}{x}} x^m}{\sqrt{a-b x}} \, dx","Integrate[(Sqrt[b - a/x]*x^m)/Sqrt[a - b*x],x]","\frac{x^{m+1} \sqrt{b-\frac{a}{x}}}{\left(m+\frac{1}{2}\right) \sqrt{a-b x}}","\frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}}",1,"(Sqrt[b - a/x]*x^(1 + m))/((1/2 + m)*Sqrt[a - b*x])","A",1
582,1,29,29,0.0143621,"\int \frac{\sqrt{b-\frac{a}{x}} x^2}{\sqrt{a-b x}} \, dx","Integrate[(Sqrt[b - a/x]*x^2)/Sqrt[a - b*x],x]","\frac{2 x^3 \sqrt{b-\frac{a}{x}}}{5 \sqrt{a-b x}}","\frac{2 x^3 \sqrt{b-\frac{a}{x}}}{5 \sqrt{a-b x}}",1,"(2*Sqrt[b - a/x]*x^3)/(5*Sqrt[a - b*x])","A",1
583,1,29,29,0.01235,"\int \frac{\sqrt{b-\frac{a}{x}} x}{\sqrt{a-b x}} \, dx","Integrate[(Sqrt[b - a/x]*x)/Sqrt[a - b*x],x]","\frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}}","\frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}}",1,"(2*Sqrt[b - a/x]*x^2)/(3*Sqrt[a - b*x])","A",1
584,1,25,25,0.0108242,"\int \frac{\sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \, dx","Integrate[Sqrt[b - a/x]/Sqrt[a - b*x],x]","\frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}}","\frac{2 x \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}}",1,"(2*Sqrt[b - a/x]*x)/Sqrt[a - b*x]","A",1
585,1,24,24,0.0131292,"\int \frac{\sqrt{b-\frac{a}{x}}}{x \sqrt{a-b x}} \, dx","Integrate[Sqrt[b - a/x]/(x*Sqrt[a - b*x]),x]","-\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}}","-\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}}",1,"(-2*Sqrt[b - a/x])/Sqrt[a - b*x]","A",1
586,1,29,29,0.0109181,"\int \frac{\sqrt{b-\frac{a}{x}}}{x^2 \sqrt{a-b x}} \, dx","Integrate[Sqrt[b - a/x]/(x^2*Sqrt[a - b*x]),x]","-\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}}","-\frac{2 \sqrt{b-\frac{a}{x}}}{3 x \sqrt{a-b x}}",1,"(-2*Sqrt[b - a/x])/(3*x*Sqrt[a - b*x])","A",1
587,0,0,80,0.0645629,"\int \left(a+\frac{b}{x}\right)^m (c+d x)^n \, dx","Integrate[(a + b/x)^m*(c + d*x)^n,x]","\int \left(a+\frac{b}{x}\right)^m (c+d x)^n \, dx","\frac{x \left(a+\frac{b}{x}\right)^m \left(\frac{a x}{b}+1\right)^{-m} (c+d x)^n \left(\frac{d x}{c}+1\right)^{-n} F_1\left(1-m;-m,-n;2-m;-\frac{a x}{b},-\frac{d x}{c}\right)}{1-m}",1,"Integrate[(a + b/x)^m*(c + d*x)^n, x]","F",-1
588,1,112,138,0.0762218,"\int \left(a+\frac{b}{x}\right)^m (c+d x)^2 \, dx","Integrate[(a + b/x)^m*(c + d*x)^2,x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^m \left(a^2 d (m+1) x^2 (2 a (3 c+d x)+b d (m-2))-b \left(6 a^2 c^2+6 a b c d (m-1)+b^2 d^2 \left(m^2-3 m+2\right)\right) \, _2F_1\left(2,m+1;m+2;\frac{b}{a x}+1\right)\right)}{6 a^4 (m+1) x}","\frac{d x^2 \left(a+\frac{b}{x}\right)^{m+1} (6 a c-b d (2-m))}{6 a^2}-\frac{b \left(a+\frac{b}{x}\right)^{m+1} \left(6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left(m^2-3 m+2\right)\right) \, _2F_1\left(2,m+1;m+2;\frac{b}{a x}+1\right)}{6 a^4 (m+1)}+\frac{d^2 x^3 \left(a+\frac{b}{x}\right)^{m+1}}{3 a}",1,"((a + b/x)^m*(b + a*x)*(a^2*d*(1 + m)*x^2*(b*d*(-2 + m) + 2*a*(3*c + d*x)) - b*(6*a^2*c^2 + 6*a*b*c*d*(-1 + m) + b^2*d^2*(2 - 3*m + m^2))*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)]))/(6*a^4*(1 + m)*x)","A",1
589,1,73,79,0.0275422,"\int \left(a+\frac{b}{x}\right)^m (c+d x) \, dx","Integrate[(a + b/x)^m*(c + d*x),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^m \left(a^2 d (m+1) x^2+b (-2 a c-b d (m-1)) \, _2F_1\left(2,m+1;m+2;\frac{b}{a x}+1\right)\right)}{2 a^3 (m+1) x}","\frac{d x^2 \left(a+\frac{b}{x}\right)^{m+1}}{2 a}-\frac{b \left(a+\frac{b}{x}\right)^{m+1} (2 a c-b d (1-m)) \, _2F_1\left(2,m+1;m+2;\frac{b}{a x}+1\right)}{2 a^3 (m+1)}",1,"((a + b/x)^m*(b + a*x)*(a^2*d*(1 + m)*x^2 + b*(-2*a*c - b*d*(-1 + m))*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + b/(a*x)]))/(2*a^3*(1 + m)*x)","A",1
590,1,50,40,0.0173666,"\int \left(a+\frac{b}{x}\right)^m \, dx","Integrate[(a + b/x)^m,x]","-\frac{x \left(a+\frac{b}{x}\right)^m \left(\frac{a x}{b}+1\right)^{-m} \, _2F_1\left(1-m,-m;2-m;-\frac{a x}{b}\right)}{m-1}","-\frac{b \left(a+\frac{b}{x}\right)^{m+1} \, _2F_1\left(2,m+1;m+2;\frac{b}{a x}+1\right)}{a^2 (m+1)}",1,"-(((a + b/x)^m*x*Hypergeometric2F1[1 - m, -m, 2 - m, -((a*x)/b)])/((-1 + m)*(1 + (a*x)/b)^m))","A",1
591,1,97,101,0.0434546,"\int \frac{\left(a+\frac{b}{x}\right)^m}{c+d x} \, dx","Integrate[(a + b/x)^m/(c + d*x),x]","\frac{(a x+b) \left(a+\frac{b}{x}\right)^m \left(a c \, _2F_1\left(1,m+1;m+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)+(b d-a c) \, _2F_1\left(1,m+1;m+2;\frac{b}{a x}+1\right)\right)}{a d (m+1) x (b d-a c)}","\frac{\left(a+\frac{b}{x}\right)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{b}{a x}+1\right)}{a d (m+1)}-\frac{c \left(a+\frac{b}{x}\right)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{d (m+1) (a c-b d)}",1,"((a + b/x)^m*(b + a*x)*(a*c*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(a + b/x))/(a*c - b*d)] + (-(a*c) + b*d)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + b/(a*x)]))/(a*d*(-(a*c) + b*d)*(1 + m)*x)","A",1
592,1,57,56,0.0241655,"\int \frac{\left(a+\frac{b}{x}\right)^m}{(c+d x)^2} \, dx","Integrate[(a + b/x)^m/(c + d*x)^2,x]","-\frac{b \left(a+\frac{b}{x}\right)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{c \left(a+\frac{b}{x}\right)}{b d-a c}\right)}{(m+1) (b d-a c)^2}","-\frac{b \left(a+\frac{b}{x}\right)^{m+1} \, _2F_1\left(2,m+1;m+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{(m+1) (a c-b d)^2}",1,"-((b*(a + b/x)^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((c*(a + b/x))/(-(a*c) + b*d))])/((-(a*c) + b*d)^2*(1 + m)))","A",1
593,1,99,112,0.0756533,"\int \frac{\left(a+\frac{b}{x}\right)^m}{(c+d x)^3} \, dx","Integrate[(a + b/x)^m/(c + d*x)^3,x]","\frac{\left(a+\frac{b}{x}\right)^{m+1} \left(\frac{b (b d (m+1)-2 a c) \, _2F_1\left(2,m+1;m+2;\frac{b c+a x c}{a c x-b d x}\right)}{(m+1) (a c-b d)^2}-\frac{d x^2}{(c+d x)^2}\right)}{2 c (a c-b d)}","-\frac{b \left(a+\frac{b}{x}\right)^{m+1} (2 a c-b d (m+1)) \, _2F_1\left(2,m+1;m+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{2 c (m+1) (a c-b d)^3}-\frac{d \left(a+\frac{b}{x}\right)^{m+1}}{2 c \left(\frac{c}{x}+d\right)^2 (a c-b d)}",1,"((a + b/x)^(1 + m)*(-((d*x^2)/(c + d*x)^2) + (b*(-2*a*c + b*d*(1 + m))*Hypergeometric2F1[2, 1 + m, 2 + m, (b*c + a*c*x)/(a*c*x - b*d*x)])/((a*c - b*d)^2*(1 + m))))/(2*c*(a*c - b*d))","A",1
594,1,155,185,0.1460005,"\int \frac{\left(a+\frac{b}{x}\right)^m}{(c+d x)^4} \, dx","Integrate[(a + b/x)^m/(c + d*x)^4,x]","\frac{\left(a+\frac{b}{x}\right)^{m+1} \left(-\frac{b \left(6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left(m^2+3 m+2\right)\right) \, _2F_1\left(2,m+1;m+2;\frac{b c+a x c}{a c x-b d x}\right)}{(m+1) (a c-b d)^2}+\frac{2 d^2 x^3 (a c-b d)}{(c+d x)^3}+\frac{d x^2 (b d (m+4)-6 a c)}{(c+d x)^2}\right)}{6 c^2 (a c-b d)^2}","-\frac{b \left(a+\frac{b}{x}\right)^{m+1} \left(6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left(m^2+3 m+2\right)\right) \, _2F_1\left(2,m+1;m+2;\frac{c \left(a+\frac{b}{x}\right)}{a c-b d}\right)}{6 c^2 (m+1) (a c-b d)^4}+\frac{d^2 \left(a+\frac{b}{x}\right)^{m+1}}{3 c^2 \left(\frac{c}{x}+d\right)^3 (a c-b d)}-\frac{d \left(a+\frac{b}{x}\right)^{m+1} (6 a c-b d (m+4))}{6 c^2 \left(\frac{c}{x}+d\right)^2 (a c-b d)^2}",1,"((a + b/x)^(1 + m)*((2*d^2*(a*c - b*d)*x^3)/(c + d*x)^3 + (d*(-6*a*c + b*d*(4 + m))*x^2)/(c + d*x)^2 - (b*(6*a^2*c^2 - 6*a*b*c*d*(1 + m) + b^2*d^2*(2 + 3*m + m^2))*Hypergeometric2F1[2, 1 + m, 2 + m, (b*c + a*c*x)/(a*c*x - b*d*x)])/((a*c - b*d)^2*(1 + m))))/(6*c^2*(a*c - b*d)^2)","A",1
595,1,33,33,0.0179048,"\int \frac{\sqrt{b-\frac{a}{x^2}} x^m}{\sqrt{a-b x^2}} \, dx","Integrate[(Sqrt[b - a/x^2]*x^m)/Sqrt[a - b*x^2],x]","\frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}}","\frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}}",1,"(Sqrt[b - a/x^2]*x^(1 + m))/(m*Sqrt[a - b*x^2])","A",1
596,1,31,31,0.0100898,"\int \frac{\sqrt{b-\frac{a}{x^2}} x^2}{\sqrt{a-b x^2}} \, dx","Integrate[(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2],x]","\frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}}","\frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}}",1,"(Sqrt[b - a/x^2]*x^3)/(2*Sqrt[a - b*x^2])","A",1
597,1,28,28,0.0083328,"\int \frac{\sqrt{b-\frac{a}{x^2}} x}{\sqrt{a-b x^2}} \, dx","Integrate[(Sqrt[b - a/x^2]*x)/Sqrt[a - b*x^2],x]","\frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}","\frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}",1,"(Sqrt[b - a/x^2]*x^2)/Sqrt[a - b*x^2]","A",1
598,1,28,28,0.008205,"\int \frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \, dx","Integrate[Sqrt[b - a/x^2]/Sqrt[a - b*x^2],x]","\frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}","\frac{x \log (x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}",1,"(Sqrt[b - a/x^2]*x*Log[x])/Sqrt[a - b*x^2]","A",1
599,1,26,26,0.0079134,"\int \frac{\sqrt{b-\frac{a}{x^2}}}{x \sqrt{a-b x^2}} \, dx","Integrate[Sqrt[b - a/x^2]/(x*Sqrt[a - b*x^2]),x]","-\frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}","-\frac{\sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}",1,"-(Sqrt[b - a/x^2]/Sqrt[a - b*x^2])","A",1
600,1,31,31,0.0072818,"\int \frac{\sqrt{b-\frac{a}{x^2}}}{x^2 \sqrt{a-b x^2}} \, dx","Integrate[Sqrt[b - a/x^2]/(x^2*Sqrt[a - b*x^2]),x]","-\frac{\sqrt{b-\frac{a}{x^2}}}{2 x \sqrt{a-b x^2}}","-\frac{\sqrt{b-\frac{a}{x^2}}}{2 x \sqrt{a-b x^2}}",1,"-1/2*Sqrt[b - a/x^2]/(x*Sqrt[a - b*x^2])","A",1
601,1,540,406,3.0306257,"\int \frac{(c+d x)^{3/2}}{\sqrt{a+\frac{b}{x^2}}} \, dx","Integrate[(c + d*x)^(3/2)/Sqrt[a + b/x^2],x]","\frac{\sqrt{c+d x} \left(\frac{2 \left(a x^2+b\right) (2 c+d x)}{a}+\frac{2 \left(\sqrt{a} (c+d x)^{3/2} \left(-i a^{3/2} c^3+a \sqrt{b} c^2 d+3 i \sqrt{a} b c d^2-3 b^{3/2} d^3\right) \sqrt{\frac{d \left(x+\frac{i \sqrt{b}}{\sqrt{a}}\right)}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} E\left(i \sinh ^{-1}\left(\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right)|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right)+d^2 \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}} \left(a^2 c^2 x^2+a b \left(c^2-3 d^2 x^2\right)-3 b^2 d^2\right)-\sqrt{a} \sqrt{b} d (c+d x)^{3/2} \left(4 i \sqrt{a} \sqrt{b} c d+a c^2-3 b d^2\right) \sqrt{\frac{d \left(x+\frac{i \sqrt{b}}{\sqrt{a}}\right)}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} F\left(i \sinh ^{-1}\left(\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right)|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right)\right)}{a^2 d^2 (c+d x) \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}\right)}{5 x \sqrt{a+\frac{b}{x^2}}}","-\frac{2 \sqrt{b} c \sqrt{\frac{a x^2}{b}+1} \left(a c^2+b d^2\right) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right)}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x}}+\frac{2 \sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sqrt{c+d x} \left(a c^2-3 b d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right)}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}+\frac{2 \left(a x^2+b\right) (c+d x)^{3/2}}{5 a x \sqrt{a+\frac{b}{x^2}}}+\frac{2 c \left(a x^2+b\right) \sqrt{c+d x}}{5 a x \sqrt{a+\frac{b}{x^2}}}",1,"(Sqrt[c + d*x]*((2*(2*c + d*x)*(b + a*x^2))/a + (2*(d^2*Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]*(-3*b^2*d^2 + a^2*c^2*x^2 + a*b*(c^2 - 3*d^2*x^2)) + Sqrt[a]*((-I)*a^(3/2)*c^3 + a*Sqrt[b]*c^2*d + (3*I)*Sqrt[a]*b*c*d^2 - 3*b^(3/2)*d^3)*Sqrt[(d*((I*Sqrt[b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)] - Sqrt[a]*Sqrt[b]*d*(a*c^2 + (4*I)*Sqrt[a]*Sqrt[b]*c*d - 3*b*d^2)*Sqrt[(d*((I*Sqrt[b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c + d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)]))/(a^2*d^2*Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]*(c + d*x))))/(5*Sqrt[a + b/x^2]*x)","C",1
602,1,15,15,0.0254007,"\int \frac{-1+x^3}{\left(-4 x+x^4\right)^{2/3}} \, dx","Integrate[(-1 + x^3)/(-4*x + x^4)^(2/3),x]","\frac{3}{4} \sqrt[3]{x \left(x^3-4\right)}","\frac{3}{4} \sqrt[3]{x^4-4 x}",1,"(3*(x*(-4 + x^3))^(1/3))/4","A",1
603,1,72,17,0.0451429,"\int \left(2-x^2\right) \sqrt[4]{6 x-x^3} \, dx","Integrate[(2 - x^2)*(6*x - x^3)^(1/4),x]","-\frac{4 \sqrt[4]{-x \left(x^2-6\right)} \left(5 x^3 \, _2F_1\left(-\frac{1}{4},\frac{13}{8};\frac{21}{8};\frac{x^2}{6}\right)-26 x \, _2F_1\left(-\frac{1}{4},\frac{5}{8};\frac{13}{8};\frac{x^2}{6}\right)\right)}{65 \sqrt[4]{1-\frac{x^2}{6}}}","\frac{4}{15} \left(6 x-x^3\right)^{5/4}",1,"(-4*(-(x*(-6 + x^2)))^(1/4)*(-26*x*Hypergeometric2F1[-1/4, 5/8, 13/8, x^2/6] + 5*x^3*Hypergeometric2F1[-1/4, 13/8, 21/8, x^2/6]))/(65*(1 - x^2/6)^(1/4))","C",1
604,1,15,15,0.0279295,"\int \left(1+x^4\right) \sqrt{5 x+x^5} \, dx","Integrate[(1 + x^4)*Sqrt[5*x + x^5],x]","\frac{2}{15} \left(x \left(x^4+5\right)\right)^{3/2}","\frac{2}{15} \left(x^5+5 x\right)^{3/2}",1,"(2*(x*(5 + x^4))^(3/2))/15","A",1
605,1,15,15,0.0343483,"\int \left(2+5 x^4\right) \sqrt{2 x+x^5} \, dx","Integrate[(2 + 5*x^4)*Sqrt[2*x + x^5],x]","\frac{2}{3} \left(x \left(x^4+2\right)\right)^{3/2}","\frac{2}{3} \left(x^5+2 x\right)^{3/2}",1,"(2*(x*(2 + x^4))^(3/2))/3","A",1
606,1,13,13,0.0088587,"\int \frac{x+3 x^2}{\sqrt{x^2+2 x^3}} \, dx","Integrate[(x + 3*x^2)/Sqrt[x^2 + 2*x^3],x]","\sqrt{x^2 (2 x+1)}","\sqrt{2 x^3+x^2}",1,"Sqrt[x^2*(1 + 2*x)]","A",1
607,1,44,44,0.020852,"\int \frac{2+\sqrt[3]{1-5 x}}{3+\sqrt[3]{1-5 x}} \, dx","Integrate[(2 + (1 - 5*x)^(1/3))/(3 + (1 - 5*x)^(1/3)),x]","x+\frac{3}{10} (1-5 x)^{2/3}-\frac{9}{5} \sqrt[3]{1-5 x}+\frac{27}{5} \log \left(\sqrt[3]{1-5 x}+3\right)","x+\frac{3}{10} (1-5 x)^{2/3}-\frac{9}{5} \sqrt[3]{1-5 x}+\frac{27}{5} \log \left(\sqrt[3]{1-5 x}+3\right)",1,"(-9*(1 - 5*x)^(1/3))/5 + (3*(1 - 5*x)^(2/3))/10 + x + (27*Log[3 + (1 - 5*x)^(1/3)])/5","A",1
608,1,20,21,0.0084184,"\int \frac{1+\sqrt{x}}{-1+\sqrt{x}} \, dx","Integrate[(1 + Sqrt[x])/(-1 + Sqrt[x]),x]","x+4 \left(\sqrt{x}+\log \left(1-\sqrt{x}\right)\right)","x+4 \sqrt{x}+4 \log \left(1-\sqrt{x}\right)",1,"x + 4*(Sqrt[x] + Log[1 - Sqrt[x]])","A",1
609,1,33,33,0.0134655,"\int \frac{1-\sqrt{2+3 x}}{1+\sqrt{2+3 x}} \, dx","Integrate[(1 - Sqrt[2 + 3*x])/(1 + Sqrt[2 + 3*x]),x]","-x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left(\sqrt{3 x+2}+1\right)","-x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left(\sqrt{3 x+2}+1\right)",1,"-x + (4*Sqrt[2 + 3*x])/3 - (4*Log[1 + Sqrt[2 + 3*x]])/3","A",1
610,1,33,33,0.0152525,"\int \frac{-1+\sqrt{a+b x}}{1+\sqrt{a+b x}} \, dx","Integrate[(-1 + Sqrt[a + b*x])/(1 + Sqrt[a + b*x]),x]","-\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left(\sqrt{a+b x}+1\right)}{b}+x","-\frac{4 \sqrt{a+b x}}{b}+\frac{4 \log \left(\sqrt{a+b x}+1\right)}{b}+x",1,"x - (4*Sqrt[a + b*x])/b + (4*Log[1 + Sqrt[a + b*x]])/b","A",1
611,1,17,10,0.028113,"\int \frac{a+b n x^{-1+n}}{a x+b x^n} \, dx","Integrate[(a + b*n*x^(-1 + n))/(a*x + b*x^n),x]","\log \left(a x^{1-n}+b\right)+n \log (x)","\log \left(a x+b x^n\right)",1,"n*Log[x] + Log[b + a*x^(1 - n)]","A",1
612,1,17,17,0.0158912,"\int \frac{x^{-n} \left(a+b n x^{-1+n}\right)}{b+a x^{1-n}} \, dx","Integrate[(a + b*n*x^(-1 + n))/(x^n*(b + a*x^(1 - n))),x]","\log \left(a x^{1-n}+b\right)+n \log (x)","\log \left(a x^{1-n}+b\right)+n \log (x)",1,"n*Log[x] + Log[b + a*x^(1 - n)]","A",1
613,1,34,37,0.3493964,"\int x \left(a+b x+c x^2\right)^m \left(d+e x+f x^2+g x^3\right)^n \left(2 a d+(3 b d+3 a e+b d m+a e n) x+(4 c d+4 b e+4 a f+2 c d m+b e m+b e n+2 a f n) x^2+(5 c e+5 b f+5 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(6 c f+6 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (7+2 m+3 n) x^5\right) \, dx","Integrate[x*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(2*a*d + (3*b*d + 3*a*e + b*d*m + a*e*n)*x + (4*c*d + 4*b*e + 4*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (5*c*e + 5*b*f + 5*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (6*c*f + 6*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(7 + 2*m + 3*n)*x^5),x]","x^2 (a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}","x^2 \left(a+b x+c x^2\right)^{m+1} \left(d+e x+f x^2+g x^3\right)^{n+1}",1,"x^2*(a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n)","A",1
614,1,32,35,0.3309684,"\int \left(a+b x+c x^2\right)^m \left(d+e x+f x^2+g x^3\right)^n \left(a d+(2 b d+2 a e+b d m+a e n) x+(3 c d+3 b e+3 a f+2 c d m+b e m+b e n+2 a f n) x^2+(4 c e+4 b f+4 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(5 c f+5 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (6+2 m+3 n) x^5\right) \, dx","Integrate[(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(a*d + (2*b*d + 2*a*e + b*d*m + a*e*n)*x + (3*c*d + 3*b*e + 3*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (4*c*e + 4*b*f + 4*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (5*c*f + 5*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(6 + 2*m + 3*n)*x^5),x]","x (a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}","x \left(a+b x+c x^2\right)^{m+1} \left(d+e x+f x^2+g x^3\right)^{n+1}",1,"x*(a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n)","A",1
615,1,31,34,0.3132338,"\int \left(a+b x+c x^2\right)^m \left(d+e x+f x^2+g x^3\right)^n \left(b d+a e+b d m+a e n+(2 c d+2 b e+2 a f+2 c d m+b e m+b e n+2 a f n) x+(3 c e+3 b f+3 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^2+(4 c f+4 b g+2 c f m+b g m+2 c f n+3 b g n) x^3+c g (5+2 m+3 n) x^4\right) \, dx","Integrate[(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(b*d + a*e + b*d*m + a*e*n + (2*c*d + 2*b*e + 2*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x + (3*c*e + 3*b*f + 3*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^2 + (4*c*f + 4*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^3 + c*g*(5 + 2*m + 3*n)*x^4),x]","(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}","\left(a+b x+c x^2\right)^{m+1} \left(d+e x+f x^2+g x^3\right)^{n+1}",1,"(a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n)","A",1
616,1,34,37,0.8075738,"\int \frac{\left(a+b x+c x^2\right)^m \left(d+e x+f x^2+g x^3\right)^n \left(-a d+(b d m+a e n) x+(c d+b e+a f+2 c d m+b e m+b e n+2 a f n) x^2+(2 c e+2 b f+2 a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(3 c f+3 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (4+2 m+3 n) x^5\right)}{x^2} \, dx","Integrate[((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(-(a*d) + (b*d*m + a*e*n)*x + (c*d + b*e + a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (2*c*e + 2*b*f + 2*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (3*c*f + 3*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(4 + 2*m + 3*n)*x^5))/x^2,x]","\frac{(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}}{x}","\frac{\left(a+b x+c x^2\right)^{m+1} \left(d+e x+f x^2+g x^3\right)^{n+1}}{x}",1,"((a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n))/x","A",1
617,1,34,37,1.1539491,"\int \frac{\left(a+b x+c x^2\right)^m \left(d+e x+f x^2+g x^3\right)^n \left(-2 a d+(-b d-a e+b d m+a e n) x+(2 c d m+b e m+b e n+2 a f n) x^2+(c e+b f+a g+2 c e m+b f m+c e n+2 b f n+3 a g n) x^3+(2 c f+2 b g+2 c f m+b g m+2 c f n+3 b g n) x^4+c g (3+2 m+3 n) x^5\right)}{x^3} \, dx","Integrate[((a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(-2*a*d + (-(b*d) - a*e + b*d*m + a*e*n)*x + (2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (c*e + b*f + a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (2*c*f + 2*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(3 + 2*m + 3*n)*x^5))/x^3,x]","\frac{(a+x (b+c x))^{m+1} (d+x (e+x (f+g x)))^{n+1}}{x^2}","\frac{\left(a+b x+c x^2\right)^{m+1} \left(d+e x+f x^2+g x^3\right)^{n+1}}{x^2}",1,"((a + x*(b + c*x))^(1 + m)*(d + x*(e + x*(f + g*x)))^(1 + n))/x^2","A",1
618,1,88,185,0.3065909,"\int x^3 \left(a+b \sqrt{c+d x}\right)^2 \, dx","Integrate[x^3*(a + b*Sqrt[c + d*x])^2,x]","\frac{a^2 x^4}{4}+\frac{4 a b \sqrt{c+d x} \left(-16 c^4+8 c^3 d x-6 c^2 d^2 x^2+5 c d^3 x^3+35 d^4 x^4\right)}{315 d^4}+\frac{1}{20} b^2 x^4 (5 c+4 d x)","\frac{c^2 \left(3 a^2-b^2 c\right) (c+d x)^2}{2 d^4}+\frac{\left(a^2-3 b^2 c\right) (c+d x)^4}{4 d^4}-\frac{c \left(a^2-b^2 c\right) (c+d x)^3}{d^4}-\frac{a^2 c^3 x}{d^3}-\frac{4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac{12 a b c^2 (c+d x)^{5/2}}{5 d^4}+\frac{4 a b (c+d x)^{9/2}}{9 d^4}-\frac{12 a b c (c+d x)^{7/2}}{7 d^4}+\frac{b^2 (c+d x)^5}{5 d^4}",1,"(a^2*x^4)/4 + (b^2*x^4*(5*c + 4*d*x))/20 + (4*a*b*Sqrt[c + d*x]*(-16*c^4 + 8*c^3*d*x - 6*c^2*d^2*x^2 + 5*c*d^3*x^3 + 35*d^4*x^4))/(315*d^4)","A",1
619,1,77,138,0.1871632,"\int x^2 \left(a+b \sqrt{c+d x}\right)^2 \, dx","Integrate[x^2*(a + b*Sqrt[c + d*x])^2,x]","\frac{a^2 x^3}{3}+\frac{4 a b \sqrt{c+d x} \left(8 c^3-4 c^2 d x+3 c d^2 x^2+15 d^3 x^3\right)}{105 d^3}+\frac{1}{12} b^2 x^3 (4 c+3 d x)","\frac{\left(a^2-2 b^2 c\right) (c+d x)^3}{3 d^3}-\frac{c \left(2 a^2-b^2 c\right) (c+d x)^2}{2 d^3}+\frac{a^2 c^2 x}{d^2}+\frac{4 a b c^2 (c+d x)^{3/2}}{3 d^3}+\frac{4 a b (c+d x)^{7/2}}{7 d^3}-\frac{8 a b c (c+d x)^{5/2}}{5 d^3}+\frac{b^2 (c+d x)^4}{4 d^3}",1,"(a^2*x^3)/3 + (b^2*x^3*(4*c + 3*d*x))/12 + (4*a*b*Sqrt[c + d*x]*(8*c^3 - 4*c^2*d*x + 3*c*d^2*x^2 + 15*d^3*x^3))/(105*d^3)","A",1
620,1,63,89,0.1049742,"\int x \left(a+b \sqrt{c+d x}\right)^2 \, dx","Integrate[x*(a + b*Sqrt[c + d*x])^2,x]","\frac{1}{30} \left(15 a^2 x^2+\frac{8 a b \sqrt{c+d x} \left(-2 c^2+c d x+3 d^2 x^2\right)}{d^2}+5 b^2 x^2 (3 c+2 d x)\right)","\frac{\left(a^2-b^2 c\right) (c+d x)^2}{2 d^2}-\frac{a^2 c x}{d}+\frac{4 a b (c+d x)^{5/2}}{5 d^2}-\frac{4 a b c (c+d x)^{3/2}}{3 d^2}+\frac{b^2 (c+d x)^3}{3 d^2}",1,"(15*a^2*x^2 + 5*b^2*x^2*(3*c + 2*d*x) + (8*a*b*Sqrt[c + d*x]*(-2*c^2 + c*d*x + 3*d^2*x^2))/d^2)/30","A",1
621,1,40,41,0.0261115,"\int \left(a+b \sqrt{c+d x}\right)^2 \, dx","Integrate[(a + b*Sqrt[c + d*x])^2,x]","\frac{6 a^2 d x+8 a b (c+d x)^{3/2}+3 b^2 (c+d x)^2}{6 d}","a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d}",1,"(6*a^2*d*x + 8*a*b*(c + d*x)^(3/2) + 3*b^2*(c + d*x)^2)/(6*d)","A",1
622,1,79,57,0.1623445,"\int \frac{\left(a+b \sqrt{c+d x}\right)^2}{x} \, dx","Integrate[(a + b*Sqrt[c + d*x])^2/x,x]","b \left(4 a \sqrt{c+d x}+b d x\right)+\left(a-b \sqrt{c}\right)^2 \log \left(\sqrt{c+d x}+\sqrt{c}\right)+\left(a+b \sqrt{c}\right)^2 \log \left(\sqrt{c}-\sqrt{c+d x}\right)","\log (x) \left(a^2+b^2 c\right)+4 a b \sqrt{c+d x}-4 a b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)+b^2 d x",1,"b*(b*d*x + 4*a*Sqrt[c + d*x]) + (a + b*Sqrt[c])^2*Log[Sqrt[c] - Sqrt[c + d*x]] + (a - b*Sqrt[c])^2*Log[Sqrt[c] + Sqrt[c + d*x]]","A",1
623,1,161,54,0.1995013,"\int \frac{\left(a+b \sqrt{c+d x}\right)^2}{x^2} \, dx","Integrate[(a + b*Sqrt[c + d*x])^2/x^2,x]","\frac{\sqrt{c} \left(a^4+2 a^3 b \sqrt{c+d x}-2 a b^3 c \sqrt{c+d x}-b^4 c (c+2 d x)\right)+b d x \left(a+b \sqrt{c}\right) \left(a-b \sqrt{c}\right)^2 \log \left(\sqrt{c+d x}+\sqrt{c}\right)+b d x \left(b \sqrt{c}-a\right) \left(a+b \sqrt{c}\right)^2 \log \left(\sqrt{c}-\sqrt{c+d x}\right)}{\sqrt{c} x \left(b^2 c-a^2\right)}","-\frac{\left(a+b \sqrt{c+d x}\right)^2}{x}-\frac{2 a b d \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{\sqrt{c}}+b^2 d \log (x)",1,"(Sqrt[c]*(a^4 + 2*a^3*b*Sqrt[c + d*x] - 2*a*b^3*c*Sqrt[c + d*x] - b^4*c*(c + 2*d*x)) + b*(-a + b*Sqrt[c])*(a + b*Sqrt[c])^2*d*x*Log[Sqrt[c] - Sqrt[c + d*x]] + b*(a - b*Sqrt[c])^2*(a + b*Sqrt[c])*d*x*Log[Sqrt[c] + Sqrt[c + d*x]])/(Sqrt[c]*(-a^2 + b^2*c)*x)","B",1
624,1,221,80,0.3480972,"\int \frac{\left(a+b \sqrt{c+d x}\right)^2}{x^3} \, dx","Integrate[(a + b*Sqrt[c + d*x])^2/x^3,x]","\frac{-\frac{2 \sqrt{c} \left(a^6 c+a^5 b \sqrt{c+d x} (2 c+d x)+a^4 b^2 \left(-c^2+2 c d x+3 d^2 x^2\right)-2 a^3 b^3 c \sqrt{c+d x} (2 c+d x)-a^2 b^4 c \left(c^2+4 c d x+2 d^2 x^2\right)+a b^5 c^2 \sqrt{c+d x} (2 c+d x)+b^6 c^2 (c+d x)^2\right)}{x^2 \left(a^2-b^2 c\right)^2}-a b d^2 \log \left(\sqrt{c}-\sqrt{c+d x}\right)+a b d^2 \log \left(\sqrt{c+d x}+\sqrt{c}\right)}{4 c^{3/2}}","\frac{a b d^2 \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{2 c^{3/2}}-\frac{\left(a+b \sqrt{c+d x}\right)^2}{2 x^2}-\frac{b d \left(a \sqrt{c+d x}+b c\right)}{2 c x}",1,"((-2*Sqrt[c]*(a^6*c + b^6*c^2*(c + d*x)^2 + a^5*b*Sqrt[c + d*x]*(2*c + d*x) - 2*a^3*b^3*c*Sqrt[c + d*x]*(2*c + d*x) + a*b^5*c^2*Sqrt[c + d*x]*(2*c + d*x) - a^2*b^4*c*(c^2 + 4*c*d*x + 2*d^2*x^2) + a^4*b^2*(-c^2 + 2*c*d*x + 3*d^2*x^2)))/((a^2 - b^2*c)^2*x^2) - a*b*d^2*Log[Sqrt[c] - Sqrt[c + d*x]] + a*b*d^2*Log[Sqrt[c] + Sqrt[c + d*x]])/(4*c^(3/2))","B",1
625,1,232,326,0.4167249,"\int x^3 \sqrt{a+b \sqrt{c+d x}} \, dx","Integrate[x^3*Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \left(a+b \sqrt{c+d x}\right)^{3/2} \left(-14336 a^7+21504 a^6 b \sqrt{c+d x}+3840 a^5 b^2 (10 c-7 d x)-640 a^4 b^3 (104 c-49 d x) \sqrt{c+d x}-48 a^3 b^4 \left(616 c^2-1080 c d x+735 d^2 x^2\right)+24 a^2 b^5 \sqrt{c+d x} \left(2960 c^2-2716 c d x+1617 d^2 x^2\right)+6 a b^6 \left(320 c^3-3936 c^2 d x+5754 c d^2 x^2-7007 d^3 x^3\right)-231 b^7 \sqrt{c+d x} \left(128 c^3-160 c^2 d x+180 c d^2 x^2-195 d^3 x^3\right)\right)}{765765 b^8 d^4}","\frac{12 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{13/2}}{13 b^8 d^4}-\frac{20 a \left(7 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{11/2}}{11 b^8 d^4}-\frac{12 a \left(7 a^2-3 b^2 c\right) \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^8 d^4}+\frac{4 \left(a^2-b^2 c\right)^2 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^8 d^4}-\frac{4 a \left(a^2-b^2 c\right)^3 \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^8 d^4}+\frac{4 \left(35 a^4-30 a^2 b^2 c+3 b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{9/2}}{9 b^8 d^4}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{17/2}}{17 b^8 d^4}-\frac{28 a \left(a+b \sqrt{c+d x}\right)^{15/2}}{15 b^8 d^4}",1,"(4*(a + b*Sqrt[c + d*x])^(3/2)*(-14336*a^7 + 3840*a^5*b^2*(10*c - 7*d*x) + 21504*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3*(104*c - 49*d*x)*Sqrt[c + d*x] - 48*a^3*b^4*(616*c^2 - 1080*c*d*x + 735*d^2*x^2) + 24*a^2*b^5*Sqrt[c + d*x]*(2960*c^2 - 2716*c*d*x + 1617*d^2*x^2) + 6*a*b^6*(320*c^3 - 3936*c^2*d*x + 5754*c*d^2*x^2 - 7007*d^3*x^3) - 231*b^7*Sqrt[c + d*x]*(128*c^3 - 160*c^2*d*x + 180*c*d^2*x^2 - 195*d^3*x^3)))/(765765*b^8*d^4)","A",1
626,1,147,224,0.237391,"\int x^2 \sqrt{a+b \sqrt{c+d x}} \, dx","Integrate[x^2*Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \left(a+b \sqrt{c+d x}\right)^{3/2} \left(-1280 a^5+1920 a^4 b \sqrt{c+d x}+32 a^3 b^2 (68 c-75 d x)+16 a^2 b^3 \sqrt{c+d x} (175 d x-254 c)-6 a b^4 \left(96 c^2-380 c d x+525 d^2 x^2\right)+77 b^5 \sqrt{c+d x} \left(32 c^2-40 c d x+45 d^2 x^2\right)\right)}{45045 b^6 d^3}","\frac{8 \left(5 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{9/2}}{9 b^6 d^3}-\frac{8 a \left(5 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^6 d^3}-\frac{4 a \left(a^2-b^2 c\right)^2 \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^6 d^3}+\frac{4 \left(5 a^4-6 a^2 b^2 c+b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^6 d^3}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{13/2}}{13 b^6 d^3}-\frac{20 a \left(a+b \sqrt{c+d x}\right)^{11/2}}{11 b^6 d^3}",1,"(4*(a + b*Sqrt[c + d*x])^(3/2)*(-1280*a^5 + 32*a^3*b^2*(68*c - 75*d*x) + 1920*a^4*b*Sqrt[c + d*x] + 16*a^2*b^3*Sqrt[c + d*x]*(-254*c + 175*d*x) + 77*b^5*Sqrt[c + d*x]*(32*c^2 - 40*c*d*x + 45*d^2*x^2) - 6*a*b^4*(96*c^2 - 380*c*d*x + 525*d^2*x^2)))/(45045*b^6*d^3)","A",1
627,1,84,133,0.1386744,"\int x \sqrt{a+b \sqrt{c+d x}} \, dx","Integrate[x*Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \left(a+b \sqrt{c+d x}\right)^{3/2} \left(-16 a^3+24 a^2 b \sqrt{c+d x}+6 a b^2 (2 c-5 d x)+7 b^3 \sqrt{c+d x} (5 d x-4 c)\right)}{315 b^4 d^2}","\frac{4 \left(3 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^4 d^2}-\frac{4 a \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^4 d^2}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{9/2}}{9 b^4 d^2}-\frac{12 a \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^4 d^2}",1,"(4*(a + b*Sqrt[c + d*x])^(3/2)*(-16*a^3 + 6*a*b^2*(2*c - 5*d*x) + 24*a^2*b*Sqrt[c + d*x] + 7*b^3*Sqrt[c + d*x]*(-4*c + 5*d*x)))/(315*b^4*d^2)","A",1
628,1,43,56,0.0280024,"\int \sqrt{a+b \sqrt{c+d x}} \, dx","Integrate[Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \left(a+b \sqrt{c+d x}\right)^{3/2} \left(3 b \sqrt{c+d x}-2 a\right)}{15 b^2 d}","\frac{4 \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^2 d}-\frac{4 a \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^2 d}",1,"(4*(a + b*Sqrt[c + d*x])^(3/2)*(-2*a + 3*b*Sqrt[c + d*x]))/(15*b^2*d)","A",1
629,1,116,116,0.130817,"\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x} \, dx","Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x,x]","4 \sqrt{a+b \sqrt{c+d x}}-2 \sqrt{a-b \sqrt{c}} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)-2 \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)","4 \sqrt{a+b \sqrt{c+d x}}-2 \sqrt{a-b \sqrt{c}} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)-2 \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)",1,"4*Sqrt[a + b*Sqrt[c + d*x]] - 2*Sqrt[a - b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]] - 2*Sqrt[a + b*Sqrt[c]]*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]","A",1
630,1,181,137,0.1764297,"\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x^2} \, dx","Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^2,x]","\frac{\left(a-b \sqrt{c}\right) \left(2 \sqrt{c} \left(a+b \sqrt{c}\right) \sqrt{a+b \sqrt{c+d x}}+b d x \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)\right)-b d x \sqrt{a-b \sqrt{c}} \left(a+b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{2 \sqrt{c} x \left(b^2 c-a^2\right)}","-\frac{\sqrt{a+b \sqrt{c+d x}}}{x}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{2 \sqrt{c} \sqrt{a-b \sqrt{c}}}-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{2 \sqrt{c} \sqrt{a+b \sqrt{c}}}",1,"(-(b*Sqrt[a - b*Sqrt[c]]*(a + b*Sqrt[c])*d*x*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) + (a - b*Sqrt[c])*(2*(a + b*Sqrt[c])*Sqrt[c]*Sqrt[a + b*Sqrt[c + d*x]] + b*Sqrt[a + b*Sqrt[c]]*d*x*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(2*Sqrt[c]*(-a^2 + b^2*c)*x)","A",1
631,1,258,224,0.4220889,"\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x^3} \, dx","Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^3,x]","\frac{-\left(a-b \sqrt{c}\right) \left(2 \sqrt{c} \left(a+b \sqrt{c}\right) \sqrt{a+b \sqrt{c+d x}} \left(4 a^2 c+a b d x \sqrt{c+d x}-b^2 c (4 c+d x)\right)-b d^2 x^2 \sqrt{a+b \sqrt{c}} \left(2 a^2+a b \sqrt{c}-3 b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)\right)-b d^2 x^2 \left(2 a-3 b \sqrt{c}\right) \sqrt{a-b \sqrt{c}} \left(a+b \sqrt{c}\right)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{16 c^{3/2} x^2 \left(a^2-b^2 c\right)^2}","\frac{b d \left(b c-a \sqrt{c+d x}\right) \sqrt{a+b \sqrt{c+d x}}}{8 c x \left(a^2-b^2 c\right)}-\frac{b d^2 \left(2 a-3 b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{16 c^{3/2} \left(a-b \sqrt{c}\right)^{3/2}}+\frac{b d^2 \left(2 a+3 b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{16 c^{3/2} \left(a+b \sqrt{c}\right)^{3/2}}-\frac{\sqrt{a+b \sqrt{c+d x}}}{2 x^2}",1,"(-(b*(2*a - 3*b*Sqrt[c])*Sqrt[a - b*Sqrt[c]]*(a + b*Sqrt[c])^2*d^2*x^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) - (a - b*Sqrt[c])*(2*(a + b*Sqrt[c])*Sqrt[c]*Sqrt[a + b*Sqrt[c + d*x]]*(4*a^2*c + a*b*d*x*Sqrt[c + d*x] - b^2*c*(4*c + d*x)) - b*Sqrt[a + b*Sqrt[c]]*(2*a^2 + a*b*Sqrt[c] - 3*b^2*c)*d^2*x^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(16*c^(3/2)*(a^2 - b^2*c)^2*x^2)","A",1
632,1,213,230,0.2006563,"\int \frac{x^3}{a+b \sqrt{c+d x}} \, dx","Integrate[x^3/(a + b*Sqrt[c + d*x]),x]","\frac{b \left(420 a^6 \sqrt{c+d x}-210 a^5 b d x-140 a^4 b^2 (8 c-d x) \sqrt{c+d x}-105 a^3 b^3 d x (d x-4 c)+84 a^2 b^4 \sqrt{c+d x} \left(11 c^2-3 c d x+d^2 x^2\right)-35 a b^5 d x \left(6 c^2-3 c d x+2 d^2 x^2\right)+12 b^6 \sqrt{c+d x} \left(-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right)\right)-420 a \left(a^2-b^2 c\right)^3 \log \left(a+b \sqrt{c+d x}\right)}{210 b^8 d^4}","-\frac{2 a \left(a^2-b^2 c\right)^3 \log \left(a+b \sqrt{c+d x}\right)}{b^8 d^4}+\frac{2 \left(a^2-b^2 c\right)^3 \sqrt{c+d x}}{b^7 d^4}-\frac{a \left(a^2-3 b^2 c\right) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left(a^2-3 b^2 c\right) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a x \left(a^4-3 a^2 b^2 c+3 b^4 c^2\right)}{b^6 d^3}+\frac{2 \left(a^4-3 a^2 b^2 c+3 b^4 c^2\right) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4}",1,"(b*(-210*a^5*b*d*x - 105*a^3*b^3*d*x*(-4*c + d*x) + 420*a^6*Sqrt[c + d*x] - 140*a^4*b^2*(8*c - d*x)*Sqrt[c + d*x] + 84*a^2*b^4*Sqrt[c + d*x]*(11*c^2 - 3*c*d*x + d^2*x^2) - 35*a*b^5*d*x*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 12*b^6*Sqrt[c + d*x]*(-16*c^3 + 8*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3)) - 420*a*(a^2 - b^2*c)^3*Log[a + b*Sqrt[c + d*x]])/(210*b^8*d^4)","A",1
633,1,138,151,0.1325244,"\int \frac{x^2}{a+b \sqrt{c+d x}} \, dx","Integrate[x^2/(a + b*Sqrt[c + d*x]),x]","\frac{b \left(60 a^4 \sqrt{c+d x}-30 a^3 b d x-20 a^2 b^2 (5 c-d x) \sqrt{c+d x}-15 a b^3 d x (d x-2 c)+4 b^4 \sqrt{c+d x} \left(8 c^2-4 c d x+3 d^2 x^2\right)\right)-60 a \left(a^2-b^2 c\right)^2 \log \left(a+b \sqrt{c+d x}\right)}{30 b^6 d^3}","-\frac{2 a \left(a^2-b^2 c\right)^2 \log \left(a+b \sqrt{c+d x}\right)}{b^6 d^3}+\frac{2 \left(a^2-b^2 c\right)^2 \sqrt{c+d x}}{b^5 d^3}-\frac{a x \left(a^2-2 b^2 c\right)}{b^4 d^2}+\frac{2 \left(a^2-2 b^2 c\right) (c+d x)^{3/2}}{3 b^3 d^3}-\frac{a (c+d x)^2}{2 b^2 d^3}+\frac{2 (c+d x)^{5/2}}{5 b d^3}",1,"(b*(-30*a^3*b*d*x - 15*a*b^3*d*x*(-2*c + d*x) + 60*a^4*Sqrt[c + d*x] - 20*a^2*b^2*(5*c - d*x)*Sqrt[c + d*x] + 4*b^4*Sqrt[c + d*x]*(8*c^2 - 4*c*d*x + 3*d^2*x^2)) - 60*a*(a^2 - b^2*c)^2*Log[a + b*Sqrt[c + d*x]])/(30*b^6*d^3)","A",1
634,1,82,90,0.0667919,"\int \frac{x}{a+b \sqrt{c+d x}} \, dx","Integrate[x/(a + b*Sqrt[c + d*x]),x]","\frac{b \left(6 a^2 \sqrt{c+d x}-3 a b d x+2 b^2 (d x-2 c) \sqrt{c+d x}\right)-6 \left(a^3-a b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{3 b^4 d^2}","-\frac{2 a \left(a^2-b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{b^4 d^2}+\frac{2 \left(a^2-b^2 c\right) \sqrt{c+d x}}{b^3 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2}",1,"(b*(-3*a*b*d*x + 6*a^2*Sqrt[c + d*x] + 2*b^2*(-2*c + d*x)*Sqrt[c + d*x]) - 6*(a^3 - a*b^2*c)*Log[a + b*Sqrt[c + d*x]])/(3*b^4*d^2)","A",1
635,1,39,41,0.0194806,"\int \frac{1}{a+b \sqrt{c+d x}} \, dx","Integrate[(a + b*Sqrt[c + d*x])^(-1),x]","\frac{2 \left(\frac{\sqrt{c+d x}}{b}-\frac{a \log \left(a+b \sqrt{c+d x}\right)}{b^2}\right)}{d}","\frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left(a+b \sqrt{c+d x}\right)}{b^2 d}",1,"(2*(Sqrt[c + d*x]/b - (a*Log[a + b*Sqrt[c + d*x]])/b^2))/d","A",1
636,1,61,82,0.0899993,"\int \frac{1}{x \left(a+b \sqrt{c+d x}\right)} \, dx","Integrate[1/(x*(a + b*Sqrt[c + d*x])),x]","\frac{-2 a \log \left(a+b \sqrt{c+d x}\right)+a \log (d x)+2 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{a^2-b^2 c}","-\frac{2 a \log \left(a+b \sqrt{c+d x}\right)}{a^2-b^2 c}+\frac{2 b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{a^2-b^2 c}+\frac{a \log (x)}{a^2-b^2 c}",1,"(2*b*Sqrt[c]*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + a*Log[d*x] - 2*a*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)","A",1
637,1,144,130,0.1936188,"\int \frac{1}{x^2 \left(a+b \sqrt{c+d x}\right)} \, dx","Integrate[1/(x^2*(a + b*Sqrt[c + d*x])),x]","\frac{\sqrt{c} \left(-\left(a^2-b^2 c\right) \left(a-b \sqrt{c+d x}\right)-a b^2 d x \log \left(a^2-b^2 (c+d x)\right)+a b^2 d x \log (x)\right)+b d x \left(a^2+b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)-2 a b^2 \sqrt{c} d x \tanh ^{-1}\left(\frac{b \sqrt{c+d x}}{a}\right)}{\sqrt{c} x \left(a^2-b^2 c\right)^2}","-\frac{a-b \sqrt{c+d x}}{x \left(a^2-b^2 c\right)}+\frac{a b^2 d \log (x)}{\left(a^2-b^2 c\right)^2}-\frac{2 a b^2 d \log \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^2}+\frac{b d \left(a^2+b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{\sqrt{c} \left(a^2-b^2 c\right)^2}",1,"(-2*a*b^2*Sqrt[c]*d*x*ArcTanh[(b*Sqrt[c + d*x])/a] + b*(a^2 + b^2*c)*d*x*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + Sqrt[c]*(-((a^2 - b^2*c)*(a - b*Sqrt[c + d*x])) + a*b^2*d*x*Log[x] - a*b^2*d*x*Log[a^2 - b^2*(c + d*x)]))/(Sqrt[c]*(a^2 - b^2*c)^2*x)","A",1
638,1,228,204,0.4286305,"\int \frac{1}{x^3 \left(a+b \sqrt{c+d x}\right)} \, dx","Integrate[1/(x^3*(a + b*Sqrt[c + d*x])),x]","\frac{b d^2 x^2 \left(a^4-6 a^2 b^2 c-3 b^4 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)+\sqrt{c} \left(4 a b^4 c d^2 x^2 \log \left(a^2-b^2 (c+d x)\right)+\left(a^2-b^2 c\right) \left(2 a^3 c-a^2 b \sqrt{c+d x} (2 c+d x)-2 a b^2 c (c-2 d x)+b^3 c (2 c-3 d x) \sqrt{c+d x}\right)-4 a b^4 c d^2 x^2 \log (x)\right)+8 a b^4 c^{3/2} d^2 x^2 \tanh ^{-1}\left(\frac{b \sqrt{c+d x}}{a}\right)}{4 c^{3/2} x^2 \left(b^2 c-a^2\right)^3}","-\frac{a-b \sqrt{c+d x}}{2 x^2 \left(a^2-b^2 c\right)}-\frac{b d \left(4 a b c-\left(a^2+3 b^2 c\right) \sqrt{c+d x}\right)}{4 c x \left(a^2-b^2 c\right)^2}+\frac{a b^4 d^2 \log (x)}{\left(a^2-b^2 c\right)^3}-\frac{2 a b^4 d^2 \log \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^3}-\frac{b d^2 \left(a^4-6 a^2 b^2 c-3 b^4 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{4 c^{3/2} \left(a^2-b^2 c\right)^3}",1,"(8*a*b^4*c^(3/2)*d^2*x^2*ArcTanh[(b*Sqrt[c + d*x])/a] + b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*x^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + Sqrt[c]*((a^2 - b^2*c)*(2*a^3*c - 2*a*b^2*c*(c - 2*d*x) + b^3*c*(2*c - 3*d*x)*Sqrt[c + d*x] - a^2*b*Sqrt[c + d*x]*(2*c + d*x)) - 4*a*b^4*c*d^2*x^2*Log[x] + 4*a*b^4*c*d^2*x^2*Log[a^2 - b^2*(c + d*x)]))/(4*c^(3/2)*(-a^2 + b^2*c)^3*x^2)","A",1
639,1,273,240,0.2720899,"\int \frac{x^3}{\left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[x^3/(a + b*Sqrt[c + d*x])^2,x]","\frac{96 a^7-324 a^6 b \sqrt{c+d x}-6 a^5 b^2 (102 c+35 d x)+2 a^4 b^3 \sqrt{c+d x} (284 c+35 d x)+a^3 b^4 \left(856 c^2+380 c d x-35 d^2 x^2\right)-3 a^2 b^5 \sqrt{c+d x} \left(76 c^2+36 c d x-7 d^2 x^2\right)+60 \left(a^2-b^2 c\right)^2 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right) \log \left(a+b \sqrt{c+d x}\right)-a b^6 \left(324 c^3+162 c^2 d x-33 c d^2 x^2+14 d^3 x^3\right)+5 b^7 d x \sqrt{c+d x} \left(6 c^2-3 c d x+2 d^2 x^2\right)}{30 b^8 d^4 \left(a+b \sqrt{c+d x}\right)}","\frac{2 a \left(a^2-b^2 c\right)^3}{b^8 d^4 \left(a+b \sqrt{c+d x}\right)}+\frac{2 \left(7 a^2-b^2 c\right) \left(a^2-b^2 c\right)^2 \log \left(a+b \sqrt{c+d x}\right)}{b^8 d^4}-\frac{12 a \left(a^2-b^2 c\right)^2 \sqrt{c+d x}}{b^7 d^4}-\frac{4 a \left(2 a^2-3 b^2 c\right) (c+d x)^{3/2}}{3 b^5 d^4}+\frac{3 \left(a^2-b^2 c\right) (c+d x)^2}{2 b^4 d^4}+\frac{x \left(5 a^4-9 a^2 b^2 c+3 b^4 c^2\right)}{b^6 d^3}-\frac{4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac{(c+d x)^3}{3 b^2 d^4}",1,"(96*a^7 - 324*a^6*b*Sqrt[c + d*x] - 6*a^5*b^2*(102*c + 35*d*x) + 2*a^4*b^3*Sqrt[c + d*x]*(284*c + 35*d*x) + a^3*b^4*(856*c^2 + 380*c*d*x - 35*d^2*x^2) - 3*a^2*b^5*Sqrt[c + d*x]*(76*c^2 + 36*c*d*x - 7*d^2*x^2) + 5*b^7*d*x*Sqrt[c + d*x]*(6*c^2 - 3*c*d*x + 2*d^2*x^2) - a*b^6*(324*c^3 + 162*c^2*d*x - 33*c*d^2*x^2 + 14*d^3*x^3) + 60*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/(30*b^8*d^4*(a + b*Sqrt[c + d*x]))","A",1
640,1,185,166,0.1693874,"\int \frac{x^2}{\left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[x^2/(a + b*Sqrt[c + d*x])^2,x]","\frac{16 a^5-44 a^4 b \sqrt{c+d x}-2 a^3 b^2 (38 c+15 d x)+2 a^2 b^3 \sqrt{c+d x} (18 c+5 d x)+12 \left(5 a^4-6 a^2 b^2 c+b^4 c^2\right) \left(a+b \sqrt{c+d x}\right) \log \left(a+b \sqrt{c+d x}\right)+a b^4 \left(52 c^2+26 c d x-5 d^2 x^2\right)+3 b^5 d x (d x-2 c) \sqrt{c+d x}}{6 b^6 d^3 \left(a+b \sqrt{c+d x}\right)}","\frac{2 a \left(a^2-b^2 c\right)^2}{b^6 d^3 \left(a+b \sqrt{c+d x}\right)}-\frac{8 a \left(a^2-b^2 c\right) \sqrt{c+d x}}{b^5 d^3}+\frac{x \left(3 a^2-2 b^2 c\right)}{b^4 d^2}+\frac{2 \left(5 a^4-6 a^2 b^2 c+b^4 c^2\right) \log \left(a+b \sqrt{c+d x}\right)}{b^6 d^3}-\frac{4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac{(c+d x)^2}{2 b^2 d^3}",1,"(16*a^5 - 44*a^4*b*Sqrt[c + d*x] + 3*b^5*d*x*(-2*c + d*x)*Sqrt[c + d*x] + 2*a^2*b^3*Sqrt[c + d*x]*(18*c + 5*d*x) - 2*a^3*b^2*(38*c + 15*d*x) + a*b^4*(52*c^2 + 26*c*d*x - 5*d^2*x^2) + 12*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/(6*b^6*d^3*(a + b*Sqrt[c + d*x]))","A",1
641,1,112,95,0.0880062,"\int \frac{x}{\left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[x/(a + b*Sqrt[c + d*x])^2,x]","\frac{2 a^3+2 \left(3 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right) \log \left(a+b \sqrt{c+d x}\right)-4 a^2 b \sqrt{c+d x}-3 a b^2 (2 c+d x)+b^3 d x \sqrt{c+d x}}{b^4 d^2 \left(a+b \sqrt{c+d x}\right)}","\frac{2 a \left(a^2-b^2 c\right)}{b^4 d^2 \left(a+b \sqrt{c+d x}\right)}+\frac{2 \left(3 a^2-b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{b^4 d^2}-\frac{4 a \sqrt{c+d x}}{b^3 d^2}+\frac{x}{b^2 d}",1,"(2*a^3 - 4*a^2*b*Sqrt[c + d*x] + b^3*d*x*Sqrt[c + d*x] - 3*a*b^2*(2*c + d*x) + 2*(3*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2*(a + b*Sqrt[c + d*x]))","A",1
642,1,40,47,0.0336575,"\int \frac{1}{\left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[(a + b*Sqrt[c + d*x])^(-2),x]","\frac{2 \left(\frac{a}{a+b \sqrt{c+d x}}+\log \left(a+b \sqrt{c+d x}\right)\right)}{b^2 d}","\frac{2 a}{b^2 d \left(a+b \sqrt{c+d x}\right)}+\frac{2 \log \left(a+b \sqrt{c+d x}\right)}{b^2 d}",1,"(2*(a/(a + b*Sqrt[c + d*x]) + Log[a + b*Sqrt[c + d*x]]))/(b^2*d)","A",1
643,1,164,129,0.276916,"\int \frac{1}{x \left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[1/(x*(a + b*Sqrt[c + d*x])^2),x]","\frac{2 a^3-2 \left(a^2+b^2 c\right) \left(a+b \sqrt{c+d x}\right) \log \left(a+b \sqrt{c+d x}\right)-2 a b^2 c+\left(a-b \sqrt{c}\right)^2 \log \left(\sqrt{c}-\sqrt{c+d x}\right) \left(a+b \sqrt{c+d x}\right)+\left(a+b \sqrt{c}\right)^2 \log \left(\sqrt{c+d x}+\sqrt{c}\right) \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^2 \left(a+b \sqrt{c+d x}\right)}","\frac{2 a}{\left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)}-\frac{2 \left(a^2+b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^2}+\frac{4 a b \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{\left(a^2-b^2 c\right)^2}+\frac{\log (x) \left(a^2+b^2 c\right)}{\left(a^2-b^2 c\right)^2}",1,"(2*a^3 - 2*a*b^2*c + (a - b*Sqrt[c])^2*(a + b*Sqrt[c + d*x])*Log[Sqrt[c] - Sqrt[c + d*x]] + (a + b*Sqrt[c])^2*(a + b*Sqrt[c + d*x])*Log[Sqrt[c] + Sqrt[c + d*x]] - 2*(a^2 + b^2*c)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/((a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x]))","A",1
644,1,230,202,0.7583656,"\int \frac{1}{x^2 \left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[1/(x^2*(a + b*Sqrt[c + d*x])^2),x]","\frac{\frac{\sqrt{c} \left(2 b^2 \sqrt{c} d \left(3 a^2+b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)+\frac{\sqrt{c} \left(a^2-b^2 c\right) \left(a^3-a^2 b \sqrt{c+d x}-a b^2 (c+4 d x)+b^3 c \sqrt{c+d x}\right)}{x \left(a+b \sqrt{c+d x}\right)}-b d \left(a+b \sqrt{c}\right)^3 \log \left(\sqrt{c+d x}+\sqrt{c}\right)\right)}{\left(a^2-b^2 c\right)^2}+\frac{\left(a b \sqrt{c} d-b^2 c d\right) \log \left(\sqrt{c}-\sqrt{c+d x}\right)}{\left(a+b \sqrt{c}\right)^2}}{c \left(b^2 c-a^2\right)}","\frac{4 a b^2 d}{\left(a^2-b^2 c\right)^2 \left(a+b \sqrt{c+d x}\right)}-\frac{a-b \sqrt{c+d x}}{x \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)}+\frac{b^2 d \log (x) \left(3 a^2+b^2 c\right)}{\left(a^2-b^2 c\right)^3}-\frac{2 b^2 d \left(3 a^2+b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^3}+\frac{2 a b d \left(a^2+3 b^2 c\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{\sqrt{c} \left(a^2-b^2 c\right)^3}",1,"(((a*b*Sqrt[c]*d - b^2*c*d)*Log[Sqrt[c] - Sqrt[c + d*x]])/(a + b*Sqrt[c])^2 + (Sqrt[c]*((Sqrt[c]*(a^2 - b^2*c)*(a^3 - a^2*b*Sqrt[c + d*x] + b^3*c*Sqrt[c + d*x] - a*b^2*(c + 4*d*x)))/(x*(a + b*Sqrt[c + d*x])) - b*(a + b*Sqrt[c])^3*d*Log[Sqrt[c] + Sqrt[c + d*x]] + 2*b^2*Sqrt[c]*(3*a^2 + b^2*c)*d*Log[a + b*Sqrt[c + d*x]]))/(a^2 - b^2*c)^2)/(c*(-a^2 + b^2*c))","A",1
645,1,401,306,0.8259185,"\int \frac{1}{x^3 \left(a+b \sqrt{c+d x}\right)^2} \, dx","Integrate[1/(x^3*(a + b*Sqrt[c + d*x])^2),x]","\frac{\frac{d^2 \left(\frac{2 b \sqrt{c} \left(a^2+2 b^2 c\right) \left(\left(b \sqrt{c}-a\right) \log \left(\sqrt{c}-\sqrt{c+d x}\right)+\left(a+b \sqrt{c}\right) \log \left(\sqrt{c+d x}+\sqrt{c}\right)-2 b \sqrt{c} \log \left(a+b \sqrt{c+d x}\right)\right)}{b^2 c-a^2}-a b c \left(a^2+11 b^2 c\right) \left(\frac{2 b \left(\frac{b^2 c-a^2}{a+b \sqrt{c+d x}}+2 a \log \left(a+b \sqrt{c+d x}\right)\right)}{\left(a^2-b^2 c\right)^2}+\frac{\log \left(\sqrt{c}-\sqrt{c+d x}\right)}{\sqrt{c} \left(a+b \sqrt{c}\right)^2}-\frac{\log \left(\sqrt{c+d x}+\sqrt{c}\right)}{\sqrt{c} \left(a-b \sqrt{c}\right)^2}\right)\right)}{2 c \left(a^2-b^2 c\right)}+\frac{b d \left(a^2 \sqrt{c+d x}-3 a b c+2 b^2 c \sqrt{c+d x}\right)}{x \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)}-\frac{c \left(a-b \sqrt{c+d x}\right)}{x^2 \left(a+b \sqrt{c+d x}\right)}}{2 c \left(a^2-b^2 c\right)}","\frac{a b^2 d^2 \left(a^2+11 b^2 c\right)}{2 c \left(a^2-b^2 c\right)^3 \left(a+b \sqrt{c+d x}\right)}-\frac{a-b \sqrt{c+d x}}{2 x^2 \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)}-\frac{b d \left(3 a b c-\left(a^2+2 b^2 c\right) \sqrt{c+d x}\right)}{2 c x \left(a^2-b^2 c\right)^2 \left(a+b \sqrt{c+d x}\right)}+\frac{b^4 d^2 \log (x) \left(5 a^2+b^2 c\right)}{\left(a^2-b^2 c\right)^4}-\frac{2 b^4 d^2 \left(5 a^2+b^2 c\right) \log \left(a+b \sqrt{c+d x}\right)}{\left(a^2-b^2 c\right)^4}-\frac{a b d^2 \left(a^4-10 a^2 b^2 c-15 b^4 c^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d x}}{\sqrt{c}}\right)}{2 c^{3/2} \left(a^2-b^2 c\right)^4}",1,"(-((c*(a - b*Sqrt[c + d*x]))/(x^2*(a + b*Sqrt[c + d*x]))) + (b*d*(-3*a*b*c + a^2*Sqrt[c + d*x] + 2*b^2*c*Sqrt[c + d*x]))/((a^2 - b^2*c)*x*(a + b*Sqrt[c + d*x])) + (d^2*((2*b*Sqrt[c]*(a^2 + 2*b^2*c)*((-a + b*Sqrt[c])*Log[Sqrt[c] - Sqrt[c + d*x]] + (a + b*Sqrt[c])*Log[Sqrt[c] + Sqrt[c + d*x]] - 2*b*Sqrt[c]*Log[a + b*Sqrt[c + d*x]]))/(-a^2 + b^2*c) - a*b*c*(a^2 + 11*b^2*c)*(Log[Sqrt[c] - Sqrt[c + d*x]]/((a + b*Sqrt[c])^2*Sqrt[c]) - Log[Sqrt[c] + Sqrt[c + d*x]]/((a - b*Sqrt[c])^2*Sqrt[c]) + (2*b*((-a^2 + b^2*c)/(a + b*Sqrt[c + d*x]) + 2*a*Log[a + b*Sqrt[c + d*x]]))/(a^2 - b^2*c)^2)))/(2*c*(a^2 - b^2*c)))/(2*c*(a^2 - b^2*c))","A",1
646,1,232,324,0.3485097,"\int \frac{x^3}{\sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[x^3/Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \sqrt{a+b \sqrt{c+d x}} \left(-14336 a^7+7168 a^6 b \sqrt{c+d x}+768 a^5 b^2 (58 c-7 d x)-640 a^4 b^3 (32 c-7 d x) \sqrt{c+d x}-16 a^3 b^4 \left(2936 c^2-680 c d x+245 d^2 x^2\right)+24 a^2 b^5 \sqrt{c+d x} \left(784 c^2-356 c d x+147 d^2 x^2\right)+6 a b^6 \left(2880 c^3-928 c^2 d x+658 c d^2 x^2-539 d^3 x^3\right)-39 b^7 \sqrt{c+d x} \left(128 c^3-96 c^2 d x+84 c d^2 x^2-77 d^3 x^3\right)\right)}{45045 b^8 d^4}","\frac{12 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{11/2}}{11 b^8 d^4}-\frac{20 a \left(7 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{9/2}}{9 b^8 d^4}-\frac{12 a \left(7 a^2-3 b^2 c\right) \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^8 d^4}+\frac{4 \left(a^2-b^2 c\right)^2 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^8 d^4}-\frac{4 a \left(a^2-b^2 c\right)^3 \sqrt{a+b \sqrt{c+d x}}}{b^8 d^4}+\frac{4 \left(35 a^4-30 a^2 b^2 c+3 b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^8 d^4}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{15/2}}{15 b^8 d^4}-\frac{28 a \left(a+b \sqrt{c+d x}\right)^{13/2}}{13 b^8 d^4}",1,"(4*Sqrt[a + b*Sqrt[c + d*x]]*(-14336*a^7 + 768*a^5*b^2*(58*c - 7*d*x) + 7168*a^6*b*Sqrt[c + d*x] - 640*a^4*b^3*(32*c - 7*d*x)*Sqrt[c + d*x] + 24*a^2*b^5*Sqrt[c + d*x]*(784*c^2 - 356*c*d*x + 147*d^2*x^2) - 16*a^3*b^4*(2936*c^2 - 680*c*d*x + 245*d^2*x^2) + 6*a*b^6*(2880*c^3 - 928*c^2*d*x + 658*c*d^2*x^2 - 539*d^3*x^3) - 39*b^7*Sqrt[c + d*x]*(128*c^3 - 96*c^2*d*x + 84*c*d^2*x^2 - 77*d^3*x^3)))/(45045*b^8*d^4)","A",1
647,1,147,222,0.1699021,"\int \frac{x^2}{\sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[x^2/Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \sqrt{a+b \sqrt{c+d x}} \left(-1280 a^5+640 a^4 b \sqrt{c+d x}+96 a^3 b^2 (28 c-5 d x)-16 a^2 b^3 (74 c-25 d x) \sqrt{c+d x}-2 a b^4 \left(736 c^2-244 c d x+175 d^2 x^2\right)+15 b^5 \sqrt{c+d x} \left(32 c^2-24 c d x+21 d^2 x^2\right)\right)}{3465 b^6 d^3}","\frac{8 \left(5 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^6 d^3}-\frac{8 a \left(5 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^6 d^3}-\frac{4 a \left(a^2-b^2 c\right)^2 \sqrt{a+b \sqrt{c+d x}}}{b^6 d^3}+\frac{4 \left(5 a^4-6 a^2 b^2 c+b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^6 d^3}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{11/2}}{11 b^6 d^3}-\frac{20 a \left(a+b \sqrt{c+d x}\right)^{9/2}}{9 b^6 d^3}",1,"(4*Sqrt[a + b*Sqrt[c + d*x]]*(-1280*a^5 + 96*a^3*b^2*(28*c - 5*d*x) + 640*a^4*b*Sqrt[c + d*x] - 16*a^2*b^3*(74*c - 25*d*x)*Sqrt[c + d*x] + 15*b^5*Sqrt[c + d*x]*(32*c^2 - 24*c*d*x + 21*d^2*x^2) - 2*a*b^4*(736*c^2 - 244*c*d*x + 175*d^2*x^2)))/(3465*b^6*d^3)","A",1
648,1,84,131,0.0976956,"\int \frac{x}{\sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[x/Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \sqrt{a+b \sqrt{c+d x}} \left(-48 a^3+24 a^2 b \sqrt{c+d x}+2 a b^2 (26 c-9 d x)+5 b^3 \sqrt{c+d x} (3 d x-4 c)\right)}{105 b^4 d^2}","\frac{4 \left(3 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^4 d^2}-\frac{4 a \left(a^2-b^2 c\right) \sqrt{a+b \sqrt{c+d x}}}{b^4 d^2}+\frac{4 \left(a+b \sqrt{c+d x}\right)^{7/2}}{7 b^4 d^2}-\frac{12 a \left(a+b \sqrt{c+d x}\right)^{5/2}}{5 b^4 d^2}",1,"(4*Sqrt[a + b*Sqrt[c + d*x]]*(-48*a^3 + 2*a*b^2*(26*c - 9*d*x) + 24*a^2*b*Sqrt[c + d*x] + 5*b^3*Sqrt[c + d*x]*(-4*c + 3*d*x)))/(105*b^4*d^2)","A",1
649,1,42,54,0.0221721,"\int \frac{1}{\sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[1/Sqrt[a + b*Sqrt[c + d*x]],x]","\frac{4 \left(b \sqrt{c+d x}-2 a\right) \sqrt{a+b \sqrt{c+d x}}}{3 b^2 d}","\frac{4 \left(a+b \sqrt{c+d x}\right)^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d}",1,"(4*(-2*a + b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/(3*b^2*d)","A",1
650,1,97,97,0.10322,"\int \frac{1}{x \sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[1/(x*Sqrt[a + b*Sqrt[c + d*x]]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{\sqrt{a+b \sqrt{c}}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{\sqrt{a-b \sqrt{c}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{\sqrt{a+b \sqrt{c}}}",1,"(-2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/Sqrt[a - b*Sqrt[c]] - (2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/Sqrt[a + b*Sqrt[c]]","A",1
651,1,216,163,0.2527906,"\int \frac{1}{x^2 \sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[1/(x^2*Sqrt[a + b*Sqrt[c + d*x]]),x]","\frac{\sqrt{a-b \sqrt{c}} \left(b d x \left(a-b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)-2 \sqrt{c} \sqrt{a+b \sqrt{c}} \left(a-b \sqrt{c+d x}\right) \sqrt{a+b \sqrt{c+d x}}\right)-b d x \left(a+b \sqrt{c}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{2 \sqrt{c} x \sqrt{a-b \sqrt{c}} \sqrt{a+b \sqrt{c}} \left(a^2-b^2 c\right)}","-\frac{\sqrt{a+b \sqrt{c+d x}} \left(a-b \sqrt{c+d x}\right)}{x \left(a^2-b^2 c\right)}-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{2 \sqrt{c} \left(a-b \sqrt{c}\right)^{3/2}}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{2 \sqrt{c} \left(a+b \sqrt{c}\right)^{3/2}}",1,"(-(b*(a + b*Sqrt[c])^(3/2)*d*x*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) + Sqrt[a - b*Sqrt[c]]*(-2*Sqrt[a + b*Sqrt[c]]*Sqrt[c]*(a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]] + b*(a - b*Sqrt[c])*d*x*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(2*Sqrt[a - b*Sqrt[c]]*Sqrt[a + b*Sqrt[c]]*Sqrt[c]*(a^2 - b^2*c)*x)","A",1
652,1,281,261,0.7504776,"\int \frac{1}{x^3 \sqrt{a+b \sqrt{c+d x}}} \, dx","Integrate[1/(x^3*Sqrt[a + b*Sqrt[c + d*x]]),x]","-\frac{\frac{8 \left(a^2-b^2 c\right) \left(a-b \sqrt{c+d x}\right) \sqrt{a+b \sqrt{c+d x}}}{x^2}-\frac{2 b d \sqrt{a+b \sqrt{c+d x}} \left(a^2 \sqrt{c+d x}-6 a b c+5 b^2 c \sqrt{c+d x}\right)}{c x}+\frac{b d^2 \left(\left(a-b \sqrt{c}\right)^{5/2} \left(2 a+5 b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)-\left(2 a-5 b \sqrt{c}\right) \left(a+b \sqrt{c}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)\right)}{c^{3/2} \sqrt{a-b \sqrt{c}} \sqrt{a+b \sqrt{c}}}}{16 \left(a^2-b^2 c\right)^2}","-\frac{\left(a-b \sqrt{c+d x}\right) \sqrt{a+b \sqrt{c+d x}}}{2 x^2 \left(a^2-b^2 c\right)}-\frac{b d \sqrt{a+b \sqrt{c+d x}} \left(6 a b c-\left(a^2+5 b^2 c\right) \sqrt{c+d x}\right)}{8 c x \left(a^2-b^2 c\right)^2}+\frac{b d^2 \left(2 a-5 b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right)}{16 c^{3/2} \left(a-b \sqrt{c}\right)^{5/2}}-\frac{b d^2 \left(2 a+5 b \sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right)}{16 c^{3/2} \left(a+b \sqrt{c}\right)^{5/2}}",1,"-1/16*((8*(a^2 - b^2*c)*(a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/x^2 - (2*b*d*Sqrt[a + b*Sqrt[c + d*x]]*(-6*a*b*c + a^2*Sqrt[c + d*x] + 5*b^2*c*Sqrt[c + d*x]))/(c*x) + (b*d^2*(-((2*a - 5*b*Sqrt[c])*(a + b*Sqrt[c])^(5/2)*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) + (a - b*Sqrt[c])^(5/2)*(2*a + 5*b*Sqrt[c])*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(Sqrt[a - b*Sqrt[c]]*Sqrt[a + b*Sqrt[c]]*c^(3/2)))/(a^2 - b^2*c)^2","A",1
653,1,555,350,0.8519415,"\int x^3 \left(a+b \sqrt{c+d x}\right)^p \, dx","Integrate[x^3*(a + b*Sqrt[c + d*x])^p,x]","-\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+1} \left(5040 a^7-5040 a^6 b (p+1) \sqrt{c+d x}+360 a^5 b^2 \left(6 c \left(p^2+p-7\right)+7 d \left(p^2+3 p+2\right) x\right)-120 a^4 b^3 (p+1) \sqrt{c+d x} \left(2 c \left(2 p^2-5 p-63\right)+7 d \left(p^2+5 p+6\right) x\right)+6 a^3 b^4 \left(8 c^2 \left(p^4-14 p^3-139 p^2-124 p+315\right)+40 c d \left(p^4+4 p^3-16 p^2-61 p-42\right) x+35 d^2 \left(p^4+10 p^3+35 p^2+50 p+24\right) x^2\right)-6 a^2 b^5 (p+1) \sqrt{c+d x} \left(-24 c^2 \left(p^3+5 p^2-24 p-105\right)+4 c d \left(p^4-p^3-94 p^2-386 p-420\right) x+7 d^2 \left(p^4+14 p^3+71 p^2+154 p+120\right) x^2\right)+a b^6 \left(48 c^3 \left(3 p^4+38 p^3+138 p^2+103 p-105\right)-24 c^2 d \left(2 p^5+24 p^4+74 p^3-21 p^2-283 p-210\right) x+6 c d^2 \left(p^6+11 p^5+10 p^4-265 p^3-1151 p^2-1726 p-840\right) x^2+7 d^3 \left(p^6+21 p^5+175 p^4+735 p^3+1624 p^2+1764 p+720\right) x^3\right)-b^7 \left(p^4+16 p^3+86 p^2+176 p+105\right) \sqrt{c+d x} \left(-48 c^3+24 c^2 d (p+2) x-6 c d^2 \left(p^2+6 p+8\right) x^2+d^3 \left(p^3+12 p^2+44 p+48\right) x^3\right)\right)}{b^8 d^4 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6) (p+7) (p+8)}","-\frac{2 a \left(a^2-b^2 c\right)^3 \left(a+b \sqrt{c+d x}\right)^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left(a^2-b^2 c\right)^2 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left(7 a^2-3 b^2 c\right) \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left(7 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left(7 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+6}}{b^8 d^4 (p+6)}+\frac{2 \left(35 a^4-30 a^2 b^2 c+3 b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{p+4}}{b^8 d^4 (p+4)}-\frac{14 a \left(a+b \sqrt{c+d x}\right)^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+8}}{b^8 d^4 (p+8)}",1,"(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(5040*a^7 - 5040*a^6*b*(1 + p)*Sqrt[c + d*x] + 360*a^5*b^2*(6*c*(-7 + p + p^2) + 7*d*(2 + 3*p + p^2)*x) - 120*a^4*b^3*(1 + p)*Sqrt[c + d*x]*(2*c*(-63 - 5*p + 2*p^2) + 7*d*(6 + 5*p + p^2)*x) + 6*a^3*b^4*(8*c^2*(315 - 124*p - 139*p^2 - 14*p^3 + p^4) + 40*c*d*(-42 - 61*p - 16*p^2 + 4*p^3 + p^4)*x + 35*d^2*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2) - 6*a^2*b^5*(1 + p)*Sqrt[c + d*x]*(-24*c^2*(-105 - 24*p + 5*p^2 + p^3) + 4*c*d*(-420 - 386*p - 94*p^2 - p^3 + p^4)*x + 7*d^2*(120 + 154*p + 71*p^2 + 14*p^3 + p^4)*x^2) - b^7*(105 + 176*p + 86*p^2 + 16*p^3 + p^4)*Sqrt[c + d*x]*(-48*c^3 + 24*c^2*d*(2 + p)*x - 6*c*d^2*(8 + 6*p + p^2)*x^2 + d^3*(48 + 44*p + 12*p^2 + p^3)*x^3) + a*b^6*(48*c^3*(-105 + 103*p + 138*p^2 + 38*p^3 + 3*p^4) - 24*c^2*d*(-210 - 283*p - 21*p^2 + 74*p^3 + 24*p^4 + 2*p^5)*x + 6*c*d^2*(-840 - 1726*p - 1151*p^2 - 265*p^3 + 10*p^4 + 11*p^5 + p^6)*x^2 + 7*d^3*(720 + 1764*p + 1624*p^2 + 735*p^3 + 175*p^4 + 21*p^5 + p^6)*x^3)))/(b^8*d^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p)*(7 + p)*(8 + p))","A",1
654,1,284,242,0.3741217,"\int x^2 \left(a+b \sqrt{c+d x}\right)^p \, dx","Integrate[x^2*(a + b*Sqrt[c + d*x])^p,x]","-\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+1} \left(120 a^5-120 a^4 b (p+1) \sqrt{c+d x}+12 a^3 b^2 \left(4 c \left(p^2+p-5\right)+5 d \left(p^2+3 p+2\right) x\right)-4 a^2 b^3 (p+1) \sqrt{c+d x} \left(2 c \left(p^2-4 p-30\right)+5 d \left(p^2+5 p+6\right) x\right)+a b^4 \left(-8 c^2 \left(2 p^3+12 p^2+10 p-15\right)+4 c d \left(p^4+4 p^3-10 p^2-43 p-30\right) x+5 d^2 \left(p^4+10 p^3+35 p^2+50 p+24\right) x^2\right)-b^5 \left(p^3+9 p^2+23 p+15\right) \sqrt{c+d x} \left(8 c^2-4 c d (p+2) x+d^2 \left(p^2+6 p+8\right) x^2\right)\right)}{b^6 d^3 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6)}","-\frac{2 a \left(a^2-b^2 c\right)^2 \left(a+b \sqrt{c+d x}\right)^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left(5 a^2-3 b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left(5 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+4}}{b^6 d^3 (p+4)}+\frac{2 \left(5 a^4-6 a^2 b^2 c+b^4 c^2\right) \left(a+b \sqrt{c+d x}\right)^{p+2}}{b^6 d^3 (p+2)}-\frac{10 a \left(a+b \sqrt{c+d x}\right)^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+6}}{b^6 d^3 (p+6)}",1,"(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(120*a^5 - 120*a^4*b*(1 + p)*Sqrt[c + d*x] + 12*a^3*b^2*(4*c*(-5 + p + p^2) + 5*d*(2 + 3*p + p^2)*x) - 4*a^2*b^3*(1 + p)*Sqrt[c + d*x]*(2*c*(-30 - 4*p + p^2) + 5*d*(6 + 5*p + p^2)*x) - b^5*(15 + 23*p + 9*p^2 + p^3)*Sqrt[c + d*x]*(8*c^2 - 4*c*d*(2 + p)*x + d^2*(8 + 6*p + p^2)*x^2) + a*b^4*(-8*c^2*(-15 + 10*p + 12*p^2 + 2*p^3) + 4*c*d*(-30 - 43*p - 10*p^2 + 4*p^3 + p^4)*x + 5*d^2*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2)))/(b^6*d^3*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p))","A",1
655,1,128,145,0.1627359,"\int x \left(a+b \sqrt{c+d x}\right)^p \, dx","Integrate[x*(a + b*Sqrt[c + d*x])^p,x]","-\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+1} \left(6 a^3-6 a^2 b (p+1) \sqrt{c+d x}+a b^2 \left(2 c \left(p^2+p-3\right)+3 d \left(p^2+3 p+2\right) x\right)-b^3 \left(p^2+4 p+3\right) \sqrt{c+d x} (d (p+2) x-2 c)\right)}{b^4 d^2 (p+1) (p+2) (p+3) (p+4)}","-\frac{2 a \left(a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+1}}{b^4 d^2 (p+1)}+\frac{2 \left(3 a^2-b^2 c\right) \left(a+b \sqrt{c+d x}\right)^{p+2}}{b^4 d^2 (p+2)}-\frac{6 a \left(a+b \sqrt{c+d x}\right)^{p+3}}{b^4 d^2 (p+3)}+\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+4}}{b^4 d^2 (p+4)}",1,"(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(6*a^3 - 6*a^2*b*(1 + p)*Sqrt[c + d*x] - b^3*(3 + 4*p + p^2)*Sqrt[c + d*x]*(-2*c + d*(2 + p)*x) + a*b^2*(2*c*(-3 + p + p^2) + 3*d*(2 + 3*p + p^2)*x)))/(b^4*d^2*(1 + p)*(2 + p)*(3 + p)*(4 + p))","A",1
656,1,53,62,0.0358906,"\int \left(a+b \sqrt{c+d x}\right)^p \, dx","Integrate[(a + b*Sqrt[c + d*x])^p,x]","\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+1} \left(b (p+1) \sqrt{c+d x}-a\right)}{b^2 d (p+1) (p+2)}","\frac{2 \left(a+b \sqrt{c+d x}\right)^{p+2}}{b^2 d (p+2)}-\frac{2 a \left(a+b \sqrt{c+d x}\right)^{p+1}}{b^2 d (p+1)}",1,"(2*(a + b*Sqrt[c + d*x])^(1 + p)*(-a + b*(1 + p)*Sqrt[c + d*x]))/(b^2*d*(1 + p)*(2 + p))","A",1
657,1,136,139,0.1060411,"\int \frac{\left(a+b \sqrt{c+d x}\right)^p}{x} \, dx","Integrate[(a + b*Sqrt[c + d*x])^p/x,x]","-\frac{\left(a+b \sqrt{c+d x}\right)^{p+1} \left(\left(a+b \sqrt{c}\right) \, _2F_1\left(1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a-b \sqrt{c}}\right)+\left(a-b \sqrt{c}\right) \, _2F_1\left(1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a+b \sqrt{c}}\right)\right)}{(p+1) \left(a-b \sqrt{c}\right) \left(a+b \sqrt{c}\right)}","-\frac{\left(a+b \sqrt{c+d x}\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a-b \sqrt{c}}\right)}{(p+1) \left(a-b \sqrt{c}\right)}-\frac{\left(a+b \sqrt{c+d x}\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{a+b \sqrt{c+d x}}{a+b \sqrt{c}}\right)}{(p+1) \left(a+b \sqrt{c}\right)}",1,"-(((a + b*Sqrt[c + d*x])^(1 + p)*((a + b*Sqrt[c])*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[c + d*x])/(a - b*Sqrt[c])] + (a - b*Sqrt[c])*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[c + d*x])/(a + b*Sqrt[c])]))/((a - b*Sqrt[c])*(a + b*Sqrt[c])*(1 + p)))","A",1
658,1,77,93,0.0754439,"\int \frac{\left(a+b (c x)^n\right)^{5/2}}{x} \, dx","Integrate[(a + b*(c*x)^n)^(5/2)/x,x]","\frac{2 \sqrt{a+b (c x)^n} \left(23 a^2+11 a b (c x)^n+3 b^2 (c x)^{2 n}\right)-30 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{15 n}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{n}+\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left(a+b (c x)^n\right)^{3/2}}{3 n}+\frac{2 \left(a+b (c x)^n\right)^{5/2}}{5 n}",1,"(2*Sqrt[a + b*(c*x)^n]*(23*a^2 + 11*a*b*(c*x)^n + 3*b^2*(c*x)^(2*n)) - 30*a^(5/2)*ArcTanh[Sqrt[a + b*(c*x)^n]/Sqrt[a]])/(15*n)","A",1
659,1,61,70,0.0404777,"\int \frac{\left(a+b (c x)^n\right)^{3/2}}{x} \, dx","Integrate[(a + b*(c*x)^n)^(3/2)/x,x]","\frac{2 \sqrt{a+b (c x)^n} \left(4 a+b (c x)^n\right)-6 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{3 n}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{n}+\frac{2 a \sqrt{a+b (c x)^n}}{n}+\frac{2 \left(a+b (c x)^n\right)^{3/2}}{3 n}",1,"(2*Sqrt[a + b*(c*x)^n]*(4*a + b*(c*x)^n) - 6*a^(3/2)*ArcTanh[Sqrt[a + b*(c*x)^n]/Sqrt[a]])/(3*n)","A",1
660,1,47,49,0.0221415,"\int \frac{\sqrt{a+b (c x)^n}}{x} \, dx","Integrate[Sqrt[a + b*(c*x)^n]/x,x]","\frac{2 \sqrt{a+b (c x)^n}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{n}","\frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{n}",1,"(2*Sqrt[a + b*(c*x)^n] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*(c*x)^n]/Sqrt[a]])/n","A",1
661,1,30,30,0.018435,"\int \frac{1}{x \sqrt{a+b (c x)^n}} \, dx","Integrate[1/(x*Sqrt[a + b*(c*x)^n]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{\sqrt{a} n}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{\sqrt{a} n}",1,"(-2*ArcTanh[Sqrt[a + b*(c*x)^n]/Sqrt[a]])/(Sqrt[a]*n)","A",1
662,1,41,52,0.0275172,"\int \frac{1}{x \left(a+b (c x)^n\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*(c*x)^n)^(3/2)),x]","\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b (c x)^n}{a}+1\right)}{a n \sqrt{a+b (c x)^n}}","\frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{a^{3/2} n}",1,"(2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*(c*x)^n)/a])/(a*n*Sqrt[a + b*(c*x)^n])","C",1
663,1,43,75,0.033716,"\int \frac{1}{x \left(a+b (c x)^n\right)^{5/2}} \, dx","Integrate[1/(x*(a + b*(c*x)^n)^(5/2)),x]","\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b (c x)^n}{a}+1\right)}{3 a n \left(a+b (c x)^n\right)^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right)}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b (c x)^n}}+\frac{2}{3 a n \left(a+b (c x)^n\right)^{3/2}}",1,"(2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*(c*x)^n)/a])/(3*a*n*(a + b*(c*x)^n)^(3/2))","C",1
664,1,81,101,0.0788107,"\int \frac{\left(-a+b (c x)^n\right)^{5/2}}{x} \, dx","Integrate[(-a + b*(c*x)^n)^(5/2)/x,x]","\frac{2 \sqrt{b (c x)^n-a} \left(23 a^2-11 a b (c x)^n+3 b^2 (c x)^{2 n}\right)-30 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{15 n}","-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{n}+\frac{2 a^2 \sqrt{b (c x)^n-a}}{n}-\frac{2 a \left(b (c x)^n-a\right)^{3/2}}{3 n}+\frac{2 \left(b (c x)^n-a\right)^{5/2}}{5 n}",1,"(2*Sqrt[-a + b*(c*x)^n]*(23*a^2 - 11*a*b*(c*x)^n + 3*b^2*(c*x)^(2*n)) - 30*a^(5/2)*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/(15*n)","A",1
665,1,66,76,0.0472375,"\int \frac{\left(-a+b (c x)^n\right)^{3/2}}{x} \, dx","Integrate[(-a + b*(c*x)^n)^(3/2)/x,x]","\frac{6 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)-2 \left(4 a-b (c x)^n\right) \sqrt{b (c x)^n-a}}{3 n}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{n}-\frac{2 a \sqrt{b (c x)^n-a}}{n}+\frac{2 \left(b (c x)^n-a\right)^{3/2}}{3 n}",1,"(-2*(4*a - b*(c*x)^n)*Sqrt[-a + b*(c*x)^n] + 6*a^(3/2)*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/(3*n)","A",1
666,1,51,53,0.0226954,"\int \frac{\sqrt{-a+b (c x)^n}}{x} \, dx","Integrate[Sqrt[-a + b*(c*x)^n]/x,x]","\frac{2 \sqrt{b (c x)^n-a}-2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{n}","\frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{n}",1,"(2*Sqrt[-a + b*(c*x)^n] - 2*Sqrt[a]*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/n","A",1
667,1,32,32,0.0209265,"\int \frac{1}{x \sqrt{-a+b (c x)^n}} \, dx","Integrate[1/(x*Sqrt[-a + b*(c*x)^n]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{\sqrt{a} n}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{\sqrt{a} n}",1,"(2*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/(Sqrt[a]*n)","A",1
668,1,44,56,0.0295688,"\int \frac{1}{x \left(-a+b (c x)^n\right)^{3/2}} \, dx","Integrate[1/(x*(-a + b*(c*x)^n)^(3/2)),x]","-\frac{2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{b (c x)^n}{a}\right)}{a n \sqrt{b (c x)^n-a}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{a^{3/2} n}-\frac{2}{a n \sqrt{b (c x)^n-a}}",1,"(-2*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (b*(c*x)^n)/a])/(a*n*Sqrt[-a + b*(c*x)^n])","C",1
669,1,46,81,0.0374333,"\int \frac{1}{x \left(-a+b (c x)^n\right)^{5/2}} \, dx","Integrate[1/(x*(-a + b*(c*x)^n)^(5/2)),x]","-\frac{2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{b (c x)^n}{a}\right)}{3 a n \left(b (c x)^n-a\right)^{3/2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right)}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{b (c x)^n-a}}-\frac{2}{3 a n \left(b (c x)^n-a\right)^{3/2}}",1,"(-2*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (b*(c*x)^n)/a])/(3*a*n*(-a + b*(c*x)^n)^(3/2))","C",1
670,1,23,23,0.0045852,"\int \frac{1}{x \sqrt{a+b x}} \, dx","Integrate[1/(x*Sqrt[a + b*x]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b x}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(-2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a]","A",1
671,1,30,30,0.0322426,"\int \frac{1}{x \sqrt{a+b (c x)^m}} \, dx","Integrate[1/(x*Sqrt[a + b*(c*x)^m]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^m}}{\sqrt{a}}\right)}{\sqrt{a} m}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b (c x)^m}}{\sqrt{a}}\right)}{\sqrt{a} m}",1,"(-2*ArcTanh[Sqrt[a + b*(c*x)^m]/Sqrt[a]])/(Sqrt[a]*m)","A",1
672,1,37,37,0.0654038,"\int \frac{1}{x \sqrt{a+b \left(c (d x)^m\right)^n}} \, dx","Integrate[1/(x*Sqrt[a + b*(c*(d*x)^m)^n]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c (d x)^m\right)^n}}{\sqrt{a}}\right)}{\sqrt{a} m n}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c (d x)^m\right)^n}}{\sqrt{a}}\right)}{\sqrt{a} m n}",1,"(-2*ArcTanh[Sqrt[a + b*(c*(d*x)^m)^n]/Sqrt[a]])/(Sqrt[a]*m*n)","A",1
673,1,44,44,0.145819,"\int \frac{1}{x \sqrt{a+b \left(c \left(d (e x)^m\right)^n\right)^p}} \, dx","Integrate[1/(x*Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c \left(d (e x)^m\right)^n\right)^p}}{\sqrt{a}}\right)}{\sqrt{a} m n p}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c \left(d (e x)^m\right)^n\right)^p}}{\sqrt{a}}\right)}{\sqrt{a} m n p}",1,"(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*x)^m)^n)^p]/Sqrt[a]])/(Sqrt[a]*m*n*p)","A",1
674,1,51,51,0.2600854,"\int \frac{1}{x \sqrt{a+b \left(c \left(d \left(e (f x)^m\right)^n\right)^p\right)^q}} \, dx","Integrate[1/(x*Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c \left(d \left(e (f x)^m\right)^n\right)^p\right)^q}}{\sqrt{a}}\right)}{\sqrt{a} m n p q}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \left(c \left(d \left(e (f x)^m\right)^n\right)^p\right)^q}}{\sqrt{a}}\right)}{\sqrt{a} m n p q}",1,"(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]/Sqrt[a]])/(Sqrt[a]*m*n*p*q)","A",1
675,1,34,76,0.0104192,"\int \frac{\sqrt{-1+\frac{1}{x^2}} \left(-1+x^2\right)^3}{x} \, dx","Integrate[(Sqrt[-1 + x^(-2)]*(-1 + x^2)^3)/x,x]","\frac{\sqrt{\frac{1}{x^2}-1} \, _2F_1\left(-\frac{7}{2},-\frac{1}{2};\frac{1}{2};x^2\right)}{\sqrt{1-x^2}}","-\frac{35}{48} \left(\frac{1}{x^2}-1\right)^{3/2} x^2+\frac{35}{16} \sqrt{\frac{1}{x^2}-1}-\frac{35}{16} \tan ^{-1}\left(\sqrt{\frac{1}{x^2}-1}\right)-\frac{1}{6} \left(\frac{1}{x^2}-1\right)^{7/2} x^6-\frac{7}{24} \left(\frac{1}{x^2}-1\right)^{5/2} x^4",1,"(Sqrt[-1 + x^(-2)]*Hypergeometric2F1[-7/2, -1/2, 1/2, x^2])/Sqrt[1 - x^2]","C",1
676,1,35,60,0.0071869,"\int \frac{\sqrt{-1+\frac{1}{x^2}} \left(-1+x^2\right)^2}{x} \, dx","Integrate[(Sqrt[-1 + x^(-2)]*(-1 + x^2)^2)/x,x]","-\frac{\sqrt{\frac{1}{x^2}-1} \, _2F_1\left(-\frac{5}{2},-\frac{1}{2};\frac{1}{2};x^2\right)}{\sqrt{1-x^2}}","\frac{5}{8} \left(\frac{1}{x^2}-1\right)^{3/2} x^2-\frac{15}{8} \sqrt{\frac{1}{x^2}-1}+\frac{15}{8} \tan ^{-1}\left(\sqrt{\frac{1}{x^2}-1}\right)+\frac{1}{4} \left(\frac{1}{x^2}-1\right)^{5/2} x^4",1,"-((Sqrt[-1 + x^(-2)]*Hypergeometric2F1[-5/2, -1/2, 1/2, x^2])/Sqrt[1 - x^2])","C",1
677,1,34,44,0.0065557,"\int \frac{\sqrt{-1+\frac{1}{x^2}} \left(-1+x^2\right)}{x} \, dx","Integrate[(Sqrt[-1 + x^(-2)]*(-1 + x^2))/x,x]","\frac{\sqrt{\frac{1}{x^2}-1} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};x^2\right)}{\sqrt{1-x^2}}","-\frac{1}{2} \left(\frac{1}{x^2}-1\right)^{3/2} x^2+\frac{3}{2} \sqrt{\frac{1}{x^2}-1}-\frac{3}{2} \tan ^{-1}\left(\sqrt{\frac{1}{x^2}-1}\right)",1,"(Sqrt[-1 + x^(-2)]*Hypergeometric2F1[-3/2, -1/2, 1/2, x^2])/Sqrt[1 - x^2]","C",1
678,1,9,9,0.0028154,"\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left(-1+x^2\right)} \, dx","Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)),x]","\sqrt{\frac{1}{x^2}-1}","\sqrt{\frac{1}{x^2}-1}",1,"Sqrt[-1 + x^(-2)]","A",1
679,1,24,21,0.0060058,"\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left(-1+x^2\right)^2} \, dx","Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^2),x]","\frac{\sqrt{\frac{1}{x^2}-1} \left(1-2 x^2\right)}{x^2-1}","\frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1}",1,"(Sqrt[-1 + x^(-2)]*(1 - 2*x^2))/(-1 + x^2)","A",1
680,1,32,34,0.0070918,"\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left(-1+x^2\right)^3} \, dx","Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^3),x]","\frac{\sqrt{\frac{1}{x^2}-1} \left(8 x^4-12 x^2+3\right)}{3 \left(x^2-1\right)^2}","\sqrt{\frac{1}{x^2}-1}-\frac{2}{\sqrt{\frac{1}{x^2}-1}}-\frac{1}{3 \left(\frac{1}{x^2}-1\right)^{3/2}}",1,"(Sqrt[-1 + x^(-2)]*(3 - 12*x^2 + 8*x^4))/(3*(-1 + x^2)^2)","A",1
681,1,20,9,0.0059831,"\int \frac{\sqrt{1+\frac{1}{x^2}} x}{\left(1+x^2\right)^2} \, dx","Integrate[(Sqrt[1 + x^(-2)]*x)/(1 + x^2)^2,x]","\frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1}","\frac{1}{\sqrt{\frac{1}{x^2}+1}}",1,"(Sqrt[1 + x^(-2)]*x^2)/(1 + x^2)","B",1
682,1,20,9,0.0039676,"\int \frac{1}{\sqrt{1+\frac{1}{x^2}} x \left(1+x^2\right)} \, dx","Integrate[1/(Sqrt[1 + x^(-2)]*x*(1 + x^2)),x]","\frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1}","\frac{1}{\sqrt{\frac{1}{x^2}+1}}",1,"(Sqrt[1 + x^(-2)]*x^2)/(1 + x^2)","B",1
683,1,18,18,0.0419537,"\int \frac{x}{a+b x^2+\sqrt{a+b x^2}} \, dx","Integrate[x/(a + b*x^2 + Sqrt[a + b*x^2]),x]","\frac{\log \left(\sqrt{a+b x^2}+1\right)}{b}","\frac{\log \left(\sqrt{a+b x^2}+1\right)}{b}",1,"Log[1 + Sqrt[a + b*x^2]]/b","A",1
684,1,16,16,0.02564,"\int \frac{x}{x^2-\sqrt[3]{x^2}} \, dx","Integrate[x/(x^2 - (x^2)^(1/3)),x]","\frac{3}{4} \log \left(1-\left(x^2\right)^{2/3}\right)","\frac{3}{4} \log \left(1-\left(x^2\right)^{2/3}\right)",1,"(3*Log[1 - (x^2)^(2/3)])/4","A",1
685,1,30,44,0.0133794,"\int x \left(1+x^2\right)^3 \sqrt{2+2 x^2+x^4} \, dx","Integrate[x*(1 + x^2)^3*Sqrt[2 + 2*x^2 + x^4],x]","\frac{1}{30} \left(x^4+2 x^2+2\right)^{3/2} \left(3 x^4+6 x^2+1\right)","\frac{1}{10} \left(x^2+1\right)^2 \left(x^4+2 x^2+2\right)^{3/2}-\frac{1}{15} \left(x^4+2 x^2+2\right)^{3/2}",1,"((2 + 2*x^2 + x^4)^(3/2)*(1 + 6*x^2 + 3*x^4))/30","A",1
686,1,57,121,0.042774,"\int x^5 \sqrt{1-x^3} \left(1+x^9\right)^2 \, dx","Integrate[x^5*Sqrt[1 - x^3]*(1 + x^9)^2,x]","\frac{2 \sqrt{1-x^3} \left(45045 x^{24}-3003 x^{21}-3234 x^{18}+135702 x^{15}-19390 x^{12}-22160 x^9+126561 x^6-86507 x^3-173014\right)}{2297295}","\frac{2}{51} \left(1-x^3\right)^{17/2}-\frac{14}{45} \left(1-x^3\right)^{15/2}+\frac{14}{13} \left(1-x^3\right)^{13/2}-\frac{74}{33} \left(1-x^3\right)^{11/2}+\frac{86}{27} \left(1-x^3\right)^{9/2}-\frac{22}{7} \left(1-x^3\right)^{7/2}+\frac{32}{15} \left(1-x^3\right)^{5/2}-\frac{8}{9} \left(1-x^3\right)^{3/2}",1,"(2*Sqrt[1 - x^3]*(-173014 - 86507*x^3 + 126561*x^6 - 22160*x^9 - 19390*x^12 + 135702*x^15 - 3234*x^18 - 3003*x^21 + 45045*x^24))/2297295","A",1
687,1,71,50,0.030419,"\int \left(\frac{x}{\left(a+b x^2\right)^{3/2}}+\frac{x}{\left(1+x^2\right) \sqrt{a+b x^2}}\right) \, dx","Integrate[x/(a + b*x^2)^(3/2) + x/((1 + x^2)*Sqrt[a + b*x^2]),x]","\frac{b \sqrt{a-b} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)+a-b}{b (b-a) \sqrt{a+b x^2}}","-\frac{1}{b \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(a - b + Sqrt[a - b]*b*Sqrt[a + b*x^2]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a - b]])/(b*(-a + b)*Sqrt[a + b*x^2])","A",1
688,1,71,50,0.0098412,"\int \frac{x \left(1+a+x^2+b x^2\right)}{\left(1+x^2\right) \left(a+b x^2\right)^{3/2}} \, dx","Integrate[(x*(1 + a + x^2 + b*x^2))/((1 + x^2)*(a + b*x^2)^(3/2)),x]","\frac{b \sqrt{a-b} \sqrt{a+b x^2} \tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)+a-b}{b (b-a) \sqrt{a+b x^2}}","-\frac{1}{b \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(a - b + Sqrt[a - b]*b*Sqrt[a + b*x^2]*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a - b]])/(b*(-a + b)*Sqrt[a + b*x^2])","A",1
689,1,63,68,0.189793,"\int \left(\frac{x}{\left(a+b x^2\right)^{5/2}}+\frac{x}{\left(a+b x^2\right)^{3/2}}+\frac{x}{\left(1+x^2\right) \sqrt{a+b x^2}}\right) \, dx","Integrate[x/(a + b*x^2)^(5/2) + x/(a + b*x^2)^(3/2) + x/((1 + x^2)*Sqrt[a + b*x^2]),x]","\frac{-3 a-3 b x^2-1}{3 b \left(a+b x^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}","-\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left(a+b x^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(-1 - 3*a - 3*b*x^2)/(3*b*(a + b*x^2)^(3/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a - b]]/Sqrt[a - b]","A",1
690,1,63,68,0.0723899,"\int \frac{x \left(1+a+a^2+x^2+a x^2+b x^2+2 a b x^2+b x^4+b^2 x^4\right)}{\left(1+x^2\right) \left(a+b x^2\right)^{5/2}} \, dx","Integrate[(x*(1 + a + a^2 + x^2 + a*x^2 + b*x^2 + 2*a*b*x^2 + b*x^4 + b^2*x^4))/((1 + x^2)*(a + b*x^2)^(5/2)),x]","\frac{-3 a-3 b x^2-1}{3 b \left(a+b x^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}","-\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left(a+b x^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(-1 - 3*a - 3*b*x^2)/(3*b*(a + b*x^2)^(3/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a - b]]/Sqrt[a - b]","A",1
691,1,39,34,0.0327885,"\int \frac{1}{\sqrt{\sqrt{x}+x}} \, dx","Integrate[1/Sqrt[Sqrt[x] + x],x]","2 \sqrt{x+\sqrt{x}} \left(1-\frac{\sinh ^{-1}\left(\sqrt[4]{x}\right)}{\sqrt{\sqrt{x}+1} \sqrt[4]{x}}\right)","2 \sqrt{x+\sqrt{x}}-2 \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right)",1,"2*Sqrt[Sqrt[x] + x]*(1 - ArcSinh[x^(1/4)]/(Sqrt[1 + Sqrt[x]]*x^(1/4)))","A",1
692,1,51,74,0.0408229,"\int \sqrt{\sqrt{x}+x} \, dx","Integrate[Sqrt[Sqrt[x] + x],x]","\frac{1}{12} \sqrt{x+\sqrt{x}} \left(8 x+2 \sqrt{x}+\frac{3 \sinh ^{-1}\left(\sqrt[4]{x}\right)}{\sqrt{\sqrt{x}+1} \sqrt[4]{x}}-3\right)","\frac{2}{3} \sqrt{x+\sqrt{x}} x+\frac{1}{6} \sqrt{x+\sqrt{x}} \sqrt{x}-\frac{\sqrt{x+\sqrt{x}}}{4}+\frac{1}{4} \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right)",1,"(Sqrt[Sqrt[x] + x]*(-3 + 2*Sqrt[x] + 8*x + (3*ArcSinh[x^(1/4)])/(Sqrt[1 + Sqrt[x]]*x^(1/4))))/12","A",1
693,1,19,19,0.0077113,"\int \sqrt{-x} \left(\sqrt{-x}+x\right) \, dx","Integrate[Sqrt[-x]*(Sqrt[-x] + x),x]","\frac{2}{5} (-x)^{5/2}-\frac{x^2}{2}","\frac{2}{5} (-x)^{5/2}-\frac{x^2}{2}",1,"(2*(-x)^(5/2))/5 - x^2/2","A",1
694,1,107,54,0.075482,"\int \frac{5+\sqrt[4]{x}}{-6+x} \, dx","Integrate[(5 + x^(1/4))/(-6 + x),x]","4 \sqrt[4]{x}+\left(5+\sqrt[4]{6}\right) \log \left(\sqrt[4]{6}-\sqrt[4]{x}\right)+\left(5-i \sqrt[4]{6}\right) \log \left(\sqrt[4]{6}-i \sqrt[4]{x}\right)+\left(5+i \sqrt[4]{6}\right) \log \left(\sqrt[4]{6}+i \sqrt[4]{x}\right)-\left(\sqrt[4]{6}-5\right) \log \left(\sqrt[4]{x}+\sqrt[4]{6}\right)","4 \sqrt[4]{x}+5 \log (6-x)-2 \sqrt[4]{6} \tan ^{-1}\left(\frac{\sqrt[4]{x}}{\sqrt[4]{6}}\right)-2 \sqrt[4]{6} \tanh ^{-1}\left(\frac{\sqrt[4]{x}}{\sqrt[4]{6}}\right)",1,"4*x^(1/4) + (5 + 6^(1/4))*Log[6^(1/4) - x^(1/4)] + (5 - I*6^(1/4))*Log[6^(1/4) - I*x^(1/4)] + (5 + I*6^(1/4))*Log[6^(1/4) + I*x^(1/4)] - (-5 + 6^(1/4))*Log[6^(1/4) + x^(1/4)]","C",1
695,1,14,14,0.0052522,"\int \frac{1}{4+\sqrt{4-x}-x} \, dx","Integrate[(4 + Sqrt[4 - x] - x)^(-1),x]","-2 \log \left(\sqrt{4-x}+1\right)","-2 \log \left(\sqrt{4-x}+1\right)",1,"-2*Log[1 + Sqrt[4 - x]]","A",1
696,1,58,61,0.0415353,"\int \frac{1}{1+x-\sqrt{2+x}} \, dx","Integrate[(1 + x - Sqrt[2 + x])^(-1),x]","\frac{1}{5} \left(\left(5+\sqrt{5}\right) \log \left(-2 \sqrt{x+2}+\sqrt{5}+1\right)-\left(\sqrt{5}-5\right) \log \left(-2 \sqrt{x+2}-\sqrt{5}+1\right)\right)","\frac{1}{5} \left(5-\sqrt{5}\right) \log \left(-2 \sqrt{x+2}-\sqrt{5}+1\right)+\frac{1}{5} \left(5+\sqrt{5}\right) \log \left(-2 \sqrt{x+2}+\sqrt{5}+1\right)",1,"(-((-5 + Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]]) + (5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[2 + x]])/5","A",1
697,1,37,37,0.0145575,"\int \frac{1}{4+x+\sqrt{1+x}} \, dx","Integrate[(4 + x + Sqrt[1 + x])^(-1),x]","\log \left(x+\sqrt{x+1}+4\right)-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt{x+1}+1}{\sqrt{11}}\right)}{\sqrt{11}}","\log \left(x+\sqrt{x+1}+4\right)-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt{x+1}+1}{\sqrt{11}}\right)}{\sqrt{11}}",1,"(-2*ArcTan[(1 + 2*Sqrt[1 + x])/Sqrt[11]])/Sqrt[11] + Log[4 + x + Sqrt[1 + x]]","A",1
698,1,58,61,0.0238689,"\int \frac{1}{x-\sqrt{1+x}} \, dx","Integrate[(x - Sqrt[1 + x])^(-1),x]","\frac{1}{5} \left(\left(5+\sqrt{5}\right) \log \left(-2 \sqrt{x+1}+\sqrt{5}+1\right)-\left(\sqrt{5}-5\right) \log \left(-2 \sqrt{x+1}-\sqrt{5}+1\right)\right)","\frac{1}{5} \left(5-\sqrt{5}\right) \log \left(-2 \sqrt{x+1}-\sqrt{5}+1\right)+\frac{1}{5} \left(5+\sqrt{5}\right) \log \left(-2 \sqrt{x+1}+\sqrt{5}+1\right)",1,"(-((-5 + Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + x]]) + (5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + x]])/5","A",1
699,1,31,31,0.0098402,"\int \frac{1}{x-\sqrt{2+x}} \, dx","Integrate[(x - Sqrt[2 + x])^(-1),x]","\frac{4}{3} \log \left(2-\sqrt{x+2}\right)+\frac{2}{3} \log \left(\sqrt{x+2}+1\right)","\frac{4}{3} \log \left(2-\sqrt{x+2}\right)+\frac{2}{3} \log \left(\sqrt{x+2}+1\right)",1,"(4*Log[2 - Sqrt[2 + x]])/3 + (2*Log[1 + Sqrt[2 + x]])/3","A",1
700,1,62,65,0.0265975,"\int \frac{1}{-\sqrt{1-x}+x} \, dx","Integrate[(-Sqrt[1 - x] + x)^(-1),x]","\frac{1}{5} \left(\left(5+\sqrt{5}\right) \log \left(2 \sqrt{1-x}+\sqrt{5}+1\right)-\left(\sqrt{5}-5\right) \log \left(2 \sqrt{1-x}-\sqrt{5}+1\right)\right)","\frac{1}{5} \left(5-\sqrt{5}\right) \log \left(2 \sqrt{1-x}-\sqrt{5}+1\right)+\frac{1}{5} \left(5+\sqrt{5}\right) \log \left(2 \sqrt{1-x}+\sqrt{5}+1\right)",1,"(-((-5 + Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 - x]]) + (5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 - x]])/5","A",1
701,1,49,62,0.0184646,"\int \sqrt{1+\sqrt{x}+x} \, dx","Integrate[Sqrt[1 + Sqrt[x] + x],x]","\frac{1}{24} \left(2 \sqrt{x+\sqrt{x}+1} \left(8 x+2 \sqrt{x}+5\right)-9 \sinh ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)\right)","\frac{2}{3} \left(x+\sqrt{x}+1\right)^{3/2}-\frac{1}{4} \left(2 \sqrt{x}+1\right) \sqrt{x+\sqrt{x}+1}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)",1,"(2*Sqrt[1 + Sqrt[x] + x]*(5 + 2*Sqrt[x] + 8*x) - 9*ArcSinh[(1 + 2*Sqrt[x])/Sqrt[3]])/24","A",1
702,1,62,75,0.0532465,"\int \sqrt{1+x+\sqrt{1+x}} \, dx","Integrate[Sqrt[1 + x + Sqrt[1 + x]],x]","\frac{1}{12} \sqrt{x+\sqrt{x+1}+1} \left(8 x+2 \sqrt{x+1}+\frac{3 \sinh ^{-1}\left(\sqrt[4]{x+1}\right)}{\sqrt[4]{x+1} \sqrt{\sqrt{x+1}+1}}+5\right)","\frac{2}{3} \left(x+\sqrt{x+1}+1\right)^{3/2}-\frac{1}{4} \left(2 \sqrt{x+1}+1\right) \sqrt{x+\sqrt{x+1}+1}+\frac{1}{4} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{x+\sqrt{x+1}+1}}\right)",1,"(Sqrt[1 + x + Sqrt[1 + x]]*(5 + 8*x + 2*Sqrt[1 + x] + (3*ArcSinh[(1 + x)^(1/4)])/((1 + x)^(1/4)*Sqrt[1 + Sqrt[1 + x]])))/12","A",1
703,1,54,68,0.0245968,"\int \sqrt{\sqrt{-1+x}+x} \, dx","Integrate[Sqrt[Sqrt[-1 + x] + x],x]","\frac{1}{24} \left(2 \sqrt{x+\sqrt{x-1}} \left(8 x+2 \sqrt{x-1}-3\right)-9 \sinh ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)\right)","\frac{2}{3} \left(x+\sqrt{x-1}\right)^{3/2}-\frac{1}{4} \left(2 \sqrt{x-1}+1\right) \sqrt{x+\sqrt{x-1}}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)",1,"(2*Sqrt[Sqrt[-1 + x] + x]*(-3 + 2*Sqrt[-1 + x] + 8*x) - 9*ArcSinh[(1 + 2*Sqrt[-1 + x])/Sqrt[3]])/24","A",1
704,1,62,80,0.0301513,"\int \sqrt{2 x+\sqrt{-1+2 x}} \, dx","Integrate[Sqrt[2*x + Sqrt[-1 + 2*x]],x]","\frac{1}{48} \left(2 \sqrt{2 x+\sqrt{2 x-1}} \left(16 x+2 \sqrt{2 x-1}-3\right)-9 \sinh ^{-1}\left(\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right)\right)","\frac{1}{3} \left(2 x+\sqrt{2 x-1}\right)^{3/2}-\frac{1}{8} \left(2 \sqrt{2 x-1}+1\right) \sqrt{2 x+\sqrt{2 x-1}}-\frac{3}{16} \sinh ^{-1}\left(\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right)",1,"(2*Sqrt[2*x + Sqrt[-1 + 2*x]]*(-3 + 16*x + 2*Sqrt[-1 + 2*x]) - 9*ArcSinh[(1 + 2*Sqrt[-1 + 2*x])/Sqrt[3]])/48","A",1
705,1,65,109,0.0514024,"\int \sqrt{3 x+\sqrt{-7+8 x}} \, dx","Integrate[Sqrt[3*x + Sqrt[-7 + 8*x]],x]","\frac{1}{216} \left(12 \sqrt{3 x+\sqrt{8 x-7}} \left(12 x+\sqrt{8 x-7}-4\right)-47 \sqrt{6} \sinh ^{-1}\left(\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right)\right)","\frac{\left(-3 (7-8 x)+8 \sqrt{8 x-7}+21\right)^{3/2}}{72 \sqrt{2}}-\frac{\left(3 \sqrt{8 x-7}+4\right) \sqrt{-3 (7-8 x)+8 \sqrt{8 x-7}+21}}{36 \sqrt{2}}-\frac{47 \sinh ^{-1}\left(\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right)}{36 \sqrt{6}}",1,"(12*Sqrt[3*x + Sqrt[-7 + 8*x]]*(-4 + 12*x + Sqrt[-7 + 8*x]) - 47*Sqrt[6]*ArcSinh[(4 + 3*Sqrt[-7 + 8*x])/Sqrt[47]])/216","A",1
706,1,47,47,0.0134044,"\int \frac{1}{\sqrt{x+\sqrt{1+x}}} \, dx","Integrate[1/Sqrt[x + Sqrt[1 + x]],x]","2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left(\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right)","2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left(\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right)",1,"2*Sqrt[x + Sqrt[1 + x]] - ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]","A",1
707,1,56,67,0.07023,"\int \frac{1+x}{4+x+\sqrt{-9+6 x}} \, dx","Integrate[(1 + x)/(4 + x + Sqrt[-9 + 6*x]),x]","x-2 \sqrt{6 x-9}+3 \log \left(x+\sqrt{6 x-9}+4\right)+4 \sqrt{6} \tan ^{-1}\left(\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right)","x-2 \sqrt{3} \sqrt{2 x-3}+3 \log \left(x+\sqrt{3} \sqrt{2 x-3}+4\right)+4 \sqrt{6} \tan ^{-1}\left(\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right)",1,"x - 2*Sqrt[-9 + 6*x] + 4*Sqrt[6]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt[6])] + 3*Log[4 + x + Sqrt[-9 + 6*x]]","A",1
708,1,60,71,0.0375673,"\int \frac{12-x}{4+x+\sqrt{-9+6 x}} \, dx","Integrate[(12 - x)/(4 + x + Sqrt[-9 + 6*x]),x]","-x+2 \sqrt{6 x-9}+10 \log \left(x+\sqrt{6 x-9}+4\right)-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right)","-x+2 \sqrt{3} \sqrt{2 x-3}+10 \log \left(x+\sqrt{3} \sqrt{2 x-3}+4\right)-21 \sqrt{\frac{3}{2}} \tan ^{-1}\left(\frac{\sqrt{6 x-9}+3}{2 \sqrt{6}}\right)",1,"-x + 2*Sqrt[-9 + 6*x] - 21*Sqrt[3/2]*ArcTan[(3 + Sqrt[-9 + 6*x])/(2*Sqrt[6])] + 10*Log[4 + x + Sqrt[-9 + 6*x]]","A",1
709,1,52,52,0.0244456,"\int \frac{-1+x^3}{\sqrt{x} \left(1+x^2\right)} \, dx","Integrate[(-1 + x^3)/(Sqrt[x]*(1 + x^2)),x]","\frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)","\frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{x}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{x}+1\right)",1,"(2*x^(3/2))/3 + Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]]","A",1
710,1,18,20,0.0371003,"\int \frac{1}{2 \sqrt{-1+x} \sqrt{-\sqrt{-1+x}+x}} \, dx","Integrate[1/(2*Sqrt[-1 + x]*Sqrt[-Sqrt[-1 + x] + x]),x]","\sinh ^{-1}\left(\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right)","-\sinh ^{-1}\left(\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right)",1,"ArcSinh[(-1 + 2*Sqrt[-1 + x])/Sqrt[3]]","A",1
711,1,67,43,0.0289231,"\int \frac{1+x^{7/2}}{1-x^2} \, dx","Integrate[(1 + x^(7/2))/(1 - x^2),x]","-\frac{2 x^{5/2}}{5}-2 \sqrt{x}+\left(\frac{1}{2}-\frac{i}{2}\right) \log \left(-\sqrt{x}+i\right)-\log \left(1-\sqrt{x}\right)+\left(\frac{1}{2}+\frac{i}{2}\right) \log \left(\sqrt{x}+i\right)","-\frac{2 x^{5/2}}{5}-2 \sqrt{x}-\log \left(1-\sqrt{x}\right)+\frac{1}{2} \log (x+1)+\tan ^{-1}\left(\sqrt{x}\right)",1,"-2*Sqrt[x] - (2*x^(5/2))/5 + (1/2 - I/2)*Log[I - Sqrt[x]] - Log[1 - Sqrt[x]] + (1/2 + I/2)*Log[I + Sqrt[x]]","C",1
712,1,127,116,0.0793838,"\int \frac{4+2 x}{\sqrt[3]{-1+2 x}+\sqrt{-1+2 x}} \, dx","Integrate[(4 + 2*x)/((-1 + 2*x)^(1/3) + Sqrt[-1 + 2*x]),x]","2 \left(x \left(\frac{1}{3} \sqrt{2 x-1}-\frac{3}{8} \sqrt[3]{2 x-1}+\frac{3}{7} \sqrt[6]{2 x-1}-\frac{1}{2}\right)+\frac{3}{10} (2 x-1)^{5/6}-\frac{3}{8} (2 x-1)^{2/3}+\frac{17}{6} \sqrt{2 x-1}-\frac{69}{16} \sqrt[3]{2 x-1}+\frac{123}{14} \sqrt[6]{2 x-1}-9 \log \left(\sqrt[6]{2 x-1}+1\right)\right)","\frac{1}{3} (2 x-1)^{3/2}-\frac{3}{8} (2 x-1)^{4/3}+\frac{3}{7} (2 x-1)^{7/6}+\frac{3}{5} (2 x-1)^{5/6}-\frac{3}{4} (2 x-1)^{2/3}+6 \sqrt{2 x-1}-9 \sqrt[3]{2 x-1}+18 \sqrt[6]{2 x-1}-x-18 \log \left(\sqrt[6]{2 x-1}+1\right)",1,"2*((123*(-1 + 2*x)^(1/6))/14 - (69*(-1 + 2*x)^(1/3))/16 + (17*Sqrt[-1 + 2*x])/6 - (3*(-1 + 2*x)^(2/3))/8 + (3*(-1 + 2*x)^(5/6))/10 + x*(-1/2 + (3*(-1 + 2*x)^(1/6))/7 - (3*(-1 + 2*x)^(1/3))/8 + Sqrt[-1 + 2*x]/3) - 9*Log[1 + (-1 + 2*x)^(1/6)])","A",1
713,1,58,83,0.0551696,"\int \frac{1}{\sqrt{2+\sqrt{1+\sqrt{x}}}} \, dx","Integrate[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]","\frac{8}{105} \sqrt{\sqrt{\sqrt{x}+1}+2} \left(3 \sqrt{x} \left(5 \sqrt{\sqrt{x}+1}-12\right)+76 \sqrt{\sqrt{x}+1}-280\right)","\frac{8}{7} \left(\sqrt{\sqrt{x}+1}+2\right)^{7/2}-\frac{48}{5} \left(\sqrt{\sqrt{x}+1}+2\right)^{5/2}+\frac{88}{3} \left(\sqrt{\sqrt{x}+1}+2\right)^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2}",1,"(8*Sqrt[2 + Sqrt[1 + Sqrt[x]]]*(-280 + 76*Sqrt[1 + Sqrt[x]] + 3*(-12 + 5*Sqrt[1 + Sqrt[x]])*Sqrt[x]))/105","A",1
714,1,43,64,0.0285413,"\int \sqrt{2+\sqrt{4+\sqrt{x}}} \, dx","Integrate[Sqrt[2 + Sqrt[4 + Sqrt[x]]],x]","-\frac{8}{315} \left(\sqrt{\sqrt{x}+4}+2\right)^{5/2} \left(130 \sqrt{\sqrt{x}+4}-35 \sqrt{x}-244\right)","\frac{8}{9} \left(\sqrt{\sqrt{x}+4}+2\right)^{9/2}-\frac{48}{7} \left(\sqrt{\sqrt{x}+4}+2\right)^{7/2}+\frac{64}{5} \left(\sqrt{\sqrt{x}+4}+2\right)^{5/2}",1,"(-8*(2 + Sqrt[4 + Sqrt[x]])^(5/2)*(-244 + 130*Sqrt[4 + Sqrt[x]] - 35*Sqrt[x]))/315","A",1
715,1,57,82,0.0469015,"\int \sqrt{2-\sqrt{4+\sqrt{-9+5 x}}} \, dx","Integrate[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]","\frac{8 \left(2-\sqrt{\sqrt{5 x-9}+4}\right)^{5/2} \left(35 \sqrt{5 x-9}+130 \sqrt{\sqrt{5 x-9}+4}+244\right)}{1575}","\frac{8}{45} \left(2-\sqrt{\sqrt{5 x-9}+4}\right)^{9/2}-\frac{48}{35} \left(2-\sqrt{\sqrt{5 x-9}+4}\right)^{7/2}+\frac{64}{25} \left(2-\sqrt{\sqrt{5 x-9}+4}\right)^{5/2}",1,"(8*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2)*(244 + 35*Sqrt[-9 + 5*x] + 130*Sqrt[4 + Sqrt[-9 + 5*x]]))/1575","A",1
716,1,58,83,0.0212644,"\int \frac{1}{\sqrt{2+\sqrt{1+\sqrt{x}}}} \, dx","Integrate[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]","\frac{8}{105} \sqrt{\sqrt{\sqrt{x}+1}+2} \left(3 \sqrt{x} \left(5 \sqrt{\sqrt{x}+1}-12\right)+76 \sqrt{\sqrt{x}+1}-280\right)","\frac{8}{7} \left(\sqrt{\sqrt{x}+1}+2\right)^{7/2}-\frac{48}{5} \left(\sqrt{\sqrt{x}+1}+2\right)^{5/2}+\frac{88}{3} \left(\sqrt{\sqrt{x}+1}+2\right)^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2}",1,"(8*Sqrt[2 + Sqrt[1 + Sqrt[x]]]*(-280 + 76*Sqrt[1 + Sqrt[x]] + 3*(-12 + 5*Sqrt[1 + Sqrt[x]])*Sqrt[x]))/105","A",1
717,1,135,190,0.10217,"\int \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{x}}}} \, dx","Integrate[Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]],x]","\frac{16 \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{5/2} \left(231 \sqrt{x} \left(-377 \sqrt{\sqrt{\sqrt{x}+1}+1}+195 \sqrt{\sqrt{x}+1}+365\right)+8 \left(252 \sqrt{\sqrt{x}+1} \sqrt{\sqrt{\sqrt{x}+1}+1}+8642 \sqrt{\sqrt{\sqrt{x}+1}+1}-4865 \sqrt{\sqrt{x}+1}-8221\right)\right)}{765765}","\frac{16}{17} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{17/2}-\frac{112}{15} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{15/2}+\frac{288}{13} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{13/2}-\frac{320}{11} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{11/2}+\frac{112}{9} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{9/2}+\frac{48}{7} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{7/2}-\frac{32}{5} \left(\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right)^{5/2}",1,"(16*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(5/2)*(8*(-8221 + 8642*Sqrt[1 + Sqrt[1 + Sqrt[x]]] - 4865*Sqrt[1 + Sqrt[x]] + 252*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqrt[x]]) + 231*(365 - 377*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 195*Sqrt[1 + Sqrt[x]])*Sqrt[x]))/765765","A",1
718,1,183,233,0.1305891,"\int \sqrt{2+\sqrt{3+\sqrt{-1+2 \sqrt{x}}}} \, dx","Integrate[Sqrt[2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]],x]","\frac{8 \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{3/2} \left(7 \sqrt{x} \left(2145 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}+1452 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4004 \sqrt{2 \sqrt{x}-1}-3576\right)+4 \left(3843 \sqrt{2 \sqrt{x}-1} \sqrt{\sqrt{2 \sqrt{x}-1}+3}-2535 \sqrt{\sqrt{2 \sqrt{x}-1}+3}-4286 \sqrt{2 \sqrt{x}-1}-9786\right)\right)}{255255}","\frac{4}{17} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{17/2}-\frac{56}{15} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{15/2}+\frac{300}{13} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{13/2}-\frac{760}{11} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{11/2}+\frac{304}{3} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{9/2}-\frac{480}{7} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{7/2}+\frac{136}{5} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{5/2}-\frac{16}{3} \left(\sqrt{\sqrt{2 \sqrt{x}-1}+3}+2\right)^{3/2}",1,"(8*(2 + Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]])^(3/2)*(4*(-9786 - 2535*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]] - 4286*Sqrt[-1 + 2*Sqrt[x]] + 3843*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]]) + 7*(-3576 + 1452*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]] - 4004*Sqrt[-1 + 2*Sqrt[x]] + 2145*Sqrt[3 + Sqrt[-1 + 2*Sqrt[x]]]*Sqrt[-1 + 2*Sqrt[x]])*Sqrt[x]))/255255","A",1
719,1,103,160,0.0879322,"\int \sqrt{1+\sqrt{1+\sqrt{-1+x}}} x \, dx","Integrate[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]","\frac{8 \left(\sqrt{\sqrt{x-1}+1}+1\right)^{5/2} \left(8 \left(84 \sqrt{x-1} \sqrt{\sqrt{x-1}+1}-3030 \sqrt{\sqrt{x-1}+1}+1715 \sqrt{x-1}+2591\right)+77 \left(-377 \sqrt{\sqrt{x-1}+1}+195 \sqrt{x-1}+365\right) x\right)}{255255}","\frac{8}{17} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{17/2}-\frac{56}{15} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{15/2}+\frac{144}{13} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{13/2}-\frac{160}{11} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{11/2}+8 \left(\sqrt{\sqrt{x-1}+1}+1\right)^{9/2}-\frac{24}{7} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{7/2}+\frac{16}{5} \left(\sqrt{\sqrt{x-1}+1}+1\right)^{5/2}",1,"(8*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2)*(8*(2591 - 3030*Sqrt[1 + Sqrt[-1 + x]] + 1715*Sqrt[-1 + x] + 84*Sqrt[1 + Sqrt[-1 + x]]*Sqrt[-1 + x]) + 77*(365 - 377*Sqrt[1 + Sqrt[-1 + x]] + 195*Sqrt[-1 + x])*x))/255255","A",1
720,1,20,20,0.0189667,"\int \frac{1}{\sqrt{-1+x} \sqrt{-\sqrt{-1+x}+x}} \, dx","Integrate[1/(Sqrt[-1 + x]*Sqrt[-Sqrt[-1 + x] + x]),x]","2 \sinh ^{-1}\left(\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right)","-2 \sinh ^{-1}\left(\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right)",1,"2*ArcSinh[(-1 + 2*Sqrt[-1 + x])/Sqrt[3]]","A",1
721,1,44,44,0.0178994,"\int \frac{1}{\sqrt{1+x+\sqrt{-1+2 x}}} \, dx","Integrate[1/Sqrt[1 + x + Sqrt[-1 + 2*x]],x]","2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left(\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right)","2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left(\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right)",1,"2*Sqrt[1 + x + Sqrt[-1 + 2*x]] - Sqrt[2]*ArcSinh[(1 + Sqrt[-1 + 2*x])/Sqrt[2]]","A",1
722,1,50,54,0.0807781,"\int \frac{q+p x}{\sqrt{b+a x} \left(f+\sqrt{b+a x}\right)} \, dx","Integrate[(q + p*x)/(Sqrt[b + a*x]*(f + Sqrt[b + a*x])),x]","\frac{2 \left(a q-b p+f^2 p\right) \log \left(\sqrt{a x+b}+f\right)+p \left(a x-2 f \sqrt{a x+b}\right)}{a^2}","-\frac{2 \left(-a q+b p+f^2 (-p)\right) \log \left(\sqrt{a x+b}+f\right)}{a^2}-\frac{2 f p \sqrt{a x+b}}{a^2}+\frac{p x}{a}",1,"(p*(a*x - 2*f*Sqrt[b + a*x]) + 2*(-(b*p) + f^2*p + a*q)*Log[f + Sqrt[b + a*x]])/a^2","A",1
723,1,53,70,0.0279681,"\int \sqrt{1-\sqrt{x}-x} \, dx","Integrate[Sqrt[1 - Sqrt[x] - x],x]","\frac{1}{12} \sqrt{-x-\sqrt{x}+1} \left(8 x+2 \sqrt{x}-11\right)+\frac{5}{8} \sin ^{-1}\left(\frac{-2 \sqrt{x}-1}{\sqrt{5}}\right)","-\frac{2}{3} \left(-x-\sqrt{x}+1\right)^{3/2}-\frac{1}{4} \left(2 \sqrt{x}+1\right) \sqrt{-x-\sqrt{x}+1}-\frac{5}{8} \sin ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{5}}\right)",1,"(Sqrt[1 - Sqrt[x] - x]*(-11 + 2*Sqrt[x] + 8*x))/12 + (5*ArcSin[(-1 - 2*Sqrt[x])/Sqrt[5]])/8","A",1
724,1,19,19,0.0102346,"\int \frac{9+6 \sqrt{x}+x}{4 \sqrt{x}+x} \, dx","Integrate[(9 + 6*Sqrt[x] + x)/(4*Sqrt[x] + x),x]","x+4 \sqrt{x}+2 \log \left(\sqrt{x}+4\right)","x+4 \sqrt{x}+2 \log \left(\sqrt{x}+4\right)",1,"4*Sqrt[x] + x + 2*Log[4 + Sqrt[x]]","A",1
725,1,66,77,0.0515965,"\int \frac{6-8 x^{7/2}}{5-9 \sqrt{x}} \, dx","Integrate[(6 - 8*x^(7/2))/(5 - 9*Sqrt[x]),x]","\frac{2 \left(9 \left(21257640 x^{7/2}+9185400 x^{5/2}+4725000 x^{3/2}+33480783 x^4+13778100 x^3+6378750 x^2+3937500 x-196509698 \sqrt{x}\right)-982548490 \log \left(5-9 \sqrt{x}\right)\right)}{2711943423}","\frac{80 x^{7/2}}{567}+\frac{400 x^{5/2}}{6561}+\frac{50000 x^{3/2}}{1594323}+\frac{2 x^4}{9}+\frac{200 x^3}{2187}+\frac{2500 x^2}{59049}+\frac{125000 x}{4782969}-\frac{56145628 \sqrt{x}}{43046721}-\frac{280728140 \log \left(5-9 \sqrt{x}\right)}{387420489}",1,"(2*(9*(-196509698*Sqrt[x] + 3937500*x + 4725000*x^(3/2) + 6378750*x^2 + 9185400*x^(5/2) + 13778100*x^3 + 21257640*x^(7/2) + 33480783*x^4) - 982548490*Log[5 - 9*Sqrt[x]]))/2711943423","A",1
726,1,68,80,0.1031376,"\int \frac{\sqrt{1+x} \left(1+x^3\right)}{1+x^2} \, dx","Integrate[(Sqrt[1 + x]*(1 + x^3))/(1 + x^2),x]","\frac{2}{15} \sqrt{x+1} \left(3 x^2+x-17\right)+(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1+i}}\right)","\frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2}-2 \sqrt{x+1}+(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{x+1}}{\sqrt{1+i}}\right)",1,"(2*Sqrt[1 + x]*(-17 + x + 3*x^2))/15 + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + x]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + x]/Sqrt[1 + I]]","A",1
727,1,89,89,0.0589782,"\int \frac{\sqrt{-1-\sqrt{x}+x}}{(-1+x) \sqrt{x}} \, dx","Integrate[Sqrt[-1 - Sqrt[x] + x]/((-1 + x)*Sqrt[x]),x]","\tan ^{-1}\left(\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right)-2 \tanh ^{-1}\left(\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right)-\tanh ^{-1}\left(\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right)","\tan ^{-1}\left(\frac{3-\sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right)-2 \tanh ^{-1}\left(\frac{1-2 \sqrt{x}}{2 \sqrt{x-\sqrt{x}-1}}\right)-\tanh ^{-1}\left(\frac{3 \sqrt{x}+1}{2 \sqrt{x-\sqrt{x}-1}}\right)",1,"ArcTan[(3 - Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - 2*ArcTanh[(1 - 2*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])] - ArcTanh[(1 + 3*Sqrt[x])/(2*Sqrt[-1 - Sqrt[x] + x])]","A",1
728,1,61,61,0.0477082,"\int \frac{1+2 \sqrt{1+x}}{x \sqrt{1+x} \sqrt{x+\sqrt{1+x}}} \, dx","Integrate[(1 + 2*Sqrt[1 + x])/(x*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]),x]","\tan ^{-1}\left(\frac{-\sqrt{x+1}-3}{2 \sqrt{x+\sqrt{x+1}}}\right)-3 \tanh ^{-1}\left(\frac{3 \sqrt{x+1}-1}{2 \sqrt{x+\sqrt{x+1}}}\right)","3 \tanh ^{-1}\left(\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right)-\tan ^{-1}\left(\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right)",1,"ArcTan[(-3 - Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])] - 3*ArcTanh[(-1 + 3*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]","A",1
729,1,8,8,0.0024995,"\int \frac{1}{\sqrt{x} \sqrt{1+x}} \, dx","Integrate[1/(Sqrt[x]*Sqrt[1 + x]),x]","2 \sinh ^{-1}\left(\sqrt{x}\right)","2 \sinh ^{-1}\left(\sqrt{x}\right)",1,"2*ArcSinh[Sqrt[x]]","A",1
730,1,8,8,0.0046354,"\int \frac{\sqrt{\frac{x}{1+x}}}{x} \, dx","Integrate[Sqrt[x/(1 + x)]/x,x]","2 \sinh ^{-1}\left(\sqrt{x}\right)","2 \sinh ^{-1}\left(\sqrt{x}\right)",1,"2*ArcSinh[Sqrt[x]]","A",1
731,1,42,22,0.0155894,"\int \frac{\sqrt{x}}{\sqrt{1+x}} \, dx","Integrate[Sqrt[x]/Sqrt[1 + x],x]","\frac{\sqrt{\frac{x}{x+1}} \left(\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)\right)}{\sqrt{x}}","\sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]","A",1
732,1,42,22,0.0020591,"\int \sqrt{\frac{x}{1+x}} \, dx","Integrate[Sqrt[x/(1 + x)],x]","\frac{\sqrt{\frac{x}{x+1}} \left(\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)\right)}{\sqrt{x}}","\sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]","A",1
733,1,50,36,0.0181435,"\int \frac{\sqrt{-1+x}}{x^2 \sqrt{1+x}} \, dx","Integrate[Sqrt[-1 + x]/(x^2*Sqrt[1 + x]),x]","\frac{\sqrt{\frac{x-1}{x+1}} \left(-x^2+\sqrt{x^2-1} x \tan ^{-1}\left(\sqrt{x^2-1}\right)+1\right)}{(x-1) x}","\tan ^{-1}\left(\sqrt{x-1} \sqrt{x+1}\right)-\frac{\sqrt{x-1} \sqrt{x+1}}{x}",1,"(Sqrt[(-1 + x)/(1 + x)]*(1 - x^2 + x*Sqrt[-1 + x^2]*ArcTan[Sqrt[-1 + x^2]]))/((-1 + x)*x)","A",1
734,1,50,36,0.0046511,"\int \frac{\sqrt{\frac{-1+x}{1+x}}}{x^2} \, dx","Integrate[Sqrt[(-1 + x)/(1 + x)]/x^2,x]","\frac{\sqrt{\frac{x-1}{x+1}} \left(-x^2+\sqrt{x^2-1} x \tan ^{-1}\left(\sqrt{x^2-1}\right)+1\right)}{(x-1) x}","\tan ^{-1}\left(\sqrt{x-1} \sqrt{x+1}\right)-\frac{\sqrt{x-1} \sqrt{x+1}}{x}",1,"(Sqrt[(-1 + x)/(1 + x)]*(1 - x^2 + x*Sqrt[-1 + x^2]*ArcTan[Sqrt[-1 + x^2]]))/((-1 + x)*x)","A",1
735,1,76,69,0.0710856,"\int \frac{\sqrt{-1+x} x^3}{\sqrt{1+x}} \, dx","Integrate[(Sqrt[-1 + x]*x^3)/Sqrt[1 + x],x]","\frac{\sqrt{\frac{x-1}{x+1}} \left(6 x^5-8 x^4+3 x^3-8 x^2-18 \sqrt{1-x^2} \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)-9 x+16\right)}{24 (x-1)}","\frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x)",1,"(Sqrt[(-1 + x)/(1 + x)]*(16 - 9*x - 8*x^2 + 3*x^3 - 8*x^4 + 6*x^5 - 18*Sqrt[1 - x^2]*ArcSin[Sqrt[1 - x]/Sqrt[2]]))/(24*(-1 + x))","A",1
736,1,76,69,0.0368007,"\int x^3 \sqrt{\frac{-1+x}{1+x}} \, dx","Integrate[x^3*Sqrt[(-1 + x)/(1 + x)],x]","\frac{\sqrt{\frac{x-1}{x+1}} \left(6 x^5-8 x^4+3 x^3-8 x^2-18 \sqrt{1-x^2} \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)-9 x+16\right)}{24 (x-1)}","\frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x)",1,"(Sqrt[(-1 + x)/(1 + x)]*(16 - 9*x - 8*x^2 + 3*x^3 - 8*x^4 + 6*x^5 - 18*Sqrt[1 - x^2]*ArcSin[Sqrt[1 - x]/Sqrt[2]]))/(24*(-1 + x))","A",1
737,1,32,15,0.0182497,"\int \frac{\sqrt{-\frac{x}{1+x}}}{x} \, dx","Integrate[Sqrt[-(x/(1 + x))]/x,x]","\frac{2 \sqrt{-\frac{x}{x+1}} \sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}}","2 \tan ^{-1}\left(\sqrt{-\frac{x}{x+1}}\right)",1,"(2*Sqrt[-(x/(1 + x))]*Sqrt[1 + x]*ArcSinh[Sqrt[x]])/Sqrt[x]","B",1
738,1,34,18,0.018547,"\int \frac{\sqrt{\frac{1-x}{1+x}}}{-1+x} \, dx","Integrate[Sqrt[(1 - x)/(1 + x)]/(-1 + x),x]","\frac{\sqrt{\frac{1-x}{x+1}} \sqrt{1-x^2} \sin ^{-1}(x)}{x-1}","2 \tan ^{-1}\left(\sqrt{\frac{1-x}{x+1}}\right)",1,"(Sqrt[(1 - x)/(1 + x)]*Sqrt[1 - x^2]*ArcSin[x])/(-1 + x)","A",1
739,1,93,24,0.2287657,"\int \frac{\sqrt{\frac{a+b x}{c-b x}}}{a+b x} \, dx","Integrate[Sqrt[(a + b*x)/(c - b*x)]/(a + b*x),x]","\frac{2 b \sqrt{c-b x} \sqrt{\frac{a+b x}{c-b x}} \sin ^{-1}\left(\frac{b \sqrt{c-b x}}{\sqrt{-b} \sqrt{-b (a+c)}}\right)}{(-b)^{3/2} \sqrt{-b (a+c)} \sqrt{\frac{a+b x}{a+c}}}","\frac{2 \tan ^{-1}\left(\sqrt{\frac{a+b x}{c-b x}}\right)}{b}",1,"(2*b*Sqrt[c - b*x]*Sqrt[(a + b*x)/(c - b*x)]*ArcSin[(b*Sqrt[c - b*x])/(Sqrt[-b]*Sqrt[-(b*(a + c))])])/((-b)^(3/2)*Sqrt[-(b*(a + c))]*Sqrt[(a + b*x)/(a + c)])","B",1
740,1,97,41,0.0752398,"\int \frac{\sqrt{\frac{a+b x}{c+d x}}}{a+b x} \, dx","Integrate[Sqrt[(a + b*x)/(c + d*x)]/(a + b*x),x]","\frac{2 \sqrt{b c-a d} \sqrt{\frac{a+b x}{c+d x}} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right)}{b \sqrt{d} \sqrt{a+b x}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{d}}",1,"(2*Sqrt[b*c - a*d]*Sqrt[(a + b*x)/(c + d*x)]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(b*Sqrt[d]*Sqrt[a + b*x])","B",1
741,1,43,32,0.0134308,"\int \sqrt{-\frac{x}{1+x}} \, dx","Integrate[Sqrt[-(x/(1 + x))],x]","\frac{\sqrt{-\frac{x}{x+1}} \left(\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)\right)}{\sqrt{x}}","\sqrt{-\frac{x}{x+1}} (x+1)-\tan ^{-1}\left(\sqrt{-\frac{x}{x+1}}\right)",1,"(Sqrt[-(x/(1 + x))]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]","A",1
742,1,67,38,0.0211412,"\int \sqrt{\frac{1-x}{1+x}} \, dx","Integrate[Sqrt[(1 - x)/(1 + x)],x]","\frac{\sqrt{\frac{1-x}{x+1}} \sqrt{x+1} \left(\sqrt{x+1} (x-1)+2 \sqrt{1-x} \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)\right)}{x-1}","\sqrt{\frac{1-x}{x+1}} (x+1)-2 \tan ^{-1}\left(\sqrt{\frac{1-x}{x+1}}\right)",1,"(Sqrt[(1 - x)/(1 + x)]*Sqrt[1 + x]*((-1 + x)*Sqrt[1 + x] + 2*Sqrt[1 - x]*ArcSin[Sqrt[1 - x]/Sqrt[2]]))/(-1 + x)","A",1
743,1,83,42,0.0522579,"\int \sqrt{\frac{a+x}{a-x}} \, dx","Integrate[Sqrt[(a + x)/(a - x)],x]","\frac{\sqrt{x-a} \sqrt{\frac{a+x}{a-x}} \left(2 a^{3/2} \sqrt{\frac{a+x}{a}} \sinh ^{-1}\left(\frac{\sqrt{x-a}}{\sqrt{2} \sqrt{a}}\right)+\sqrt{x-a} (a+x)\right)}{a+x}","2 a \tan ^{-1}\left(\sqrt{\frac{a+x}{a-x}}\right)-(a-x) \sqrt{\frac{a+x}{a-x}}",1,"(Sqrt[-a + x]*Sqrt[(a + x)/(a - x)]*(Sqrt[-a + x]*(a + x) + 2*a^(3/2)*Sqrt[(a + x)/a]*ArcSinh[Sqrt[-a + x]/(Sqrt[2]*Sqrt[a])]))/(a + x)","A",1
744,1,78,41,0.0682198,"\int \sqrt{\frac{-a+x}{a+x}} \, dx","Integrate[Sqrt[(-a + x)/(a + x)],x]","\frac{\sqrt{\frac{x-a}{a+x}} \left(\sqrt{x-a} (a+x)-2 a^{3/2} \sqrt{\frac{a+x}{a}} \sinh ^{-1}\left(\frac{\sqrt{x-a}}{\sqrt{2} \sqrt{a}}\right)\right)}{\sqrt{x-a}}","\sqrt{-\frac{a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left(\sqrt{-\frac{a-x}{a+x}}\right)",1,"(Sqrt[(-a + x)/(a + x)]*(Sqrt[-a + x]*(a + x) - 2*a^(3/2)*Sqrt[(a + x)/a]*ArcSinh[Sqrt[-a + x]/(Sqrt[2]*Sqrt[a])]))/Sqrt[-a + x]","A",1
745,1,123,76,0.2716465,"\int \sqrt{\frac{a+b x}{c+d x}} \, dx","Integrate[Sqrt[(a + b*x)/(c + d*x)],x]","\frac{\sqrt{\frac{a+b x}{c+d x}} \left(b \sqrt{d} (a+b x) (c+d x)-\sqrt{a+b x} (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right)\right)}{b d^{3/2} (a+b x)}","\frac{(c+d x) \sqrt{\frac{a+b x}{c+d x}}}{d}-\frac{(b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{\frac{a+b x}{c+d x}}}{\sqrt{b}}\right)}{\sqrt{b} d^{3/2}}",1,"(Sqrt[(a + b*x)/(c + d*x)]*(b*Sqrt[d]*(a + b*x)*(c + d*x) - (b*c - a*d)^(3/2)*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]]))/(b*d^(3/2)*(a + b*x))","A",1
746,1,76,49,0.0405455,"\int \sqrt{\frac{-1+x}{5+3 x}} \, dx","Integrate[Sqrt[(-1 + x)/(5 + 3*x)],x]","\frac{3 (x-1) \sqrt{3 x+5}-8 \sqrt{3} \sqrt{x-1} \sinh ^{-1}\left(\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right)}{9 \sqrt{\frac{x-1}{3 x+5}} \sqrt{3 x+5}}","\frac{1}{3} \sqrt{x-1} \sqrt{3 x+5}-\frac{8 \sinh ^{-1}\left(\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right)}{3 \sqrt{3}}",1,"(3*(-1 + x)*Sqrt[5 + 3*x] - 8*Sqrt[3]*Sqrt[-1 + x]*ArcSinh[(Sqrt[3/2]*Sqrt[-1 + x])/2])/(9*Sqrt[(-1 + x)/(5 + 3*x)]*Sqrt[5 + 3*x])","A",1
747,1,79,46,0.0341548,"\int \frac{\sqrt{\frac{-1+5 x}{1+7 x}}}{x^2} \, dx","Integrate[Sqrt[(-1 + 5*x)/(1 + 7*x)]/x^2,x]","\frac{\sqrt{\frac{5 x-1}{7 x+1}} \left(12 x \sqrt{7 x+1} \tan ^{-1}\left(\frac{\sqrt{5 x-1}}{\sqrt{7 x+1}}\right)-\sqrt{5 x-1} (7 x+1)\right)}{x \sqrt{5 x-1}}","-\frac{\sqrt{5 x-1} \sqrt{7 x+1}}{x}-12 \tan ^{-1}\left(\frac{\sqrt{7 x+1}}{\sqrt{5 x-1}}\right)",1,"(Sqrt[(-1 + 5*x)/(1 + 7*x)]*(-(Sqrt[-1 + 5*x]*(1 + 7*x)) + 12*x*Sqrt[1 + 7*x]*ArcTan[Sqrt[-1 + 5*x]/Sqrt[1 + 7*x]]))/(x*Sqrt[-1 + 5*x])","A",1
748,1,19,20,0.0086186,"\int \frac{x}{\sqrt{\frac{1-x}{1+x}} (1+x)} \, dx","Integrate[x/(Sqrt[(1 - x)/(1 + x)]*(1 + x)),x]","\frac{x-1}{\sqrt{\frac{1-x}{x+1}}}","-\sqrt{\frac{1-x}{x+1}} (x+1)",1,"(-1 + x)/Sqrt[(1 - x)/(1 + x)]","A",1
749,1,17,18,0.0064213,"\int \frac{x}{(1+x) \sqrt{-1+\frac{2}{1+x}}} \, dx","Integrate[x/((1 + x)*Sqrt[-1 + 2/(1 + x)]),x]","\frac{x-1}{\sqrt{\frac{2}{x+1}-1}}","-\left((x+1) \sqrt{\frac{2}{x+1}-1}\right)",1,"(-1 + x)/Sqrt[-1 + 2/(1 + x)]","A",1
750,1,106,54,0.0728822,"\int \frac{x}{(1+x) \sqrt{\frac{2+x}{3+x}}} \, dx","Integrate[x/((1 + x)*Sqrt[(2 + x)/(3 + x)]),x]","\frac{\sqrt{x+3} \left(x^2+5 x+6\right)+2 \sqrt{2} \sqrt{x+2} \sqrt{-(x+3)^2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{x+2}}{\sqrt{-x-3}}\right)-\sqrt{x+2} (x+3) \sinh ^{-1}\left(\sqrt{x+2}\right)}{\sqrt{\frac{x+2}{x+3}} (x+3)^{3/2}}","\sqrt{x+2} \sqrt{x+3}-\sinh ^{-1}\left(\sqrt{x+2}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{x+2}}{\sqrt{x+3}}\right)",1,"(Sqrt[3 + x]*(6 + 5*x + x^2) - Sqrt[2 + x]*(3 + x)*ArcSinh[Sqrt[2 + x]] + 2*Sqrt[2]*Sqrt[2 + x]*Sqrt[-(3 + x)^2]*ArcTan[(Sqrt[2]*Sqrt[2 + x])/Sqrt[-3 - x]])/(Sqrt[(2 + x)/(3 + x)]*(3 + x)^(3/2))","A",1
751,1,11,11,0.0052413,"\int \frac{\sqrt{1+\frac{1}{x}}}{(1+x)^2} \, dx","Integrate[Sqrt[1 + x^(-1)]/(1 + x)^2,x]","\frac{2}{\sqrt{\frac{1}{x}+1}}","\frac{2}{\sqrt{\frac{1}{x}+1}}",1,"2/Sqrt[1 + x^(-1)]","A",1
752,1,41,29,0.2274307,"\int \frac{\sqrt{1+\frac{1}{x}}}{\sqrt{1-x^2}} \, dx","Integrate[Sqrt[1 + x^(-1)]/Sqrt[1 - x^2],x]","-\tan ^{-1}\left(\frac{\sqrt{\frac{x+1}{x}} (2 x-1) \sqrt{1-x^2}}{2 \left(x^2-1\right)}\right)","-\frac{\sqrt{\frac{1}{x}+1} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{x+1}}",1,"-ArcTan[(Sqrt[(1 + x)/x]*(-1 + 2*x)*Sqrt[1 - x^2])/(2*(-1 + x^2))]","A",1
753,1,197,180,0.3839159,"\int \frac{1}{x+\sqrt{3-2 x-x^2}} \, dx","Integrate[(x + Sqrt[3 - 2*x - x^2])^(-1),x]","\frac{1}{28} \left(-\sqrt{14 \left(4+\sqrt{7}\right)} \tanh ^{-1}\left(\frac{\left(\sqrt{7}-1\right) x+\sqrt{7}+7}{\sqrt{2 \left(4+\sqrt{7}\right)} \sqrt{-x^2-2 x+3}}\right)-\sqrt{56-14 \sqrt{7}} \tanh ^{-1}\left(\frac{\sqrt{7} x+x+\sqrt{7}-7}{\sqrt{2} \sqrt{\left(\sqrt{7}-4\right) \left(x^2+2 x-3\right)}}\right)-\sqrt{7} \log \left(2 x-\sqrt{7}+1\right)+7 \log \left(2 x-\sqrt{7}+1\right)+\sqrt{7} \log \left(2 x+\sqrt{7}+1\right)+7 \log \left(2 x+\sqrt{7}+1\right)+14 \sin ^{-1}\left(\frac{x+1}{2}\right)\right)","-\frac{1}{2} \log \left(-\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-x+3}{x^2}\right)+\frac{1}{14} \left(7+\sqrt{7}\right) \log \left(-\frac{\sqrt{3} \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-\sqrt{7}+\sqrt{3}+1\right)+\frac{1}{14} \left(7-\sqrt{7}\right) \log \left(-\frac{\sqrt{3} \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}+\sqrt{7}+\sqrt{3}+1\right)+\tan ^{-1}\left(\frac{\sqrt{3}-\sqrt{-x^2-2 x+3}}{x}\right)",1,"(14*ArcSin[(1 + x)/2] - Sqrt[14*(4 + Sqrt[7])]*ArcTanh[(7 + Sqrt[7] + (-1 + Sqrt[7])*x)/(Sqrt[2*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^2])] - Sqrt[56 - 14*Sqrt[7]]*ArcTanh[(-7 + Sqrt[7] + x + Sqrt[7]*x)/(Sqrt[2]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)])] + 7*Log[1 - Sqrt[7] + 2*x] - Sqrt[7]*Log[1 - Sqrt[7] + 2*x] + 7*Log[1 + Sqrt[7] + 2*x] + Sqrt[7]*Log[1 + Sqrt[7] + 2*x])/28","A",1
754,1,306,172,0.4517685,"\int \frac{1}{\left(x+\sqrt{3-2 x-x^2}\right)^2} \, dx","Integrate[(x + Sqrt[3 - 2*x - x^2])^(-2),x]","\frac{1}{98} \left(\frac{7 (3-8 x)}{2 x^2+2 x-3}-\frac{14 (x-3) \sqrt{-x^2-2 x+3}}{2 x^2+2 x-3}-2 \left(1+\sqrt{7}\right) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left(\sqrt{14 \left(4+\sqrt{7}\right)} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right)-\frac{2}{3} \left(\sqrt{7}-1\right) \sqrt{14 \left(4+\sqrt{7}\right)} \log \left(-\sqrt{14} \sqrt{\left(\sqrt{7}-4\right) \left(x^2+2 x-3\right)}+\left(7+\sqrt{7}\right) x-7 \sqrt{7}+7\right)-4 \sqrt{7} \log \left(-2 x+\sqrt{7}-1\right)+\frac{2}{3} \left(\sqrt{7}-1\right) \sqrt{14 \left(4+\sqrt{7}\right)} \log \left(2 x-\sqrt{7}+1\right)+2 \left(1+\sqrt{7}\right) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left(2 x+\sqrt{7}+1\right)+4 \sqrt{7} \log \left(2 x+\sqrt{7}+1\right)\right)","\frac{2 \left(\frac{3 \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-\sqrt{3}+4\right)}{7 \left(\frac{\sqrt{3} \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)^2}{x^2}-\frac{2 \left(1+\sqrt{3}\right) \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-\sqrt{3}+2\right)}+\frac{8 \tanh ^{-1}\left(\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right)}{7 \sqrt{7}}",1,"((7*(3 - 8*x))/(-3 + 2*x + 2*x^2) - (14*(-3 + x)*Sqrt[3 - 2*x - x^2])/(-3 + 2*x + 2*x^2) - 4*Sqrt[7]*Log[-1 + Sqrt[7] - 2*x] + (2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[1 - Sqrt[7] + 2*x])/3 + 4*Sqrt[7]*Log[1 + Sqrt[7] + 2*x] + 2*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[1 + Sqrt[7] + 2*x] - 2*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[7 + 7*Sqrt[7] + 7*x - Sqrt[7]*x + Sqrt[14*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^2]] - (2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[7 - 7*Sqrt[7] + (7 + Sqrt[7])*x - Sqrt[14]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)]])/3)/98","A",0
755,1,333,307,1.1089983,"\int \frac{1}{\left(x+\sqrt{3-2 x-x^2}\right)^3} \, dx","Integrate[(x + Sqrt[3 - 2*x - x^2])^(-3),x]","\frac{\frac{7 (37-24 x)}{2 x^2+2 x-3}+\frac{98 (11 x-12)}{\left(2 x^2+2 x-3\right)^2}-6 \left(1+\sqrt{7}\right) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left(\sqrt{14 \left(4+\sqrt{7}\right)} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right)-2 \left(\sqrt{7}-1\right) \sqrt{14 \left(4+\sqrt{7}\right)} \log \left(-\sqrt{14} \sqrt{\left(\sqrt{7}-4\right) \left(x^2+2 x-3\right)}+\left(7+\sqrt{7}\right) x-7 \sqrt{7}+7\right)-\frac{14 \sqrt{-x^2-2 x+3} \left(34 x^3+58 x^2-83 x-15\right)}{\left(2 x^2+2 x-3\right)^2}-12 \sqrt{7} \log \left(-2 x+\sqrt{7}-1\right)+2 \left(\sqrt{7}-1\right) \sqrt{14 \left(4+\sqrt{7}\right)} \log \left(2 x-\sqrt{7}+1\right)+6 \left(1+\sqrt{7}\right) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left(2 x+\sqrt{7}+1\right)+12 \sqrt{7} \log \left(2 x+\sqrt{7}+1\right)}{1372}","-\frac{4 \left(\frac{\left(21+5 \sqrt{3}\right) \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-5 \sqrt{3}+9\right)}{21 \left(\frac{\sqrt{3} \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)^2}{x^2}-\frac{2 \left(1+\sqrt{3}\right) \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-\sqrt{3}+2\right)^2}+\frac{2 \left(-\frac{\left(18+49 \sqrt{3}\right) \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-43 \sqrt{3}+18\right)}{147 \left(\frac{\sqrt{3} \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)^2}{x^2}-\frac{2 \left(1+\sqrt{3}\right) \left(\sqrt{3}-\sqrt{-x^2-2 x+3}\right)}{x}-\sqrt{3}+2\right)}+\frac{12 \tanh ^{-1}\left(\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right)}{49 \sqrt{7}}",1,"((98*(-12 + 11*x))/(-3 + 2*x + 2*x^2)^2 + (7*(37 - 24*x))/(-3 + 2*x + 2*x^2) - (14*Sqrt[3 - 2*x - x^2]*(-15 - 83*x + 58*x^2 + 34*x^3))/(-3 + 2*x + 2*x^2)^2 - 12*Sqrt[7]*Log[-1 + Sqrt[7] - 2*x] + 2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[1 - Sqrt[7] + 2*x] + 12*Sqrt[7]*Log[1 + Sqrt[7] + 2*x] + 6*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[1 + Sqrt[7] + 2*x] - 6*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[7 + 7*Sqrt[7] + 7*x - Sqrt[7]*x + Sqrt[14*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^2]] - 2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[7 - 7*Sqrt[7] + (7 + Sqrt[7])*x - Sqrt[14]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)]])/1372","A",0
756,1,59,65,0.0280233,"\int \frac{1}{x+\sqrt{-3-2 x+x^2}} \, dx","Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-1),x]","2 \left(\frac{1}{\sqrt{x^2-2 x-3}+x-1}+\log \left(-\sqrt{x^2-2 x-3}-x+1\right)-\frac{3}{4} \log \left(\sqrt{x^2-2 x-3}+x\right)\right)","-\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+2 \log \left(-\sqrt{x^2-2 x-3}-x+1\right)-\frac{3}{2} \log \left(\sqrt{x^2-2 x-3}+x\right)",1,"2*((-1 + x + Sqrt[-3 - 2*x + x^2])^(-1) + Log[1 - x - Sqrt[-3 - 2*x + x^2]] - (3*Log[x + Sqrt[-3 - 2*x + x^2]])/4)","A",1
757,1,79,83,0.0337766,"\int \frac{1}{\left(x+\sqrt{-3-2 x+x^2}\right)^2} \, dx","Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-2),x]","\frac{2}{\sqrt{x^2-2 x-3}+x-1}+\frac{3}{2 \left(\sqrt{x^2-2 x-3}+x\right)}+4 \log \left(-\sqrt{x^2-2 x-3}-x+1\right)-4 \log \left(\sqrt{x^2-2 x-3}+x\right)","-\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{3}{2 \left(\sqrt{x^2-2 x-3}+x\right)}+4 \log \left(-\sqrt{x^2-2 x-3}-x+1\right)-4 \log \left(\sqrt{x^2-2 x-3}+x\right)",1,"2/(-1 + x + Sqrt[-3 - 2*x + x^2]) + 3/(2*(x + Sqrt[-3 - 2*x + x^2])) + 4*Log[1 - x - Sqrt[-3 - 2*x + x^2]] - 4*Log[x + Sqrt[-3 - 2*x + x^2]]","A",1
758,1,97,101,0.0509965,"\int \frac{1}{\left(x+\sqrt{-3-2 x+x^2}\right)^3} \, dx","Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-3),x]","\frac{2}{\sqrt{x^2-2 x-3}+x-1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left(\sqrt{x^2-2 x-3}+x\right)^2}+6 \log \left(-\sqrt{x^2-2 x-3}-x+1\right)-6 \log \left(\sqrt{x^2-2 x-3}+x\right)","-\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{4}{\sqrt{x^2-2 x-3}+x}+\frac{3}{4 \left(\sqrt{x^2-2 x-3}+x\right)^2}+6 \log \left(-\sqrt{x^2-2 x-3}-x+1\right)-6 \log \left(\sqrt{x^2-2 x-3}+x\right)",1,"2/(-1 + x + Sqrt[-3 - 2*x + x^2]) + 3/(4*(x + Sqrt[-3 - 2*x + x^2])^2) + 4/(x + Sqrt[-3 - 2*x + x^2]) + 6*Log[1 - x - Sqrt[-3 - 2*x + x^2]] - 6*Log[x + Sqrt[-3 - 2*x + x^2]]","A",1
759,1,187,108,0.421839,"\int \frac{1}{x+\sqrt{-3-4 x-x^2}} \, dx","Integrate[(x + Sqrt[-3 - 4*x - x^2])^(-1),x]","\frac{1}{4} \left(\log \left(2 x^2+4 x+3\right)+i \sqrt{1-2 i \sqrt{2}} \tanh ^{-1}\left(\frac{i \sqrt{2} x+2 x+2 i \sqrt{2}+2}{\sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)-i \sqrt{1+2 i \sqrt{2}} \tanh ^{-1}\left(\frac{\left(2-i \sqrt{2}\right) x-2 i \sqrt{2}+2}{\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}}\right)+2 \sin ^{-1}(x+2)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} (x+1)\right)\right)","\frac{1}{2} \log (x+3)+\frac{1}{2} \log \left(\frac{\sqrt{-x-1} x+\sqrt{x+3} x+3 \sqrt{-x-1}}{(x+3)^{3/2}}\right)-\tan ^{-1}\left(\frac{\sqrt{-x-1}}{\sqrt{x+3}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right)",1,"(2*ArcSin[2 + x] - 2*Sqrt[2]*ArcTan[Sqrt[2]*(1 + x)] + I*Sqrt[1 - (2*I)*Sqrt[2]]*ArcTanh[(2 + (2*I)*Sqrt[2] + 2*x + I*Sqrt[2]*x)/(Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] - I*Sqrt[1 + (2*I)*Sqrt[2]]*ArcTanh[(2 - (2*I)*Sqrt[2] + (2 - I*Sqrt[2])*x)/(Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2])] + Log[3 + 4*x + 2*x^2])/4","C",1
760,1,881,87,1.6661351,"\int \frac{1}{\left(x+\sqrt{-3-4 x-x^2}\right)^2} \, dx","Integrate[(x + Sqrt[-3 - 4*x - x^2])^(-2),x]","\frac{1}{16} \left(\frac{8 (x+3)}{2 x^2+4 x+3}+4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} (x+1)\right)-\frac{2 i \left(-2 i+\sqrt{2}\right) \tan ^{-1}\left(\frac{(x+2) \left(2 \left(9+2 i \sqrt{2}\right) x^2+16 \left(2+i \sqrt{2}\right) x+3 \left(5+4 i \sqrt{2}\right)\right)}{\left(8 i+6 \sqrt{2}\right) x^3+\left(-6 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}+36 i\right) x^2+\left(-12 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-5 \sqrt{2}+40 i\right) x-9 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 \sqrt{2}+12 i}\right)}{\sqrt{1+2 i \sqrt{2}}}+\frac{2 \left(2 i+\sqrt{2}\right) \tanh ^{-1}\left(\frac{(x+2) \left(2 \left(9 i+2 \sqrt{2}\right) x^2+16 \left(2 i+\sqrt{2}\right) x+3 \left(5 i+4 \sqrt{2}\right)\right)}{\left(-8 i+6 \sqrt{2}\right) x^3+\left(-6 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}-36 i\right) x^2-12 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-5 \left(8 i+\sqrt{2}\right) x-3 \left(3 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+2 \sqrt{2}+4 i\right)}\right)}{\sqrt{1-2 i \sqrt{2}}}-\frac{\left(2 i+\sqrt{2}\right) \log \left(4 \left(2 x^2+4 x+3\right)^2\right)}{\sqrt{1-2 i \sqrt{2}}}-\frac{\left(-2 i+\sqrt{2}\right) \log \left(4 \left(2 x^2+4 x+3\right)^2\right)}{\sqrt{1+2 i \sqrt{2}}}+\frac{\left(2 i+\sqrt{2}\right) \log \left(\left(2 x^2+4 x+3\right) \left(\left(2+2 i \sqrt{2}\right) x^2+\left(-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 i \sqrt{2}+4\right) x-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right)\right)}{\sqrt{1-2 i \sqrt{2}}}+\frac{\left(-2 i+\sqrt{2}\right) \log \left(\left(2 x^2+4 x+3\right) \left(\left(2-2 i \sqrt{2}\right) x^2-2 \left(\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+4 i \sqrt{2}-2\right) x-2 \sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right)\right)}{\sqrt{1+2 i \sqrt{2}}}+\frac{8 (2 x+3) \sqrt{-x^2-4 x-3}}{2 x^2+4 x+3}\right)","\frac{1-\frac{\sqrt{-x-1}}{\sqrt{x+3}}}{-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1}+\frac{\tan ^{-1}\left(\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right)}{\sqrt{2}}",1,"((8*(3 + x))/(3 + 4*x + 2*x^2) + (8*(3 + 2*x)*Sqrt[-3 - 4*x - x^2])/(3 + 4*x + 2*x^2) + 4*Sqrt[2]*ArcTan[Sqrt[2]*(1 + x)] - ((2*I)*(-2*I + Sqrt[2])*ArcTan[((2 + x)*(3*(5 + (4*I)*Sqrt[2]) + 16*(2 + I*Sqrt[2])*x + 2*(9 + (2*I)*Sqrt[2])*x^2))/(12*I - 6*Sqrt[2] + (8*I + 6*Sqrt[2])*x^3 - 9*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + x*(40*I - 5*Sqrt[2] - 12*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]) + x^2*(36*I + 8*Sqrt[2] - 6*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]] + (2*(2*I + Sqrt[2])*ArcTanh[((2 + x)*(3*(5*I + 4*Sqrt[2]) + 16*(2*I + Sqrt[2])*x + 2*(9*I + 2*Sqrt[2])*x^2))/(-5*(8*I + Sqrt[2])*x + (-8*I + 6*Sqrt[2])*x^3 - 12*Sqrt[1 - (2*I)*Sqrt[2]]*x*Sqrt[-3 - 4*x - x^2] + x^2*(-36*I + 8*Sqrt[2] - 6*Sqrt[1 - (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]) - 3*(4*I + 2*Sqrt[2] + 3*Sqrt[1 - (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sqrt[2]] - ((-2*I + Sqrt[2])*Log[4*(3 + 4*x + 2*x^2)^2])/Sqrt[1 + (2*I)*Sqrt[2]] - ((2*I + Sqrt[2])*Log[4*(3 + 4*x + 2*x^2)^2])/Sqrt[1 - (2*I)*Sqrt[2]] + ((2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 + (6*I)*Sqrt[2] + (2 + (2*I)*Sqrt[2])*x^2 - 2*Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + x*(4 + (8*I)*Sqrt[2] - 2*Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sqrt[2]] + ((-2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 - (6*I)*Sqrt[2] + (2 - (2*I)*Sqrt[2])*x^2 - 2*Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] - 2*x*(-2 + (4*I)*Sqrt[2] + Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]])/16","C",1
761,1,914,149,2.461819,"\int \frac{1}{\left(x+\sqrt{-3-4 x-x^2}\right)^3} \, dx","Integrate[(x + Sqrt[-3 - 4*x - x^2])^(-3),x]","\frac{1}{32} \left(\frac{8 (2 x-3)}{\left(2 x^2+4 x+3\right)^2}-\frac{8 \sqrt{-x^2-4 x-3} \left(8 x^3+22 x^2+26 x+15\right)}{\left(2 x^2+4 x+3\right)^2}-12 \sqrt{2} \tan ^{-1}\left(\sqrt{2} (x+1)\right)+\frac{6 \left(2+i \sqrt{2}\right) \tan ^{-1}\left(\frac{(x+2) \left(2 \left(9+2 i \sqrt{2}\right) x^2+16 \left(2+i \sqrt{2}\right) x+3 \left(5+4 i \sqrt{2}\right)\right)}{\left(8 i+6 \sqrt{2}\right) x^3+\left(-6 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}+36 i\right) x^2+\left(-12 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-5 \sqrt{2}+40 i\right) x-9 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 \sqrt{2}+12 i}\right)}{\sqrt{1+2 i \sqrt{2}}}-\frac{6 \left(2 i+\sqrt{2}\right) \tanh ^{-1}\left(\frac{(x+2) \left(2 \left(9 i+2 \sqrt{2}\right) x^2+16 \left(2 i+\sqrt{2}\right) x+3 \left(5 i+4 \sqrt{2}\right)\right)}{\left(-8 i+6 \sqrt{2}\right) x^3+\left(-6 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}-36 i\right) x^2-12 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-5 \left(8 i+\sqrt{2}\right) x-3 \left(3 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+2 \sqrt{2}+4 i\right)}\right)}{\sqrt{1-2 i \sqrt{2}}}+\frac{3 \left(2 i+\sqrt{2}\right) \log \left(4 \left(2 x^2+4 x+3\right)^2\right)}{\sqrt{1-2 i \sqrt{2}}}+\frac{3 \left(-2 i+\sqrt{2}\right) \log \left(4 \left(2 x^2+4 x+3\right)^2\right)}{\sqrt{1+2 i \sqrt{2}}}-\frac{3 \left(2 i+\sqrt{2}\right) \log \left(\left(2 x^2+4 x+3\right) \left(\left(2+2 i \sqrt{2}\right) x^2+\left(-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 i \sqrt{2}+4\right) x-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right)\right)}{\sqrt{1-2 i \sqrt{2}}}-\frac{3 \left(-2 i+\sqrt{2}\right) \log \left(\left(2 x^2+4 x+3\right) \left(\left(2-2 i \sqrt{2}\right) x^2-2 \left(\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+4 i \sqrt{2}-2\right) x-2 \sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right)\right)}{\sqrt{1+2 i \sqrt{2}}}-\frac{8 (3 x+2)}{2 x^2+4 x+3}\right)","-\frac{13-\frac{27 \sqrt{-x-1}}{\sqrt{x+3}}}{18 \left(-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right)}-\frac{2 \left(2-\frac{\sqrt{-x-1}}{\sqrt{x+3}}\right)}{9 \left(-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right)^2}-\frac{3 \tan ^{-1}\left(\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"((8*(-3 + 2*x))/(3 + 4*x + 2*x^2)^2 - (8*(2 + 3*x))/(3 + 4*x + 2*x^2) - (8*Sqrt[-3 - 4*x - x^2]*(15 + 26*x + 22*x^2 + 8*x^3))/(3 + 4*x + 2*x^2)^2 - 12*Sqrt[2]*ArcTan[Sqrt[2]*(1 + x)] + (6*(2 + I*Sqrt[2])*ArcTan[((2 + x)*(3*(5 + (4*I)*Sqrt[2]) + 16*(2 + I*Sqrt[2])*x + 2*(9 + (2*I)*Sqrt[2])*x^2))/(12*I - 6*Sqrt[2] + (8*I + 6*Sqrt[2])*x^3 - 9*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + x*(40*I - 5*Sqrt[2] - 12*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]) + x^2*(36*I + 8*Sqrt[2] - 6*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]] - (6*(2*I + Sqrt[2])*ArcTanh[((2 + x)*(3*(5*I + 4*Sqrt[2]) + 16*(2*I + Sqrt[2])*x + 2*(9*I + 2*Sqrt[2])*x^2))/(-5*(8*I + Sqrt[2])*x + (-8*I + 6*Sqrt[2])*x^3 - 12*Sqrt[1 - (2*I)*Sqrt[2]]*x*Sqrt[-3 - 4*x - x^2] + x^2*(-36*I + 8*Sqrt[2] - 6*Sqrt[1 - (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]) - 3*(4*I + 2*Sqrt[2] + 3*Sqrt[1 - (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sqrt[2]] + (3*(-2*I + Sqrt[2])*Log[4*(3 + 4*x + 2*x^2)^2])/Sqrt[1 + (2*I)*Sqrt[2]] + (3*(2*I + Sqrt[2])*Log[4*(3 + 4*x + 2*x^2)^2])/Sqrt[1 - (2*I)*Sqrt[2]] - (3*(2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 + (6*I)*Sqrt[2] + (2 + (2*I)*Sqrt[2])*x^2 - 2*Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + x*(4 + (8*I)*Sqrt[2] - 2*Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sqrt[2]] - (3*(-2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 - (6*I)*Sqrt[2] + (2 - (2*I)*Sqrt[2])*x^2 - 2*Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] - 2*x*(-2 + (4*I)*Sqrt[2] + Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]])/32","C",1
762,1,62,42,0.3682743,"\int x^3 (1+x)^3 (1+2 x) \sqrt{1-x^2-2 x^3-x^4} \, dx","Integrate[x^3*(1 + x)^3*(1 + 2*x)*Sqrt[1 - x^2 - 2*x^3 - x^4],x]","\frac{1}{15} \sqrt{-x^4-2 x^3-x^2+1} \left(3 x^8+12 x^7+18 x^6+12 x^5+2 x^4-2 x^3-x^2-2\right)","-\frac{1}{15} \left(-x^4-2 x^3-x^2+1\right)^{3/2} \left(3 x^4+6 x^3+3 x^2+2\right)",1,"(Sqrt[1 - x^2 - 2*x^3 - x^4]*(-2 - x^2 - 2*x^3 + 2*x^4 + 12*x^5 + 18*x^6 + 12*x^7 + 3*x^8))/15","A",1
763,1,62,42,0.3691416,"\int (1+2 x) \left(x+x^2\right)^3 \sqrt{1-\left(x+x^2\right)^2} \, dx","Integrate[(1 + 2*x)*(x + x^2)^3*Sqrt[1 - (x + x^2)^2],x]","\frac{1}{15} \sqrt{-x^4-2 x^3-x^2+1} \left(3 x^8+12 x^7+18 x^6+12 x^5+2 x^4-2 x^3-x^2-2\right)","-\frac{1}{15} \left(-x^4-2 x^3-x^2+1\right)^{3/2} \left(3 x^4+6 x^3+3 x^2+2\right)",1,"(Sqrt[1 - x^2 - 2*x^3 - x^4]*(-2 - x^2 - 2*x^3 + 2*x^4 + 12*x^5 + 18*x^6 + 12*x^7 + 3*x^8))/15","A",1
764,1,278,102,0.768001,"\int \left(8 x-8 x^2+4 x^3-x^4\right)^{3/2} \, dx","Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\frac{5 x^9-45 x^8+206 x^7-602 x^6+1152 x^5-1420 x^4+848 x^3+352 x^2-304 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)+\frac{112 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)}{\sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}}}-1056 x+896}{35 \sqrt{-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{1}{7} (x-1) \left(-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{2}{35} \left(13-3 (x-1)^2\right) (x-1) \sqrt{-(x-1)^4-2 (x-1)^2+3}-\frac{176}{35} \sqrt{3} F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)+\frac{16}{5} \sqrt{3} E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)",1,"(896 - 1056*x + 352*x^2 + 848*x^3 - 1420*x^4 + 1152*x^5 - 602*x^6 + 206*x^7 - 45*x^8 + 5*x^9 + ((112*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)] - (304*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(35*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])","C",0
765,1,256,62,0.6329971,"\int \sqrt{8 x-8 x^2+4 x^3-x^4} \, dx","Integrate[Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]","-\frac{x^5-5 x^4+14 x^3-24 x^2+8 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)-\frac{2 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)}{\sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}}}+24 x-16}{3 \sqrt{-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{1}{3} \sqrt{-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac{4 F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}+\frac{2 E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"-1/3*(-16 + 24*x - 24*x^2 + 14*x^3 - 5*x^4 + x^5 - ((2*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)] + (8*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]","C",0
766,1,156,17,0.1527455,"\int \frac{1}{\sqrt{8 x-8 x^2+4 x^3-x^4}} \, dx","Integrate[1/Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]","\frac{\sqrt{\frac{4 i}{x}+\sqrt{3}-i} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} x \left(-i \sqrt{3} x+x-4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)}{\sqrt{2} \sqrt{-\frac{4 i}{x}+\sqrt{3}+i} \sqrt{-x \left(x^3-4 x^2+8 x-8\right)}}","-\frac{F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(Sqrt[-I + Sqrt[3] + (4*I)/x]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x*(-4 + x - I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(Sqrt[2]*Sqrt[I + Sqrt[3] - (4*I)/x]*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])","C",0
767,1,261,73,0.9126595,"\int \frac{1}{\left(8 x-8 x^2+4 x^3-x^4\right)^{3/2}} \, dx","Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(-3/2),x]","\frac{\sqrt{-x \left(x^3-4 x^2+8 x-8\right)} \left(\frac{\sqrt{2} \left(\sqrt{3}-i\right) \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)}{\sqrt{\frac{x^2-2 x+4}{x^2}}}-\frac{x^2-4 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)+2}{x^2-2 x+4}\right)}{24 (x-2) x}","\frac{\left((x-1)^2+5\right) (x-1)}{24 \sqrt{-(x-1)^4-2 (x-1)^2+3}}-\frac{F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{4 \sqrt{3}}+\frac{E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{8 \sqrt{3}}",1,"(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*((Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[(4 - 2*x + x^2)/x^2] - (2 + x^2 - (4*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(4 - 2*x + x^2)))/(24*(-2 + x)*x)","C",0
768,1,298,109,1.0791783,"\int \frac{1}{\left(8 x-8 x^2+4 x^3-x^4\right)^{5/2}} \, dx","Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]","\frac{\frac{7 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)}{\sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}}}+\frac{7 x^6-37 x^5+115 x^4-226 x^3+274 x^2-19 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} \left(x^3-4 x^2+8 x-8\right) x^3 F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)-232 x+36}{x^3-4 x^2+8 x-8}}{432 x \sqrt{-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{\left(7 (x-1)^2+26\right) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left((x-1)^2+5\right) (x-1)}{72 \left(-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}-\frac{11 F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{144 \sqrt{3}}+\frac{7 E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{144 \sqrt{3}}",1,"(((7*I)*Sqrt[2]*(-2 + x)*x^2*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)] + (36 - 232*x + 274*x^2 - 226*x^3 + 115*x^4 - 37*x^5 + 7*x^6 - (19*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^3*Sqrt[(4 - 2*x + x^2)/x^2]*(-8 + 8*x - 4*x^2 + x^3)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(-8 + 8*x - 4*x^2 + x^3))/(432*x*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])","C",0
769,1,278,102,1.0252197,"\int \left((2-x) x \left(4-2 x+x^2\right)\right)^{3/2} \, dx","Integrate[((2 - x)*x*(4 - 2*x + x^2))^(3/2),x]","\frac{\sqrt{-x \left(x^3-4 x^2+8 x-8\right)} \left(\sqrt{\frac{x^2-2 x+4}{x^2}} \left(-5 x^7+35 x^6-116 x^5+230 x^4-228 x^3+44 x^2+152 x-224\right)+304 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)+112 \sqrt{2} \left(\sqrt{3}-i\right) \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)\right)}{35 (x-2) x \sqrt{\frac{x^2-2 x+4}{x^2}}}","\frac{1}{7} (x-1) \left(-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{2}{35} \left(13-3 (x-1)^2\right) (x-1) \sqrt{-(x-1)^4-2 (x-1)^2+3}-\frac{176}{35} \sqrt{3} F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)+\frac{16}{5} \sqrt{3} E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)",1,"(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-224 + 152*x + 44*x^2 - 228*x^3 + 230*x^4 - 116*x^5 + 35*x^6 - 5*x^7) + 112*Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])] + (304*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])]))/(35*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2])","C",0
770,1,256,62,0.8449674,"\int \sqrt{(2-x) x \left(4-2 x+x^2\right)} \, dx","Integrate[Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]","\frac{\sqrt{-x \left(x^3-4 x^2+8 x-8\right)} \left(\sqrt{\frac{x^2-2 x+4}{x^2}} \left(x^3-3 x^2+4 x-4\right)+8 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)+2 \sqrt{2} \left(\sqrt{3}-i\right) \sqrt{-\frac{i (x-2)}{\left(\sqrt{3}-i\right) x}} E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right)\right)}{3 (x-2) x \sqrt{\frac{x^2-2 x+4}{x^2}}}","\frac{1}{3} \sqrt{-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac{4 F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}+\frac{2 E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-4 + 4*x - 3*x^2 + x^3) + 2*Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])] + (8*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])]))/(3*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2])","C",0
771,1,100,17,0.2701082,"\int \frac{1}{\sqrt{(2-x) x \left(4-2 x+x^2\right)}} \, dx","Integrate[1/Sqrt[(2 - x)*x*(4 - 2*x + x^2)],x]","-\frac{\sqrt[3]{-1} (x-2)^2 \sqrt{\frac{x \left(x+i \sqrt{3}-1\right)}{(x-2)^2}} \sqrt{\frac{-\sqrt[3]{-1} x+x-2}{x-2}} F\left(\sin ^{-1}\left(\sqrt{-\frac{(-1)^{2/3} x}{x-2}}\right)|(-1)^{2/3}\right)}{\sqrt{-x \left(x^3-4 x^2+8 x-8\right)}}","-\frac{F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{\sqrt{3}}",1,"-(((-1)^(1/3)*(-2 + x)^2*Sqrt[(x*(-1 + I*Sqrt[3] + x))/(-2 + x)^2]*Sqrt[(-2 + x - (-1)^(1/3)*x)/(-2 + x)]*EllipticF[ArcSin[Sqrt[-(((-1)^(2/3)*x)/(-2 + x))]], (-1)^(2/3)])/Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])","C",1
772,1,298,73,0.9865197,"\int \frac{1}{\left((2-x) x \left(4-2 x+x^2\right)\right)^{3/2}} \, dx","Integrate[((2 - x)*x*(4 - 2*x + x^2))^(-3/2),x]","\frac{(x-2)^2 x \left(x^2-2 x+4\right) \left(-\frac{3 \left(x^2-2 x+4\right) x}{x-2}-3 \left(x^2-2 x+4\right)-4 (2-x) \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} \left(\sqrt{\frac{x^2-2 x+4}{(x-2)^2}} x+4 i \sqrt{2} \sqrt{\frac{i x}{\left(\sqrt{3}+i\right) (x-2)}} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right)-\sqrt{2} \left(\sqrt{3}+i\right) \sqrt{\frac{i x}{\left(\sqrt{3}+i\right) (x-2)}} E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right)\right)+2 (x-1) x\right)}{96 \left(-x \left(x^3-4 x^2+8 x-8\right)\right)^{3/2}}","\frac{\left((x-1)^2+5\right) (x-1)}{24 \sqrt{-(x-1)^4-2 (x-1)^2+3}}-\frac{F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{4 \sqrt{3}}+\frac{E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{8 \sqrt{3}}",1,"((-2 + x)^2*x*(4 - 2*x + x^2)*(2*(-1 + x)*x - 3*(4 - 2*x + x^2) - (3*x*(4 - 2*x + x^2))/(-2 + x) - 4*(2 - x)*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*(x*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2] - Sqrt[2]*(I + Sqrt[3])*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*EllipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])] + (4*I)*Sqrt[2]*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])])))/(96*(-(x*(-8 + 8*x - 4*x^2 + x^3)))^(3/2))","C",0
773,1,327,109,1.0908334,"\int \frac{1}{\left((2-x) x \left(4-2 x+x^2\right)\right)^{5/2}} \, dx","Integrate[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]","\frac{(x-2)^3 x^2 \left(x^2-2 x+4\right)^2 \left(-\frac{7 x \left(x^2-2 x+4\right)}{x-2}-19 i \sqrt{2} (x-2) \sqrt{\frac{i x}{\left(\sqrt{3}+i\right) (x-2)}} \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} F\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right)+\frac{7 i \sqrt{2} x \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt[4]{3}}\right)|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right)}{\sqrt{\frac{i x}{\left(\sqrt{3}+i\right) (x-2)}}}+\frac{7 x^7-49 x^6+187 x^5-445 x^4+670 x^3-622 x^2+216 x+36}{(x-2)^2 x \left(x^2-2 x+4\right)}\right)}{432 \left(-x \left(x^3-4 x^2+8 x-8\right)\right)^{5/2}}","\frac{\left(7 (x-1)^2+26\right) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left((x-1)^2+5\right) (x-1)}{72 \left(-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}-\frac{11 F\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{144 \sqrt{3}}+\frac{7 E\left(\sin ^{-1}(1-x)|-\frac{1}{3}\right)}{144 \sqrt{3}}",1,"((-2 + x)^3*x^2*(4 - 2*x + x^2)^2*((-7*x*(4 - 2*x + x^2))/(-2 + x) + (36 + 216*x - 622*x^2 + 670*x^3 - 445*x^4 + 187*x^5 - 49*x^6 + 7*x^7)/((-2 + x)^2*x*(4 - 2*x + x^2)) + ((7*I)*Sqrt[2]*x*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*EllipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])])/Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))] - (19*I)*Sqrt[2]*(-2 + x)*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])]))/(432*(-(x*(-8 + 8*x - 4*x^2 + x^3)))^(5/2))","C",0
774,1,10468,730,6.203058,"\int \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)^{3/2} \, dx","Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2),x]","\text{Result too large to show}","\frac{1}{7} \left(\frac{c}{d}+x\right) \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)^{3/2}-\frac{16 c^3 \left(8 a d^2+c^3\right) \left(\frac{c}{d}+x\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^2 \sqrt{4 a d^2+c^3} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)}+\frac{2 c \left(\frac{c}{d}+x\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left(20 a d^2+7 c^3-3 c d^2 \left(\frac{c}{d}+x\right)^2\right)}{35 d^2}+\frac{8 c^{7/4} \left(4 a d^2+c^3\right)^{3/4} \left(\sqrt{4 a d^2+c^3} \left(5 a d^2+c^3\right)-c^{3/2} \left(8 a d^2+c^3\right)\right) \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{16 c^{13/4} \left(4 a d^2+c^3\right)^{3/4} \left(8 a d^2+c^3\right) \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) E\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}",1,"Result too large to show","C",0
775,1,5218,622,6.0947714,"\int \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx","Integrate[Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]","\text{Result too large to show}","\frac{1}{3} \left(\frac{c}{d}+x\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}-\frac{2 c^2 \left(\frac{c}{d}+x\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 \sqrt{4 a d^2+c^3} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)}+\frac{c^{3/4} \sqrt[4]{4 a d^2+c^3} \left(-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right) \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{2 c^{9/4} \left(4 a d^2+c^3\right)^{3/4} \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) E\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{3 d^3 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}",1,"Result too large to show","C",0
776,1,822,227,2.2482068,"\int \frac{1}{\sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \, dx","Integrate[1/Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]","\frac{2 \left(-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right) \left(c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right) \sqrt{-\frac{\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \left(c+d x-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right)}{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}} \sqrt{-\frac{\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \left(c+d x+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right)}{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}}\right)|\frac{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right)^2}{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right)^2}\right)}{d \sqrt{c^2-2 i \sqrt{a} \sqrt{c} d} \sqrt{\frac{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}-\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(c+d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}{\left(\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}+\sqrt{c^2+2 i \sqrt{a} d \sqrt{c}}\right) \left(-c-d x+\sqrt{c^2-2 i \sqrt{a} \sqrt{c} d}\right)}} \sqrt{x^2 (2 c+d x)^2+4 a c}}","\frac{\sqrt[4]{4 a d^2+c^3} \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{2 \sqrt[4]{c} d \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}",1,"(2*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)*(c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x)*Sqrt[-((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*(c - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)))]*Sqrt[-((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*(c + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x)))]*EllipticF[ArcSin[Sqrt[((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x))]], (Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])^2/(Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])^2])/(d*Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d]*Sqrt[((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + d*x))/((Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] + Sqrt[c^2 + (2*I)*Sqrt[a]*Sqrt[c]*d])*(-c + Sqrt[c^2 - (2*I)*Sqrt[a]*Sqrt[c]*d] - d*x))]*Sqrt[4*a*c + x^2*(2*c + d*x)^2])","C",1
777,1,5276,674,6.134083,"\int \frac{1}{\left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)^{3/2}} \, dx","Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-3/2),x]","\text{Result too large to show}","-\frac{d^2 \left(\frac{c}{d}+x\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{8 a \left(4 a d^2+c^3\right)^{3/2} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)}-\frac{\left(\frac{c}{d}+x\right) \left(-4 a d^2+c^3-c d^2 \left(\frac{c}{d}+x\right)^2\right)}{8 a c \left(4 a d^2+c^3\right) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{\left(-c^{3/2} \sqrt{4 a d^2+c^3}+4 a d^2+c^3\right) \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) F\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{16 a c^{5/4} d \left(4 a d^2+c^3\right)^{3/4} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{\sqrt[4]{c} \sqrt{\frac{d^2 \left(4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right)}{\left(4 a d^2+c^3\right) \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right)^2}} \left(\frac{d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right) E\left(2 \tan ^{-1}\left(\frac{c+d x}{\sqrt[4]{c} \sqrt[4]{c^3+4 a d^2}}\right)|\frac{1}{2} \left(\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right)\right)}{8 a d \sqrt[4]{4 a d^2+c^3} \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}",1,"Result too large to show","C",1
778,1,7543,663,6.1237263,"\int \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx","Integrate[Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]","\text{Result too large to show}","\frac{1}{3} \left(\frac{d}{4 e}+x\right) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}-\frac{2 d^2 \left(\frac{d}{4 e}+x\right) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\sqrt{256 a e^3+5 d^4} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)}+\frac{\sqrt[4]{256 a e^3+5 d^4} \left(-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right) \sqrt{\frac{e \left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)}{\left(256 a e^3+5 d^4\right) \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)^2}} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right) F\left(2 \tan ^{-1}\left(\frac{d+4 e x}{\sqrt[4]{5 d^4+256 a e^3}}\right)|\frac{1}{2} \left(\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right)\right)}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{d^2 \left(256 a e^3+5 d^4\right)^{3/4} \sqrt{\frac{e \left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)}{\left(256 a e^3+5 d^4\right) \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)^2}} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right) E\left(2 \tan ^{-1}\left(\frac{d+4 e x}{\sqrt[4]{5 d^4+256 a e^3}}\right)|\frac{1}{2} \left(\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right)\right)}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}",1,"Result too large to show","B",0
779,1,1065,235,2.2839414,"\int \frac{1}{\sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \, dx","Integrate[1/Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]","-\frac{\left(-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right) \left(d+4 e x-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \sqrt{-\frac{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \left(d+4 e x+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right)}{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \left(-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right)}} \sqrt{\frac{3 d^2+\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) d+4 e \left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) x-2 \sqrt{d^4-64 a e^3}-\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}}{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \left(-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \left(d+4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right)}{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \left(-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right)}}\right)|\frac{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right)^2}{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right)^2}\right)}{2 e \left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \sqrt{\frac{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \left(-d-4 e x+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right)}{\left(\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right) \left(-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right)}} \sqrt{8 e^3 x^4+8 d e^2 x^3-d^3 x+8 a e^2}}","\frac{\sqrt[4]{256 a e^3+5 d^4} \sqrt{\frac{e \left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)}{\left(256 a e^3+5 d^4\right) \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)^2}} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right) F\left(2 \tan ^{-1}\left(\frac{d+4 e x}{\sqrt[4]{5 d^4+256 a e^3}}\right)|\frac{1}{2} \left(\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right)\right)}{\sqrt{2} e \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}",1,"-1/2*((-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x)*(d - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] + 4*e*x)*Sqrt[-((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]*(d + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] + 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x)))]*Sqrt[(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3] - Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]*Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] + d*(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]) + 4*e*(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*x)/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))]*EllipticF[ArcSin[Sqrt[((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))]], (Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])^2/(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])^2])/(e*(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*Sqrt[(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]*(-d + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))]*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])","B",1
780,1,7629,748,6.1537171,"\int \frac{1}{\left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)^{3/2}} \, dx","Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-3/2),x]","\text{Result too large to show}","\frac{4 e \left(\frac{d}{4 e}+x\right) \left(-256 a e^3+13 d^4-48 d^2 e^2 \left(\frac{d}{4 e}+x\right)^2\right)}{\left(-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{384 d^2 e^2 \left(\frac{d}{4 e}+x\right) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\left(d^4-64 a e^3\right) \left(256 a e^3+5 d^4\right)^{3/2} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)}-\frac{2 \sqrt{2} \left(-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right) \sqrt{\frac{e \left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)}{\left(256 a e^3+5 d^4\right) \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)^2}} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right) F\left(2 \tan ^{-1}\left(\frac{d+4 e x}{\sqrt[4]{5 d^4+256 a e^3}}\right)|\frac{1}{2} \left(\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right)\right)}{\left(d^4-64 a e^3\right) \left(256 a e^3+5 d^4\right)^{3/4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}-\frac{12 \sqrt{2} d^2 \sqrt{\frac{e \left(8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right)}{\left(256 a e^3+5 d^4\right) \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right)^2}} \left(\frac{16 e^2 \left(\frac{d}{4 e}+x\right)^2}{\sqrt{256 a e^3+5 d^4}}+1\right) E\left(2 \tan ^{-1}\left(\frac{d+4 e x}{\sqrt[4]{5 d^4+256 a e^3}}\right)|\frac{1}{2} \left(\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right)\right)}{\left(d^4-64 a e^3\right) \sqrt[4]{256 a e^3+5 d^4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}",1,"Result too large to show","B",0
781,1,6287,452,6.1266574,"\int \left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2} \, dx","Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\text{Result too large to show}","\frac{1}{7} (x-1) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{2}{35} (x-1) \left(5 a-3 (x-1)^2+13\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{16 (2 a+7) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{35 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{4 (a+3) (5 a+16) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{16 (2 a+7) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",1
782,1,3470,397,6.0607986,"\int \sqrt{a+8 x-8 x^2+4 x^3-x^4} \, dx","Integrate[Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\text{Result too large to show}","\frac{1}{3} (x-1) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{2 \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{3 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{2 (a+3) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{2 \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"(-1/3 + x/3)*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4] + (2*((4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (2*a*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - ((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]))/3","B",1
783,1,540,144,1.4866556,"\int \frac{1}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx","Integrate[1/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\frac{2 \left(\sqrt{-\sqrt{a+4}-1}-x+1\right) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(\sqrt{\sqrt{a+4}-1}-x+1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(\sqrt{-\sqrt{a+4}-1}-x+1\right)}} \left(\sqrt{-\sqrt{a+4}-1}+x-1\right) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(\sqrt{\sqrt{a+4}-1}+x-1\right)}{\left(\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right) \left(\sqrt{-\sqrt{a+4}-1}-x+1\right)}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)}{\sqrt{-\sqrt{a+4}-1} \sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(\sqrt{-\sqrt{a+4}-1}+x-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(\sqrt{-\sqrt{a+4}-1}-x+1\right)}} \sqrt{a-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"(2*(1 + Sqrt[-1 - Sqrt[4 + a]] - x)*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(1 + Sqrt[-1 + Sqrt[4 + a]] - x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(Sqrt[-1 - Sqrt[4 + a]]*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)])","B",1
784,1,3526,437,6.0884432,"\int \frac{1}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2}} \, dx","Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-3/2),x]","\text{Result too large to show}","\frac{(x-1) \left(a+(x-1)^2+5\right)}{2 \left(a^2+7 a+12\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{2 (a+3) (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{2 (a+3) (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"((6 + a - 8*x - a*x + 3*x^2 - x^3)*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4])/(2*(3 + a)*(4 + a)*(-a - 8*x + 8*x^2 - 4*x^3 + x^4)) + ((4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (2*a*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - ((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4])/(2*(3 + a)*(4 + a))","B",1
785,1,6386,517,6.223736,"\int \frac{1}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{5/2}} \, dx","Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-5/2),x]","\text{Result too large to show}","\frac{(x-1) \left(5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right)}{12 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1) \left(a+(x-1)^2+5\right)}{6 \left(a^2+7 a+12\right) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}-\frac{(2 a+7) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{3 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(5 a+16) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{12 (a+3) (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(2 a+7) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 (a+3)^2 (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",0
786,1,7235,558,6.1429435,"\int x \left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2} \, dx","Integrate[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\text{Result too large to show}","\frac{3}{16} (a+4) \left((x-1)^2+1\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{8} \left((x-1)^2+1\right) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{1}{7} (x-1) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{2}{35} (x-1) \left(5 a-3 (x-1)^2+13\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{16 (2 a+7) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{35 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{16} (a+4)^2 \tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{4 (a+3) (5 a+16) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{16 (2 a+7) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{35 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",0
787,1,4389,466,6.091259,"\int x \sqrt{a+8 x-8 x^2+4 x^3-x^4} \, dx","Integrate[x*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\text{Result too large to show}","\frac{1}{4} \left((x-1)^2+1\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{3} (x-1) \sqrt{a-(x-1)^4-2 (x-1)^2+3}-\frac{2 \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{3 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{1}{4} (a+4) \tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{2 (a+3) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{2 \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"(1/6 - x/6 + x^2/4)*Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)] + (Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)]*((-8*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (2*a*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (40*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (6*a*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - (4*((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]))/(6*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4])","B",1
788,1,813,179,2.9308465,"\int \frac{x}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx","Integrate[x/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\frac{2 \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(-x+\sqrt{\sqrt{a+4}-1}+1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}} \left(x+\sqrt{-\sqrt{a+4}-1}-1\right) \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(x+\sqrt{\sqrt{a+4}-1}-1\right)}{\left(\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}} \left(\left(\sqrt{-\sqrt{a+4}-1}+1\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)-2 \sqrt{-\sqrt{a+4}-1} \Pi \left(\frac{\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)\right)}{\sqrt{-\sqrt{a+4}-1} \sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}} \sqrt{a-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{1}{2} \tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"(2*(1 + Sqrt[-1 - Sqrt[4 + a]] - x)*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(1 + Sqrt[-1 + Sqrt[4 + a]] - x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*((1 + Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] - 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)])","B",1
789,1,3593,474,6.0839813,"\int \frac{x}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2}} \, dx","Integrate[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\text{Result too large to show}","\frac{(x-1) \left(a+(x-1)^2+5\right)}{2 \left(a^2+7 a+12\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{2 (a+3) (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{2 (a+3) (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"((-a - 2*x + a*x - a*x^2 - x^3)*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2)/(2*(3 + a)*(4 + a)*(-a - 8*x + 8*x^2 - 4*x^3 + x^4)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2)) + ((a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2)*((4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (2*a*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + (4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - ((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]))/(2*(3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2))","B",1
790,1,6452,591,6.1298697,"\int \frac{x}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{5/2}} \, dx","Integrate[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2),x]","\text{Result too large to show}","\frac{(x-1) \left(5 a^2+4 (2 a+7) (x-1)^2+47 a+104\right)}{12 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1) \left(a+(x-1)^2+5\right)}{6 \left(a^2+7 a+12\right) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}+\frac{(x-1)^2+1}{3 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{6 (a+4) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}-\frac{(2 a+7) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{3 (a+3)^2 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(5 a+16) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{12 (a+3) (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(2 a+7) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{3 (a+3)^2 (a+4)^2 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",0
791,1,8500,585,6.1702353,"\int x^2 \left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2} \, dx","Integrate[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\text{Result too large to show}","\frac{4 \left(21 a^2+111 a+140\right) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{315 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{4 \left(21 a^2+111 a+140\right) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{8} (a+4) \left((x-1)^2+1\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{4} \left((x-1)^2+1\right) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{1}{63} \left(7 (x-1)^2+15\right) (x-1) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}+\frac{2}{315} (x-1) \left(3 (7 a+20) (x-1)^2+2 (27 a+80)\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{3}{8} (a+4)^2 \tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{4 (a+3) (33 a+100) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",0
792,1,5647,485,6.1220047,"\int x^2 \sqrt{a+8 x-8 x^2+4 x^3-x^4} \, dx","Integrate[x^2*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\text{Result too large to show}","\frac{1}{2} \left((x-1)^2+1\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{15} \left(3 (x-1)^2+7\right) (x-1) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{2 (3 a+8) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{15 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{1}{2} (a+4) \tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{8 (a+3) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{15 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{2 (3 a+8) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{15 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",1
793,1,1247,388,6.0561568,"\int \frac{x^2}{\sqrt{a+8 x-8 x^2+4 x^3-x^4}} \, dx","Integrate[x^2/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]","\frac{2 \left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}} \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(x-\sqrt{\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(x-\sqrt{-\sqrt{a+4}-1}-1\right)}} \sqrt{\frac{\sqrt{-\sqrt{a+4}-1} \left(x+\sqrt{\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x-\sqrt{-\sqrt{a+4}-1}-1\right)}} \left(\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) E\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)}{2 \sqrt{-\sqrt{a+4}-1}}+\frac{\left(\left(\sqrt{-\sqrt{a+4}-1}-1\right) \left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)-\left(-\sqrt{-\sqrt{a+4}-1}-1\right) \left(-\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}-2\right)\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)}{2 \sqrt{-\sqrt{a+4}-1} \left(\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}\right)}+\frac{4 \Pi \left(\frac{\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right) \left(x+\sqrt{-\sqrt{a+4}-1}-1\right)}{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right) \left(-x+\sqrt{-\sqrt{a+4}-1}+1\right)}}\right)|\frac{\left(\sqrt{-\sqrt{a+4}-1}+\sqrt{\sqrt{a+4}-1}\right)^2}{\left(\sqrt{-\sqrt{a+4}-1}-\sqrt{\sqrt{a+4}-1}\right)^2}\right)}{\sqrt{\sqrt{a+4}-1}-\sqrt{-\sqrt{a+4}-1}}\right) \left(x-\sqrt{-\sqrt{a+4}-1}-1\right)^2+\left(x+\sqrt{-\sqrt{a+4}-1}-1\right) \left(x-\sqrt{\sqrt{a+4}-1}-1\right) \left(x+\sqrt{\sqrt{a+4}-1}-1\right)}{\sqrt{a-x \left(x^3-4 x^2+8 x-8\right)}}","\frac{\left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\tan ^{-1}\left(\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right)+\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{\sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a - x*(-8 + 8*x - 4*x^2 + x^3)]","B",1
794,1,2941,311,6.1248344,"\int \frac{x^2}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{3/2}} \, dx","Integrate[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]","\text{Result too large to show}","\frac{(a+4) \left((x-1)^2+2\right) (x-1)}{2 \left(a^2+7 a+12\right) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{(a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{\left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{2 (a+3) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{\left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{2 (a+3) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"((-a - 8*x - a*x + 6*x^2 + a*x^2 - 4*x^3 - a*x^3)*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2)/(2*(3 + a)*(4 + a)*(-a - 8*x + 8*x^2 - 4*x^3 + x^4)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2)) - ((a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2)*((2*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*EllipticF[ArcSin[Sqrt[((-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]], ((-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]))])/(Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) - (4*(-Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*((-1 - Sqrt[-1 - Sqrt[4 + a]])*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2] + 2*Sqrt[-1 - Sqrt[4 + a]]*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2]))/(Sqrt[-1 - Sqrt[4 + a]]*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]) + ((-1 + Sqrt[-1 - Sqrt[4 + a]] + x)*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x)*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x) + 2*(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x)^2*Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 - Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*Sqrt[(Sqrt[-1 - Sqrt[4 + a]]*(-1 + Sqrt[-1 + Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 - Sqrt[-1 - Sqrt[4 + a]] + x))]*(((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*EllipticE[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]) + ((-((-1 - Sqrt[-1 - Sqrt[4 + a]])*(-2 - Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])) + (-1 + Sqrt[-1 - Sqrt[4 + a]])*(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]]))*EllipticF[ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(2*Sqrt[-1 - Sqrt[4 + a]]*(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])) + (4*EllipticPi[(Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]]), ArcSin[Sqrt[((Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])*(-1 + Sqrt[-1 - Sqrt[4 + a]] + x))/((Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])*(1 + Sqrt[-1 - Sqrt[4 + a]] - x))]], (Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])^2/(Sqrt[-1 - Sqrt[4 + a]] - Sqrt[-1 + Sqrt[4 + a]])^2])/(-Sqrt[-1 - Sqrt[4 + a]] + Sqrt[-1 + Sqrt[4 + a]])))/Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4]))/(2*(3 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))^(3/2))","B",1
795,1,5812,582,6.1665408,"\int \frac{x^2}{\left(a+8 x-8 x^2+4 x^3-x^4\right)^{5/2}} \, dx","Integrate[x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2),x]","\text{Result too large to show}","\frac{(a+4) \left((x-1)^2+2\right) (x-1)}{6 \left(a^2+7 a+12\right) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}+\frac{\sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) F\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{12 \left(a^2+7 a+12\right) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{2 \left((x-1)^2+1\right)}{3 (a+4)^2 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(x-1)^2+1}{3 (a+4) \left(a-(x-1)^4-2 (x-1)^2+3\right)^{3/2}}+\frac{(x-1) \left((3 a+13) (x-1)^2+7 a+29\right)}{12 (a+3)^2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{(3 a+13) \left(1-\sqrt{a+4}\right) (x-1) \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right)}{12 (a+3)^2 (a+4) \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{(3 a+13) \left(1-\sqrt{a+4}\right) \sqrt{\sqrt{a+4}+1} \left(\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right) E\left(\tan ^{-1}\left(\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right)|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right)}{12 (a+3)^2 (a+4) \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}",1,"Result too large to show","B",0
796,1,927,129,0.8434204,"\int \frac{1}{\sqrt{8+8 x-x^3+8 x^4}} \, dx","Integrate[1/Sqrt[8 + 8*x - x^3 + 8*x^4],x]","-\frac{2 F\left(\sin ^{-1}\left(\sqrt{\frac{\left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}}\right)|\frac{\left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,3\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}{\left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,3\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right)^2 \sqrt{\frac{\left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,3\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,3\right]\right)}} \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right) \sqrt{\frac{\left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right)^2 \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)^2}}}{\sqrt{8 x^4-x^3+8 x+8} \left(-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,1\right]+\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\&,4\right]\right)}","-\frac{x^2 \sqrt{\frac{\left(\frac{4}{x}+1\right)^4-6 \left(\frac{4}{x}+1\right)^2+261}{\left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right)^2}} \left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right) F\left(2 \tan ^{-1}\left(\frac{x+4}{\sqrt{3} \sqrt[4]{29} x}\right)|\frac{1}{58} \left(29+\sqrt{29}\right)\right)}{8 \sqrt{3} \sqrt[4]{29} \sqrt{8 x^4-x^3+8 x+8}}",1,"(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])^2)])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))","C",1
797,1,4865,431,6.0464037,"\int \frac{1}{\left(8+8 x-x^3+8 x^4\right)^{3/2}} \, dx","Integrate[(8 + 8*x - x^3 + 8*x^4)^(-3/2),x]","\text{Result too large to show}","-\frac{\left(66-\left(\frac{4}{x}+1\right)^2\right) x^2}{1008 \sqrt{8 x^4-x^3+8 x+8}}+\frac{\left(216-7 \left(\frac{4}{x}+1\right)^2\right) \left(\frac{4}{x}+1\right) x^2}{12528 \sqrt{8 x^4-x^3+8 x+8}}+\frac{7 \left(\left(\frac{4}{x}+1\right)^4-6 \left(\frac{4}{x}+1\right)^2+261\right) \left(\frac{4}{x}+1\right) x^2}{432 \sqrt{29} \sqrt{8 x^4-x^3+8 x+8} \left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right)}+\frac{\left(14-5 \sqrt{29}\right) \sqrt{\frac{\left(\frac{4}{x}+1\right)^4-6 \left(\frac{4}{x}+1\right)^2+261}{\left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right)^2}} \left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right) x^2 F\left(2 \tan ^{-1}\left(\frac{x+4}{\sqrt{3} \sqrt[4]{29} x}\right)|\frac{1}{58} \left(29+\sqrt{29}\right)\right)}{576 \sqrt{3} 29^{3/4} \sqrt{8 x^4-x^3+8 x+8}}-\frac{7 \sqrt{\frac{\left(\frac{4}{x}+1\right)^4-6 \left(\frac{4}{x}+1\right)^2+261}{\left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right)^2}} \left(\frac{\sqrt{29} (x+4)^2}{x^2}+87\right) x^2 E\left(2 \tan ^{-1}\left(\frac{x+4}{\sqrt{3} \sqrt[4]{29} x}\right)|\frac{1}{58} \left(29+\sqrt{29}\right)\right)}{144 \sqrt{3} 29^{3/4} \sqrt{8 x^4-x^3+8 x+8}}",1,"(544 + 1539*x - 1146*x^2 + 784*x^3)/(21924*Sqrt[8 + 8*x - x^3 + 8*x^4]) + ((28*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]) + EllipticPi[(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]))*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) + (842*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) - (224*((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]) + (x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*((EllipticE[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]) + (EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]) - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) - (EllipticPi[(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))))/Sqrt[8 + 8*x - x^3 + 8*x^4])/6264","C",0
798,1,249,108,0.5931459,"\int \frac{1}{\sqrt{1+4 x+4 x^2+4 x^4}} \, dx","Integrate[1/Sqrt[1 + 4*x + 4*x^2 + 4*x^4],x]","\frac{(2-i) \sqrt{-\frac{1}{10}+\frac{i}{5}} \sqrt{\frac{\left(2 i+\sqrt{-1-2 i}-\sqrt{-1+2 i}\right) \left(-2 x+\sqrt{-1-2 i}-i\right)}{\left(-2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1-2 i}+i\right)}} \left(2 i x^2+2 x+1\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\left(2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1+2 i}-i\right)}{\sqrt{-1+2 i} \left(2 x+\sqrt{-1-2 i}+i\right)}}}{\sqrt{2}}\right)|\frac{1}{2} \left(5-\sqrt{5}\right)\right)}{\sqrt{\frac{(1+2 i) \left((-1+i)+\sqrt{-1-2 i}\right) \left(2 i x^2+2 x+1\right)}{\left(2 x+\sqrt{-1-2 i}+i\right)^2}} \sqrt{4 x^4+4 x^2+4 x+1}}","-\frac{\left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right) \sqrt{\frac{\left(\frac{1}{x}+1\right)^4-2 \left(\frac{1}{x}+1\right)^2+5}{\left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right)^2}} x^2 F\left(2 \tan ^{-1}\left(\frac{1+\frac{1}{x}}{\sqrt[4]{5}}\right)|\frac{1}{10} \left(5+\sqrt{5}\right)\right)}{2 \sqrt[4]{5} \sqrt{4 x^4+4 x^2+4 x+1}}",1,"((2 - I)*Sqrt[-1/10 + I/5]*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*(-I + Sqrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(I + Sqrt[-1 - 2*I] + 2*x))]*(1 + 2*x + (2*I)*x^2)*EllipticF[ArcSin[Sqrt[((2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*I] + 2*x))]/Sqrt[2]], (5 - Sqrt[5])/2])/(Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2*I])*(1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2]*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])","C",0
799,1,602,367,4.757811,"\int \frac{1}{\left(1+4 x+4 x^2+4 x^4\right)^{3/2}} \, dx","Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-3/2),x]","\frac{36 x^3-16 x^2+\frac{(6-3 i) \sqrt{-\frac{2}{5}+\frac{4 i}{5}} \sqrt{\frac{\left(2 i+\sqrt{-1-2 i}-\sqrt{-1+2 i}\right) \left(-2 x+\sqrt{-1-2 i}-i\right)}{\left(-2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1-2 i}+i\right)}} \left(2 i x^2+2 x+1\right) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\left(2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1+2 i}-i\right)}{\sqrt{-1+2 i} \left(2 x+\sqrt{-1-2 i}+i\right)}}}{\sqrt{2}}\right)|\frac{1}{2} \left(5-\sqrt{5}\right)\right)}{\sqrt{\frac{(1+2 i) \left((-1+i)+\sqrt{-1-2 i}\right) \left(2 i x^2+2 x+1\right)}{\left(2 x+\sqrt{-1-2 i}+i\right)^2}}}-\frac{9 i \sqrt{-\frac{2}{5}+\frac{4 i}{5}} \left(-2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(x+\frac{1}{2} \left(i+\sqrt{-1-2 i}\right)\right)^2 \sqrt{\frac{\left(2 i+\sqrt{-1-2 i}-\sqrt{-1+2 i}\right) \left(-2 x+\sqrt{-1-2 i}-i\right)}{\left(-2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1-2 i}+i\right)}} \sqrt{\frac{(1+2 i) \left((-1+i)+\sqrt{-1-2 i}\right) \left(2 i x^2+2 x+1\right)}{\left(2 x+\sqrt{-1-2 i}+i\right)^2}} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\left(2 i+\sqrt{-1-2 i}+\sqrt{-1+2 i}\right) \left(2 x+\sqrt{-1+2 i}-i\right)}{\sqrt{-1+2 i} \left(2 x+\sqrt{-1-2 i}+i\right)}}}{\sqrt{2}}\right)|\frac{1}{2} \left(5-\sqrt{5}\right)\right)}{(-1+i)+\sqrt{-1-2 i}}+42 x+\frac{9}{2} \left(-2 x+\sqrt{-1-2 i}-i\right) \left(2 x-\sqrt{-1+2 i}-i\right) \left(2 x+\sqrt{-1+2 i}-i\right)+19}{10 \sqrt{4 x^4+4 x^2+4 x+1}}","-\frac{\left(3-\left(\frac{1}{x}+1\right)^2\right) x^2}{\sqrt{4 x^4+4 x^2+4 x+1}}+\frac{\left(13-9 \left(\frac{1}{x}+1\right)^2\right) \left(\frac{1}{x}+1\right) x^2}{10 \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{9 \left(\left(\frac{1}{x}+1\right)^4-2 \left(\frac{1}{x}+1\right)^2+5\right) \left(\frac{1}{x}+1\right) x^2}{10 \left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right) \sqrt{4 x^4+4 x^2+4 x+1}}+\frac{3 \left(3-\sqrt{5}\right) \left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right) \sqrt{\frac{\left(\frac{1}{x}+1\right)^4-2 \left(\frac{1}{x}+1\right)^2+5}{\left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right)^2}} x^2 F\left(2 \tan ^{-1}\left(\frac{1+\frac{1}{x}}{\sqrt[4]{5}}\right)|\frac{1}{10} \left(5+\sqrt{5}\right)\right)}{4\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}}-\frac{9 \left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right) \sqrt{\frac{\left(\frac{1}{x}+1\right)^4-2 \left(\frac{1}{x}+1\right)^2+5}{\left(\left(\frac{1}{x}+1\right)^2+\sqrt{5}\right)^2}} x^2 E\left(2 \tan ^{-1}\left(\frac{1+\frac{1}{x}}{\sqrt[4]{5}}\right)|\frac{1}{10} \left(5+\sqrt{5}\right)\right)}{2\ 5^{3/4} \sqrt{4 x^4+4 x^2+4 x+1}}",1,"(19 + 42*x - 16*x^2 + 36*x^3 + (9*(-I + Sqrt[-1 - 2*I] - 2*x)*(-I - Sqrt[-1 + 2*I] + 2*x)*(-I + Sqrt[-1 + 2*I] + 2*x))/2 - ((9*I)*Sqrt[-2/5 + (4*I)/5]*(-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*((I + Sqrt[-1 - 2*I])/2 + x)^2*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*(-I + Sqrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(I + Sqrt[-1 - 2*I] + 2*x))]*Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2*I])*(1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2]*EllipticE[ArcSin[Sqrt[((2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*I] + 2*x))]/Sqrt[2]], (5 - Sqrt[5])/2])/((-1 + I) + Sqrt[-1 - 2*I]) + ((6 - 3*I)*Sqrt[-2/5 + (4*I)/5]*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*(-I + Sqrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(I + Sqrt[-1 - 2*I] + 2*x))]*(1 + 2*x + (2*I)*x^2)*EllipticF[ArcSin[Sqrt[((2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*I] + 2*x))]/Sqrt[2]], (5 - Sqrt[5])/2])/Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2*I])*(1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2])/(10*Sqrt[1 + 4*x + 4*x^2 + 4*x^4])","C",0
800,1,1148,126,0.9770915,"\int \frac{1}{\sqrt{8+24 x+8 x^2-15 x^3+8 x^4}} \, dx","Integrate[1/Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4],x]","-\frac{2 F\left(\sin ^{-1}\left(\sqrt{\frac{\left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}}\right)|\frac{\left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,3\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}{\left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,3\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right)^2 \sqrt{\frac{\left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,3\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,3\right]\right)}} \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right) \sqrt{\frac{\left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right) \left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}{\left(x-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right)^2 \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)^2}}}{\sqrt{8 x^4-15 x^3+8 x^2+24 x+8} \left(-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,1\right]+\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]\right) \left(\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,2\right]-\text{Root}\left[8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,4\right]\right)}","-\frac{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{\frac{\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517}{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right)^2}} x^2 F\left(2 \tan ^{-1}\left(\frac{3 x+4}{\sqrt[4]{517} x}\right)|\frac{517+19 \sqrt{517}}{1034}\right)}{8 \sqrt[4]{517} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}",1,"(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])^2*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])^2)])/(Sqrt[8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4]*(-Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] + Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))","C",1
801,1,6019,434,6.0767994,"\int \frac{1}{\left(8+24 x+8 x^2-15 x^3+8 x^4\right)^{3/2}} \, dx","Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-3/2),x]","\text{Result too large to show}","-\frac{\left(172-7 \left(\frac{4}{x}+3\right)^2\right) x^2}{208 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left(50896-2455 \left(\frac{4}{x}+3\right)^2\right) \left(\frac{4}{x}+3\right) x^2}{322608 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{2455 \left(\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517\right) \left(\frac{4}{x}+3\right) x^2}{322608 \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left(4910-203 \sqrt{517}\right) \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{\frac{\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517}{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right)^2}} x^2 F\left(2 \tan ^{-1}\left(\frac{3 x+4}{\sqrt[4]{517} x}\right)|\frac{517+19 \sqrt{517}}{1034}\right)}{2496\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{2455 \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{\frac{\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517}{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right)^2}} x^2 E\left(2 \tan ^{-1}\left(\frac{3 x+4}{\sqrt[4]{517} x}\right)|\frac{517+19 \sqrt{517}}{1034}\right)}{624\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}",1,"Result too large to show","C",0
802,1,6084,577,6.0575526,"\int \frac{1}{\left(8+24 x+8 x^2-15 x^3+8 x^4\right)^{5/2}} \, dx","Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-5/2),x]","\text{Result too large to show}","-\frac{\left(124415-6308 \left(\frac{4}{x}+3\right)^2\right) x^2}{97344 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left(18932921731-1086525994 \left(\frac{4}{x}+3\right)^2\right) \left(\frac{4}{x}+3\right) x^2}{78056941248 \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{543262997 \left(\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517\right) \left(\frac{4}{x}+3\right) x^2}{39028470624 \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left(11921698-359497 \left(\frac{4}{x}+3\right)^2\right) \left(\frac{4}{x}+3\right) x^2}{483912 \left(\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517\right) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{\left(64489-1399 \left(\frac{4}{x}+3\right)^2\right) x^2}{624 \left(\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517\right) \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}+\frac{\left(4346103976-175318963 \sqrt{517}\right) \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{\frac{\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517}{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right)^2}} x^2 F\left(2 \tan ^{-1}\left(\frac{3 x+4}{\sqrt[4]{517} x}\right)|\frac{517+19 \sqrt{517}}{1034}\right)}{1207844352\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}-\frac{543262997 \left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right) \sqrt{\frac{\left(\frac{4}{x}+3\right)^4-38 \left(\frac{4}{x}+3\right)^2+517}{\left(\left(\frac{4}{x}+3\right)^2+\sqrt{517}\right)^2}} x^2 E\left(2 \tan ^{-1}\left(\frac{3 x+4}{\sqrt[4]{517} x}\right)|\frac{517+19 \sqrt{517}}{1034}\right)}{75490272\ 517^{3/4} \sqrt{8 x^4-15 x^3+8 x^2+24 x+8}}",1,"Result too large to show","C",0
803,1,826,130,0.1210413,"\int \frac{1}{\sqrt{9-6 x-44 x^2+15 x^3+3 x^4}} \, dx","Integrate[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]","-\frac{2 F\left(\sin ^{-1}\left(\sqrt{\frac{\left(x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]\right)}{\left(x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]\right)}}\right)|\frac{\left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right]\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]\right)}{\left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right]\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]\right)}\right) \sqrt{\frac{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]}{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]}} \left(x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]\right)^2 \sqrt{\frac{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right]}{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]}} \sqrt{\frac{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]}{x-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]}}}{\sqrt{\left(3 x^4+15 x^3-44 x^2-6 x+9\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,1\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,3\right]\right) \left(\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,2\right]-\text{Root}\left[3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\&,4\right]\right)}}","-\frac{\sqrt{\frac{\left(\frac{6}{x}-1\right)^4-182 \left(1-\frac{6}{x}\right)^2+613}{\left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right)^2}} \left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right) x^2 F\left(2 \tan ^{-1}\left(\frac{6-x}{\sqrt[4]{613} x}\right)|\frac{613+91 \sqrt{613}}{1226}\right)}{12 \sqrt[4]{613} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}",1,"(-2*EllipticF[ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], ((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])])/Sqrt[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])]","C",1
804,1,5428,444,6.0506747,"\int \frac{1}{\left(9-6 x-44 x^2+15 x^3+3 x^4\right)^{3/2}} \, dx","Integrate[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]","\text{Result too large to show}","-\frac{\left(176-23 \left(1-\frac{6}{x}\right)^2\right) x^2}{51759 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{\left(45401-3722 \left(1-\frac{6}{x}\right)^2\right) \left(1-\frac{6}{x}\right) x^2}{31728267 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \left(\left(\frac{6}{x}-1\right)^4-182 \left(1-\frac{6}{x}\right)^2+613\right) \left(1-\frac{6}{x}\right) x^2}{31728267 \left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right) \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}-\frac{\left(7444-145 \sqrt{613}\right) \sqrt{\frac{\left(\frac{6}{x}-1\right)^4-182 \left(1-\frac{6}{x}\right)^2+613}{\left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right)^2}} \left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right) x^2 F\left(2 \tan ^{-1}\left(\frac{6-x}{\sqrt[4]{613} x}\right)|\frac{613+91 \sqrt{613}}{1226}\right)}{207036\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \sqrt{\frac{\left(\frac{6}{x}-1\right)^4-182 \left(1-\frac{6}{x}\right)^2+613}{\left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right)^2}} \left(\frac{(6-x)^2}{x^2}+\sqrt{613}\right) x^2 E\left(2 \tan ^{-1}\left(\frac{6-x}{\sqrt[4]{613} x}\right)|\frac{613+91 \sqrt{613}}{1226}\right)}{51759\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}",1,"Result too large to show","C",0
805,1,57,56,0.1076876,"\int \frac{\left(2 \sqrt{3-x}+\frac{3}{\sqrt{1+x}}\right)^2}{x} \, dx","Integrate[(2*Sqrt[3 - x] + 3/Sqrt[1 + x])^2/x,x]","-4 x+21 \log (x)-9 \log (x+1)+24 \sin ^{-1}\left(\frac{\sqrt{3-x}}{2}\right)-24 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{1-\frac{x}{3}}}{\sqrt{x+1}}\right)","-4 x+21 \log (x)-9 \log (x+1)+12 \sin ^{-1}\left(\frac{1-x}{2}\right)-24 \sqrt{3} \tanh ^{-1}\left(\frac{\sqrt{3} \sqrt{x+1}}{\sqrt{3-x}}\right)",1,"-4*x + 24*ArcSin[Sqrt[3 - x]/2] - 24*Sqrt[3]*ArcTanh[Sqrt[1 - x/3]/Sqrt[1 + x]] + 21*Log[x] - 9*Log[1 + x]","A",1
806,1,65,65,0.0755362,"\int \frac{-1+x+x^2}{1+\sqrt{1+x^2}} \, dx","Integrate[(-1 + x + x^2)/(1 + Sqrt[1 + x^2]),x]","\frac{1}{2} \sqrt{x^2+1} x+\sqrt{x^2+1}+\frac{\sqrt{x^2+1}}{x}-\log \left(\sqrt{x^2+1}+1\right)-x-\frac{1}{x}-\frac{1}{2} \sinh ^{-1}(x)","\frac{1}{2} \sqrt{x^2+1} x+\sqrt{x^2+1}+\frac{\sqrt{x^2+1}}{x}-\log \left(\sqrt{x^2+1}+1\right)-x-\frac{1}{x}-\frac{1}{2} \sinh ^{-1}(x)",1,"-x^(-1) - x + Sqrt[1 + x^2] + Sqrt[1 + x^2]/x + (x*Sqrt[1 + x^2])/2 - ArcSinh[x]/2 - Log[1 + Sqrt[1 + x^2]]","A",1
807,1,88,53,0.150334,"\int \frac{-1+x+x^2}{1+x+\sqrt{1+x^2}} \, dx","Integrate[(-1 + x + x^2)/(1 + x + Sqrt[1 + x^2]),x]","\frac{1}{12} \left(2 x^3+6 x^2-2 \left(x^2+1\right)^{3/2}+6 \left(\frac{1}{\sqrt{x^2+1}+x}+\log \left(\sqrt{x^2+1}+x\right)-2 \log \left(\sqrt{x^2+1}+x+1\right)\right)-3 \left(\sqrt{x^2+1} x+\sinh ^{-1}(x)\right)+6 x\right)","\frac{1}{12} \left(2 x^3+6 x^2+\left(-2 x^2-3 x+4\right) \sqrt{x^2+1}-6 \log \left(\sqrt{x^2+1}+1\right)-3 \sinh ^{-1}(x)\right)",1,"(6*x + 6*x^2 + 2*x^3 - 2*(1 + x^2)^(3/2) - 3*(x*Sqrt[1 + x^2] + ArcSinh[x]) + 6*((x + Sqrt[1 + x^2])^(-1) + Log[x + Sqrt[1 + x^2]] - 2*Log[1 + x + Sqrt[1 + x^2]]))/12","A",1
808,1,14,14,0.0063676,"\int \frac{2 \sqrt{-1+x}+x}{\sqrt{-1+x} x} \, dx","Integrate[(2*Sqrt[-1 + x] + x)/(Sqrt[-1 + x]*x),x]","2 \sqrt{x-1}+2 \log (x)","2 \sqrt{x-1}+2 \log (x)",1,"2*Sqrt[-1 + x] + 2*Log[x]","A",1
809,1,61,61,0.0483116,"\int \left(a+c \sqrt{x}+b x^{2/3}\right)^2 \, dx","Integrate[(a + c*Sqrt[x] + b*x^(2/3))^2,x]","a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2}","a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2}",1,"a^2*x + (4*a*c*x^(3/2))/3 + (6*a*b*x^(5/3))/5 + (c^2*x^2)/2 + (12*b*c*x^(13/6))/13 + (3*b^2*x^(7/3))/7","A",1
810,1,114,114,0.0872626,"\int \left(a+c \sqrt{x}+b x^{2/3}\right)^3 \, dx","Integrate[(a + c*Sqrt[x] + b*x^(2/3))^3,x]","a^3 x+\frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{b^3 x^3}{3}+\frac{18}{17} b^2 c x^{17/6}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2}","a^3 x+\frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{b^3 x^3}{3}+\frac{18}{17} b^2 c x^{17/6}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2}",1,"a^3*x + 2*a^2*c*x^(3/2) + (9*a^2*b*x^(5/3))/5 + (3*a*c^2*x^2)/2 + (36*a*b*c*x^(13/6))/13 + (9*a*b^2*x^(7/3))/7 + (2*c^3*x^(5/2))/5 + (9*b*c^2*x^(8/3))/8 + (18*b^2*c*x^(17/6))/17 + (b^3*x^3)/3","A",1
811,1,100,58,0.0626291,"\int \frac{-1+x^2}{\sqrt{a-b+\frac{b}{x^2}} x^3} \, dx","Integrate[(-1 + x^2)/(Sqrt[a - b + b/x^2]*x^3),x]","\frac{\sqrt{a-b} \left(a x^2-b x^2+b\right)+b x \sqrt{a x^2-b x^2+b} \tanh ^{-1}\left(\frac{x \sqrt{a-b}}{\sqrt{x^2 (a-b)+b}}\right)}{b x^2 \sqrt{a-b} \sqrt{a+b \left(\frac{1}{x^2}-1\right)}}","\frac{\sqrt{a-b \left(1-\frac{1}{x^2}\right)}}{b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b \left(1-\frac{1}{x^2}\right)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(Sqrt[a - b]*(b + a*x^2 - b*x^2) + b*x*Sqrt[b + a*x^2 - b*x^2]*ArcTanh[(Sqrt[a - b]*x)/Sqrt[b + (a - b)*x^2]])/(Sqrt[a - b]*b*Sqrt[a + b*(-1 + x^(-2))]*x^2)","A",1
812,1,100,58,0.0072803,"\int \frac{-1+x^2}{\sqrt{a+b \left(-1+\frac{1}{x^2}\right)} x^3} \, dx","Integrate[(-1 + x^2)/(Sqrt[a + b*(-1 + x^(-2))]*x^3),x]","\frac{\sqrt{a-b} \left(a x^2-b x^2+b\right)+b x \sqrt{a x^2-b x^2+b} \tanh ^{-1}\left(\frac{x \sqrt{a-b}}{\sqrt{x^2 (a-b)+b}}\right)}{b x^2 \sqrt{a-b} \sqrt{a+b \left(\frac{1}{x^2}-1\right)}}","\frac{\sqrt{a-b \left(1-\frac{1}{x^2}\right)}}{b}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b \left(1-\frac{1}{x^2}\right)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"(Sqrt[a - b]*(b + a*x^2 - b*x^2) + b*x*Sqrt[b + a*x^2 - b*x^2]*ArcTanh[(Sqrt[a - b]*x)/Sqrt[b + (a - b)*x^2]])/(Sqrt[a - b]*b*Sqrt[a + b*(-1 + x^(-2))]*x^2)","A",1
813,1,64,53,0.0438887,"\int \frac{1+x}{\left(4+x^2\right) \sqrt{9+x^2}} \, dx","Integrate[(1 + x)/((4 + x^2)*Sqrt[9 + x^2]),x]","-\frac{(2+i) \tanh ^{-1}\left(\frac{9-2 i x}{\sqrt{5} \sqrt{x^2+9}}\right)+(2-i) \tanh ^{-1}\left(\frac{9+2 i x}{\sqrt{5} \sqrt{x^2+9}}\right)}{4 \sqrt{5}}","\frac{\tan ^{-1}\left(\frac{\sqrt{5} x}{2 \sqrt{x^2+9}}\right)}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{x^2+9}}{\sqrt{5}}\right)}{\sqrt{5}}",1,"-1/4*((2 + I)*ArcTanh[(9 - (2*I)*x)/(Sqrt[5]*Sqrt[9 + x^2])] + (2 - I)*ArcTanh[(9 + (2*I)*x)/(Sqrt[5]*Sqrt[9 + x^2])])/Sqrt[5]","C",1
814,1,23,23,0.0113964,"\int x \left(1+\sqrt{1-x^2}\right) \, dx","Integrate[x*(1 + Sqrt[1 - x^2]),x]","\frac{x^2}{2}-\frac{1}{3} \left(1-x^2\right)^{3/2}","\frac{x^2}{2}-\frac{1}{3} \left(1-x^2\right)^{3/2}",1,"x^2/2 - (1 - x^2)^(3/2)/3","A",1
815,1,23,23,0.0040998,"\int x \left(1+\sqrt{1-x} \sqrt{1+x}\right) \, dx","Integrate[x*(1 + Sqrt[1 - x]*Sqrt[1 + x]),x]","\frac{x^2}{2}-\frac{1}{3} \left(1-x^2\right)^{3/2}","\frac{x^2}{2}-\frac{1}{3} \left(1-x^2\right)^{3/2}",1,"x^2/2 - (1 - x^2)^(3/2)/3","A",1
816,1,33,33,0.0124725,"\int x \left(1+\frac{1}{\sqrt{2+x} \sqrt{3+x}}\right) \, dx","Integrate[x*(1 + 1/(Sqrt[2 + x]*Sqrt[3 + x])),x]","\frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left(\sqrt{x+2}\right)","\frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left(\sqrt{x+2}\right)",1,"x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]","A",1
817,1,27,45,0.0707227,"\int \frac{x-\sqrt{x^6}}{x \left(1-x^4\right)} \, dx","Integrate[(x - Sqrt[x^6])/(x*(1 - x^4)),x]","\frac{1}{2} \left(\frac{\sqrt{x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
818,1,27,45,0.0146735,"\int \frac{1-\frac{\sqrt{x^6}}{x}}{1-x^4} \, dx","Integrate[(1 - Sqrt[x^6]/x)/(1 - x^4),x]","\frac{1}{2} \left(\frac{\sqrt{x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
819,1,27,45,0.0129478,"\int \frac{x-\sqrt{x^6}}{x-x^5} \, dx","Integrate[(x - Sqrt[x^6])/(x - x^5),x]","\frac{1}{2} \left(\frac{\sqrt{x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
820,1,27,45,0.0345807,"\int \frac{x}{x+\sqrt{x^6}} \, dx","Integrate[x/(x + Sqrt[x^6]),x]","\frac{1}{2} \left(\frac{\sqrt{x^6} \left(\tan ^{-1}(x)-\tanh ^{-1}(x)\right)}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right)","\frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"(ArcTan[x] + (Sqrt[x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2","A",1
821,1,49,52,0.0618115,"\int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx","Integrate[(Sqrt[x] - Sqrt[x^3])/(x - x^3),x]","\frac{\left(x^{3/2}+\sqrt{x^3}\right) \tan ^{-1}\left(\sqrt{x}\right)+\left(x^{3/2}-\sqrt{x^3}\right) \tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}","\frac{\sqrt{x^3} \tan ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}+\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)",1,"((x^(3/2) + Sqrt[x^3])*ArcTan[Sqrt[x]] + (x^(3/2) - Sqrt[x^3])*ArcTanh[Sqrt[x]])/x^(3/2)","A",1
822,1,49,52,0.0289517,"\int \frac{1}{\sqrt{x}+\sqrt{x^3}} \, dx","Integrate[(Sqrt[x] + Sqrt[x^3])^(-1),x]","\frac{\left(x^{3/2}+\sqrt{x^3}\right) \tan ^{-1}\left(\sqrt{x}\right)+\left(x^{3/2}-\sqrt{x^3}\right) \tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}","\frac{\sqrt{x^3} \tan ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left(\sqrt{x}\right)}{x^{3/2}}+\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)",1,"((x^(3/2) + Sqrt[x^3])*ArcTan[Sqrt[x]] + (x^(3/2) - Sqrt[x^3])*ArcTanh[Sqrt[x]])/x^(3/2)","A",1
823,1,64,68,0.1648258,"\int \frac{1}{\sqrt{-1+x}+\sqrt{(-1+x)^3}} \, dx","Integrate[(Sqrt[-1 + x] + Sqrt[(-1 + x)^3])^(-1),x]","\left(\frac{\sqrt{(x-1)^3}}{(x-1)^{3/2}}+1\right) \tan ^{-1}\left(\sqrt{x-1}\right)+\frac{\left((x-1)^{3/2}-\sqrt{(x-1)^3}\right) \tanh ^{-1}\left(\sqrt{x-1}\right)}{(x-1)^{3/2}}","\frac{\sqrt{(x-1)^3} \tan ^{-1}\left(\sqrt{x-1}\right)}{(x-1)^{3/2}}+\tan ^{-1}\left(\sqrt{x-1}\right)-\frac{\sqrt{(x-1)^3} \tanh ^{-1}\left(\sqrt{x-1}\right)}{(x-1)^{3/2}}+\tanh ^{-1}\left(\sqrt{x-1}\right)",1,"(1 + Sqrt[(-1 + x)^3]/(-1 + x)^(3/2))*ArcTan[Sqrt[-1 + x]] + (((-1 + x)^(3/2) - Sqrt[(-1 + x)^3])*ArcTanh[Sqrt[-1 + x]])/(-1 + x)^(3/2)","A",1
824,1,23,31,0.1891439,"\int \left(-\frac{3}{(4+5 x)^2}-\frac{5+4 x}{(4+5 x)^2 \sqrt{1-x^2}}\right) \, dx","Integrate[-3/(4 + 5*x)^2 - (5 + 4*x)/((4 + 5*x)^2*Sqrt[1 - x^2]),x]","\frac{5 \sqrt{1-x^2}+3}{25 x+20}","\frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)}",1,"(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)","A",1
825,1,23,31,0.1458675,"\int \frac{-5-4 x-3 \sqrt{1-x^2}}{(4+5 x)^2 \sqrt{1-x^2}} \, dx","Integrate[(-5 - 4*x - 3*Sqrt[1 - x^2])/((4 + 5*x)^2*Sqrt[1 - x^2]),x]","\frac{5 \sqrt{1-x^2}+3}{25 x+20}","\frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)}",1,"(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)","A",1
826,1,23,31,0.0915838,"\int \frac{1}{(-5-4 x) \sqrt{1-x^2}+3 \left(1-x^2\right)} \, dx","Integrate[((-5 - 4*x)*Sqrt[1 - x^2] + 3*(1 - x^2))^(-1),x]","\frac{5 \sqrt{1-x^2}+3}{25 x+20}","\frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)}",1,"(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)","A",1
827,1,23,31,0.062431,"\int \frac{1}{3-3 x^2-5 \sqrt{1-x^2}-4 x \sqrt{1-x^2}} \, dx","Integrate[(3 - 3*x^2 - 5*Sqrt[1 - x^2] - 4*x*Sqrt[1 - x^2])^(-1),x]","\frac{5 \sqrt{1-x^2}+3}{25 x+20}","\frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)}",1,"(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)","A",1
828,1,23,31,0.1841092,"\int \frac{-1+\sqrt{1-x^2}}{\sqrt{1-x^2} \left(2+x-2 \sqrt{1-x^2}\right)^2} \, dx","Integrate[(-1 + Sqrt[1 - x^2])/(Sqrt[1 - x^2]*(2 + x - 2*Sqrt[1 - x^2])^2),x]","\frac{5 \sqrt{1-x^2}+3}{25 x+20}","\frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)}",1,"(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)","A",1
829,1,38,43,0.0411991,"\int \frac{a+b x^{-1+n}}{c x+d x^n} \, dx","Integrate[(a + b*x^(-1 + n))/(c*x + d*x^n),x]","\frac{\frac{(b c-a d) \log \left(c x^{1-n}+d\right)}{c (n-1)}+b \log (x)}{d}","\frac{b \log (x)}{d}-\frac{(b c-a d) \log \left(c x^{1-n}+d\right)}{c d (1-n)}",1,"(b*Log[x] + ((b*c - a*d)*Log[d + c*x^(1 - n)])/(c*(-1 + n)))/d","A",1
830,1,42,42,0.0388338,"\int \frac{\sqrt{1+2 x^2}}{1+\sqrt{1+2 x^2}} \, dx","Integrate[Sqrt[1 + 2*x^2]/(1 + Sqrt[1 + 2*x^2]),x]","\frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}","\frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left(\sqrt{2} x\right)}{\sqrt{2}}",1,"-1/2*1/x + x + Sqrt[1 + 2*x^2]/(2*x) - ArcSinh[Sqrt[2]*x]/Sqrt[2]","A",1
831,1,54,65,0.0506712,"\int \frac{\sqrt{-1+4 x^2}}{x+\sqrt{-1+4 x^2}} \, dx","Integrate[Sqrt[-1 + 4*x^2]/(x + Sqrt[-1 + 4*x^2]),x]","\frac{1}{9} \left(-3 \sqrt{4 x^2-1}+\sqrt{3} \tanh ^{-1}\left(\sqrt{12 x^2-3}\right)+12 x-\sqrt{3} \tanh ^{-1}\left(\sqrt{3} x\right)\right)","-\frac{1}{3} \sqrt{4 x^2-1}+\frac{\tanh ^{-1}\left(\sqrt{3} \sqrt{4 x^2-1}\right)}{3 \sqrt{3}}+\frac{4 x}{3}-\frac{\tanh ^{-1}\left(\sqrt{3} x\right)}{3 \sqrt{3}}",1,"(12*x - 3*Sqrt[-1 + 4*x^2] - Sqrt[3]*ArcTanh[Sqrt[3]*x] + Sqrt[3]*ArcTanh[Sqrt[-3 + 12*x^2]])/9","A",1
832,1,240,195,0.3945531,"\int \frac{a+b x+c x^2}{(d+e x)^3 \sqrt{-1+x^2}} \, dx","Integrate[(a + b*x + c*x^2)/((d + e*x)^3*Sqrt[-1 + x^2]),x]","\frac{1}{2} \left(-\frac{\log \left(-\sqrt{x^2-1} \sqrt{d^2-e^2}+d x+e\right) \left(a \left(2 d^2+e^2\right)-3 b d e+c \left(d^2+2 e^2\right)\right)}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}+\frac{\log (d+e x) \left(a \left(2 d^2+e^2\right)-3 b d e+c \left(d^2+2 e^2\right)\right)}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}+\frac{\sqrt{x^2-1} \left(a e \left(-4 d^2-3 d e x+e^2\right)+b \left(2 d^3+d^2 e x+d e^2+2 e^3 x\right)+c d \left(d^2 x-3 d e-4 e^2 x\right)\right)}{\left(d^2-e^2\right)^2 (d+e x)^2}\right)","-\frac{\sqrt{x^2-1} \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{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right) \left(-a \left(2 d^2+e^2\right)+3 b d e-c \left(d^2+2 e^2\right)\right)}{2 \left(d^2-e^2\right)^{5/2}}+\frac{\sqrt{x^2-1} \left(c \left(d^3-4 d e^2\right)-e \left(3 a d e-b \left(d^2+2 e^2\right)\right)\right)}{2 e \left(d^2-e^2\right)^2 (d+e x)}",1,"((Sqrt[-1 + x^2]*(a*e*(-4*d^2 + e^2 - 3*d*e*x) + c*d*(-3*d*e + d^2*x - 4*e^2*x) + b*(2*d^3 + d*e^2 + d^2*e*x + 2*e^3*x)))/((d^2 - e^2)^2*(d + e*x)^2) + ((-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*Log[d + e*x])/((d - e)^2*(d + e)^2*Sqrt[d^2 - e^2]) - ((-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*Log[e + d*x - Sqrt[d^2 - e^2]*Sqrt[-1 + x^2]])/((d - e)^2*(d + e)^2*Sqrt[d^2 - e^2]))/2","A",1
833,1,28,28,0.0172243,"\int \frac{1+2 x^8}{x \left(1+x^8\right)^{3/2}} \, dx","Integrate[(1 + 2*x^8)/(x*(1 + x^8)^(3/2)),x]","-\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{x^8+1}\right)","-\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{x^8+1}\right)",1,"-1/4*1/Sqrt[1 + x^8] - ArcTanh[Sqrt[1 + x^8]]/4","A",1
834,1,28,28,0.0056363,"\int \frac{\sqrt{1+x^8} \left(1+2 x^8\right)}{x+2 x^9+x^{17}} \, dx","Integrate[(Sqrt[1 + x^8]*(1 + 2*x^8))/(x + 2*x^9 + x^17),x]","-\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{x^8+1}\right)","-\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left(\sqrt{x^8+1}\right)",1,"-1/4*1/Sqrt[1 + x^8] - ArcTanh[Sqrt[1 + x^8]]/4","A",1
835,1,22,22,0.0097591,"\int \left(1-9 x^2+\frac{x}{\sqrt{1-9 x^2}}\right) \, dx","Integrate[1 - 9*x^2 + x/Sqrt[1 - 9*x^2],x]","-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x","-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x",1,"x - 3*x^3 - Sqrt[1 - 9*x^2]/9","A",1
836,1,22,22,0.0025176,"\int \frac{x+\left(1-9 x^2\right)^{3/2}}{\sqrt{1-9 x^2}} \, dx","Integrate[(x + (1 - 9*x^2)^(3/2))/Sqrt[1 - 9*x^2],x]","-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x","-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x",1,"x - 3*x^3 - Sqrt[1 - 9*x^2]/9","A",1
837,1,17,17,0.0227781,"\int \frac{\left(-3+2 \sqrt{x}\right) \left(-3 \sqrt{x}+x\right)^{2/3}}{\sqrt{x}} \, dx","Integrate[((-3 + 2*Sqrt[x])*(-3*Sqrt[x] + x)^(2/3))/Sqrt[x],x]","\frac{6}{5} \left(x-3 \sqrt{x}\right)^{5/3}","\frac{6}{5} \left(x-3 \sqrt{x}\right)^{5/3}",1,"(6*(-3*Sqrt[x] + x)^(5/3))/5","A",1
838,1,17,17,0.035391,"\int \frac{9-9 \sqrt{x}+2 x}{\sqrt[3]{-3 \sqrt{x}+x}} \, dx","Integrate[(9 - 9*Sqrt[x] + 2*x)/(-3*Sqrt[x] + x)^(1/3),x]","\frac{6}{5} \left(x-3 \sqrt{x}\right)^{5/3}","\frac{6}{5} \left(x-3 \sqrt{x}\right)^{5/3}",1,"(6*(-3*Sqrt[x] + x)^(5/3))/5","A",1
839,1,10,10,0.0048993,"\int \frac{1}{\sqrt{4-9 x^2}} \, dx","Integrate[1/Sqrt[4 - 9*x^2],x]","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)",1,"ArcSin[(3*x)/2]/3","A",1
840,1,10,10,0.0063167,"\int \frac{1}{\sqrt{2-3 x} \sqrt{2+3 x}} \, dx","Integrate[1/(Sqrt[2 - 3*x]*Sqrt[2 + 3*x]),x]","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)",1,"ArcSin[(3*x)/2]/3","A",1
841,1,10,10,0.0054309,"\int \frac{1}{\sqrt{(2-3 x) (2+3 x)}} \, dx","Integrate[1/Sqrt[(2 - 3*x)*(2 + 3*x)],x]","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)","\frac{1}{3} \sin ^{-1}\left(\frac{3 x}{2}\right)",1,"ArcSin[(3*x)/2]/3","A",1
842,1,12,12,0.0079356,"\int \frac{1}{\sqrt{15-2 x-x^2}} \, dx","Integrate[1/Sqrt[15 - 2*x - x^2],x]","-\sin ^{-1}\left(\frac{1}{4} (-x-1)\right)","-\sin ^{-1}\left(\frac{1}{4} (-x-1)\right)",1,"-ArcSin[(-1 - x)/4]","A",1
843,1,21,12,0.0183724,"\int \frac{1}{\sqrt{3-x} \sqrt{5+x}} \, dx","Integrate[1/(Sqrt[3 - x]*Sqrt[5 + x]),x]","-2 \sin ^{-1}\left(\frac{\sqrt{3-x}}{2 \sqrt{2}}\right)","-\sin ^{-1}\left(\frac{1}{4} (-x-1)\right)",1,"-2*ArcSin[Sqrt[3 - x]/(2*Sqrt[2])]","A",1
844,1,21,12,0.0027132,"\int \frac{1}{\sqrt{(3-x) (5+x)}} \, dx","Integrate[1/Sqrt[(3 - x)*(5 + x)],x]","-2 \sin ^{-1}\left(\frac{\sqrt{3-x}}{2 \sqrt{2}}\right)","-\sin ^{-1}\left(\frac{1}{4} (-x-1)\right)",1,"-2*ArcSin[Sqrt[3 - x]/(2*Sqrt[2])]","A",1
845,1,4,4,0.007499,"\int \frac{1}{\sqrt{-15-8 x-x^2}} \, dx","Integrate[1/Sqrt[-15 - 8*x - x^2],x]","\sin ^{-1}(x+4)","\sin ^{-1}(x+4)",1,"ArcSin[4 + x]","A",1
846,1,18,4,0.0143297,"\int \frac{1}{\sqrt{-3-x} \sqrt{5+x}} \, dx","Integrate[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]","-2 \sin ^{-1}\left(\frac{\sqrt{-x-3}}{\sqrt{2}}\right)","\sin ^{-1}(x+4)",1,"-2*ArcSin[Sqrt[-3 - x]/Sqrt[2]]","B",1
847,1,18,4,0.0031383,"\int \frac{1}{\sqrt{(-3-x) (5+x)}} \, dx","Integrate[1/Sqrt[(-3 - x)*(5 + x)],x]","-2 \sin ^{-1}\left(\frac{\sqrt{-x-3}}{\sqrt{2}}\right)","\sin ^{-1}(x+4)",1,"-2*ArcSin[Sqrt[-3 - x]/Sqrt[2]]","B",1
848,1,11,11,0.0021799,"\int \left(1-\sqrt{x}\right) \, dx","Integrate[1 - Sqrt[x],x]","x-\frac{2 x^{3/2}}{3}","x-\frac{2 x^{3/2}}{3}",1,"x - (2*x^(3/2))/3","A",1
849,1,11,11,0.0005143,"\int \frac{1-x}{1+\sqrt{x}} \, dx","Integrate[(1 - x)/(1 + Sqrt[x]),x]","x-\frac{2 x^{3/2}}{3}","x-\frac{2 x^{3/2}}{3}",1,"x - (2*x^(3/2))/3","A",1
850,1,27,27,0.0083318,"\int \sqrt{\frac{1}{1-x^2}} \, dx","Integrate[Sqrt[(1 - x^2)^(-1)],x]","\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x)","\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x)",1,"Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]","A",1
851,1,27,27,0.0055685,"\int \sqrt{\frac{1+x^2}{1-x^4}} \, dx","Integrate[Sqrt[(1 + x^2)/(1 - x^4)],x]","\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x)","\sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x)",1,"Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]","A",1
852,1,56,25,0.026293,"\int \sqrt{\frac{1}{-1+x^2}} \, dx","Integrate[Sqrt[(-1 + x^2)^(-1)],x]","\frac{1}{2} \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \left(\log \left(\frac{x}{\sqrt{x^2-1}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-1}}\right)\right)","\sqrt{1-x^2} \sqrt{\frac{1}{x^2-1}} \sin ^{-1}(x)",1,"(Sqrt[(-1 + x^2)^(-1)]*Sqrt[-1 + x^2]*(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sqrt[-1 + x^2]]))/2","B",1
853,1,56,25,0.0034697,"\int \sqrt{\frac{1+x^2}{-1+x^4}} \, dx","Integrate[Sqrt[(1 + x^2)/(-1 + x^4)],x]","\frac{1}{2} \sqrt{\frac{1}{x^2-1}} \sqrt{x^2-1} \left(\log \left(\frac{x}{\sqrt{x^2-1}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-1}}\right)\right)","\sqrt{1-x^2} \sqrt{\frac{1}{x^2-1}} \sin ^{-1}(x)",1,"(Sqrt[(-1 + x^2)^(-1)]*Sqrt[-1 + x^2]*(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sqrt[-1 + x^2]]))/2","B",1
854,1,11,11,0.0030766,"\int \frac{1}{\sqrt{1-x}} \, dx","Integrate[1/Sqrt[1 - x],x]","-2 \sqrt{1-x}","-2 \sqrt{1-x}",1,"-2*Sqrt[1 - x]","A",1
855,1,23,11,0.0230168,"\int \frac{\sqrt{1+x}}{\sqrt{1-x^2}} \, dx","Integrate[Sqrt[1 + x]/Sqrt[1 - x^2],x]","\frac{2 (x-1) \sqrt{x+1}}{\sqrt{1-x^2}}","-2 \sqrt{1-x}",1,"(2*(-1 + x)*Sqrt[1 + x])/Sqrt[1 - x^2]","B",1
856,1,9,9,0.0020361,"\int \frac{1}{\sqrt{1+x}} \, dx","Integrate[1/Sqrt[1 + x],x]","2 \sqrt{x+1}","2 \sqrt{x+1}",1,"2*Sqrt[1 + x]","A",1
857,1,25,9,0.0232756,"\int \frac{\sqrt{1-x}}{\sqrt{1-x^2}} \, dx","Integrate[Sqrt[1 - x]/Sqrt[1 - x^2],x]","\frac{2 \sqrt{1-x} (x+1)}{\sqrt{1-x^2}}","2 \sqrt{x+1}",1,"(2*Sqrt[1 - x]*(1 + x))/Sqrt[1 - x^2]","B",1
858,1,13,13,0.0031945,"\int \sqrt{1-x} \, dx","Integrate[Sqrt[1 - x],x]","-\frac{2}{3} (1-x)^{3/2}","-\frac{2}{3} (1-x)^{3/2}",1,"(-2*(1 - x)^(3/2))/3","A",1
859,1,25,13,0.0210271,"\int \frac{\sqrt{1-x^2}}{\sqrt{1+x}} \, dx","Integrate[Sqrt[1 - x^2]/Sqrt[1 + x],x]","\frac{2 (x-1) \sqrt{1-x^2}}{3 \sqrt{x+1}}","-\frac{2}{3} (1-x)^{3/2}",1,"(2*(-1 + x)*Sqrt[1 - x^2])/(3*Sqrt[1 + x])","A",1
860,1,11,11,0.0023782,"\int \sqrt{1+x} \, dx","Integrate[Sqrt[1 + x],x]","\frac{2}{3} (x+1)^{3/2}","\frac{2}{3} (x+1)^{3/2}",1,"(2*(1 + x)^(3/2))/3","A",1
861,1,27,11,0.0219654,"\int \frac{\sqrt{1-x^2}}{\sqrt{1-x}} \, dx","Integrate[Sqrt[1 - x^2]/Sqrt[1 - x],x]","\frac{2 (x+1) \sqrt{1-x^2}}{3 \sqrt{1-x}}","\frac{2}{3} (x+1)^{3/2}",1,"(2*(1 + x)*Sqrt[1 - x^2])/(3*Sqrt[1 - x])","B",1
862,1,49,35,0.0194797,"\int \frac{\sqrt{2+3 x}}{\sqrt{1+x}} \, dx","Integrate[Sqrt[2 + 3*x]/Sqrt[1 + x],x]","\frac{3 \sqrt{x+1} (3 x+2)-\sqrt{9 x+6} \sinh ^{-1}\left(\sqrt{3 x+2}\right)}{3 \sqrt{3 x+2}}","\sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left(\sqrt{3 x+2}\right)}{\sqrt{3}}",1,"(3*Sqrt[1 + x]*(2 + 3*x) - Sqrt[6 + 9*x]*ArcSinh[Sqrt[2 + 3*x]])/(3*Sqrt[2 + 3*x])","A",1
863,1,49,35,0.0081841,"\int \frac{\sqrt{1-x} \sqrt{2+3 x}}{\sqrt{1-x^2}} \, dx","Integrate[(Sqrt[1 - x]*Sqrt[2 + 3*x])/Sqrt[1 - x^2],x]","\frac{3 \sqrt{x+1} (3 x+2)-\sqrt{9 x+6} \sinh ^{-1}\left(\sqrt{3 x+2}\right)}{3 \sqrt{3 x+2}}","\sqrt{x+1} \sqrt{3 x+2}-\frac{\sinh ^{-1}\left(\sqrt{3 x+2}\right)}{\sqrt{3}}",1,"(3*Sqrt[1 + x]*(2 + 3*x) - Sqrt[6 + 9*x]*ArcSinh[Sqrt[2 + 3*x]])/(3*Sqrt[2 + 3*x])","A",1
864,1,61,43,0.0387338,"\int \frac{(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx","Integrate[(1 + x)^(3/2)/((1 - x)^(3/2)*x),x]","\frac{2 \left(\sqrt{1-x^2} \sin ^{-1}\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)+2 x+2\right)}{\sqrt{1-x^2}}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)","\frac{4 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left(\sqrt{1-x} \sqrt{x+1}\right)",1,"(2*(2 + 2*x + Sqrt[1 - x^2]*ArcSin[Sqrt[1 - x]/Sqrt[2]]))/Sqrt[1 - x^2] - ArcTanh[Sqrt[1 - x^2]]","A",1
865,1,47,35,0.0231996,"\int \frac{(1+x)^3}{x \left(1-x^2\right)^{3/2}} \, dx","Integrate[(1 + x)^3/(x*(1 - x^2)^(3/2)),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-x^2\right)-\sqrt{1-x^2} \sin ^{-1}(x)+4 x+3}{\sqrt{1-x^2}}","\frac{4 (x+1)}{\sqrt{1-x^2}}-\tanh ^{-1}\left(\sqrt{1-x^2}\right)-\sin ^{-1}(x)",1,"(3 + 4*x - Sqrt[1 - x^2]*ArcSin[x] + Hypergeometric2F1[-1/2, 1, 1/2, 1 - x^2])/Sqrt[1 - x^2]","C",1
866,1,72,51,0.0604992,"\int \frac{(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx","Integrate[(1 + a*x)^(3/2)/(x*(1 - a*x)^(3/2)),x]","\frac{2 \left(\sqrt{1-a^2 x^2} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)+2 a x+2\right)}{\sqrt{1-a^2 x^2}}-\tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)","\frac{4 \sqrt{a x+1}}{\sqrt{1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)",1,"(2*(2 + 2*a*x + Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/Sqrt[1 - a^2*x^2] - ArcTanh[Sqrt[1 - a^2*x^2]]","A",1
867,1,59,45,0.0381535,"\int \frac{(1+a x)^3}{x \left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-a^2 x^2\right)-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)+4 a x+3}{\sqrt{1-a^2 x^2}}","\frac{4 (a x+1)}{\sqrt{1-a^2 x^2}}-\tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\sin ^{-1}(a x)",1,"(3 + 4*a*x - Sqrt[1 - a^2*x^2]*ArcSin[a*x] + Hypergeometric2F1[-1/2, 1, 1/2, 1 - a^2*x^2])/Sqrt[1 - a^2*x^2]","C",1
868,1,2,2,0.0036574,"\int \frac{1}{\sqrt{1-x^2}} \, dx","Integrate[1/Sqrt[1 - x^2],x]","\sin ^{-1}(x)","\sin ^{-1}(x)",1,"ArcSin[x]","A",1
869,1,32,2,0.0256216,"\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^4}} \, dx","Integrate[Sqrt[1 + x^2]/Sqrt[1 - x^4],x]","-\tan ^{-1}\left(\frac{x \sqrt{x^2+1} \sqrt{1-x^4}}{x^4-1}\right)","\sin ^{-1}(x)",1,"-ArcTan[(x*Sqrt[1 + x^2]*Sqrt[1 - x^4])/(-1 + x^4)]","B",1
870,1,2,2,0.0033631,"\int \frac{1}{\sqrt{1+x^2}} \, dx","Integrate[1/Sqrt[1 + x^2],x]","\sinh ^{-1}(x)","\sinh ^{-1}(x)",1,"ArcSinh[x]","A",1
871,1,42,2,0.0238864,"\int \frac{\sqrt{1-x^2}}{\sqrt{1-x^4}} \, dx","Integrate[Sqrt[1 - x^2]/Sqrt[1 - x^4],x]","\log \left(1-x^2\right)-\log \left(x^3+\sqrt{1-x^2} \sqrt{1-x^4}-x\right)","\sinh ^{-1}(x)",1,"Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*Sqrt[1 - x^4]]","B",1
872,1,20,23,0.0056822,"\int \sqrt{1-x^2} \, dx","Integrate[Sqrt[1 - x^2],x]","\frac{1}{2} \left(\sqrt{1-x^2} x+\sin ^{-1}(x)\right)","\frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x)",1,"(x*Sqrt[1 - x^2] + ArcSin[x])/2","A",1
873,1,50,23,0.0414552,"\int \frac{\sqrt{1-x^4}}{\sqrt{1+x^2}} \, dx","Integrate[Sqrt[1 - x^4]/Sqrt[1 + x^2],x]","\frac{1}{2} \left(\frac{\sqrt{1-x^4} x}{\sqrt{x^2+1}}+\tan ^{-1}\left(\frac{x \sqrt{x^2+1}}{\sqrt{1-x^4}}\right)\right)","\frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x)",1,"((x*Sqrt[1 - x^4])/Sqrt[1 + x^2] + ArcTan[(x*Sqrt[1 + x^2])/Sqrt[1 - x^4]])/2","B",1
874,1,18,21,0.0042976,"\int \sqrt{1+x^2} \, dx","Integrate[Sqrt[1 + x^2],x]","\frac{1}{2} \left(\sqrt{x^2+1} x+\sinh ^{-1}(x)\right)","\frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x)",1,"(x*Sqrt[1 + x^2] + ArcSinh[x])/2","A",1
875,1,70,21,0.0559377,"\int \frac{\sqrt{1-x^4}}{\sqrt{1-x^2}} \, dx","Integrate[Sqrt[1 - x^4]/Sqrt[1 - x^2],x]","\frac{1}{2} \left(\log \left(1-x^2\right)+\frac{\sqrt{1-x^4} x}{\sqrt{1-x^2}}-\log \left(x^3+\sqrt{1-x^2} \sqrt{1-x^4}-x\right)\right)","\frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x)",1,"((x*Sqrt[1 - x^4])/Sqrt[1 - x^2] + Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*Sqrt[1 - x^4]])/2","B",1
876,1,53,49,0.0126807,"\int \left(\frac{a+b+c x^2}{d}\right)^m \, dx","Integrate[((a + b + c*x^2)/d)^m,x]","x \left(\frac{c x^2}{a+b}+1\right)^{-m} \left(\frac{a+b+c x^2}{d}\right)^m \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};-\frac{c x^2}{a+b}\right)","\frac{d x \left(\frac{a+b}{d}+\frac{c x^2}{d}\right)^{m+1} \, _2F_1\left(1,m+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a+b}\right)}{a+b}",1,"(x*((a + b + c*x^2)/d)^m*Hypergeometric2F1[1/2, -m, 3/2, -((c*x^2)/(a + b))])/(1 + (c*x^2)/(a + b))^m","A",1
877,1,38,28,0.0118519,"\int \frac{1}{x-\sqrt{1+x^2}} \, dx","Integrate[(x - Sqrt[1 + x^2])^(-1),x]","\frac{1}{2} \log \left(x-\sqrt{x^2+1}\right)-\frac{1}{4 \left(x-\sqrt{x^2+1}\right)^2}","-\frac{x^2}{2}-\frac{1}{2} \sqrt{x^2+1} x-\frac{1}{2} \sinh ^{-1}(x)",1,"-1/4*1/(x - Sqrt[1 + x^2])^2 + Log[x - Sqrt[1 + x^2]]/2","A",1
878,1,37,37,0.0229734,"\int \frac{1}{x-\sqrt{1-x^2}} \, dx","Integrate[(x - Sqrt[1 - x^2])^(-1),x]","\frac{1}{4} \log \left(1-2 x^2\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)-\frac{1}{2} \sin ^{-1}(x)","\frac{1}{4} \log \left(1-2 x^2\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)-\frac{1}{2} \sin ^{-1}(x)",1,"-1/2*ArcSin[x] - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4","A",1
879,1,74,40,0.0399866,"\int \frac{1}{x-\sqrt{1+2 x^2}} \, dx","Integrate[(x - Sqrt[1 + 2*x^2])^(-1),x]","\frac{1}{4} \left(-2 \log \left(x^2+1\right)-\log \left(3 x^2-2 \sqrt{2 x^2+1} x+1\right)+\log \left(3 x^2+2 \sqrt{2 x^2+1} x+1\right)-4 \sqrt{2} \sinh ^{-1}\left(\sqrt{2} x\right)\right)","-\frac{1}{2} \log \left(x^2+1\right)+\tanh ^{-1}\left(\frac{x}{\sqrt{2 x^2+1}}\right)-\sqrt{2} \sinh ^{-1}\left(\sqrt{2} x\right)",1,"(-4*Sqrt[2]*ArcSinh[Sqrt[2]*x] - 2*Log[1 + x^2] - Log[1 + 3*x^2 - 2*x*Sqrt[1 + 2*x^2]] + Log[1 + 3*x^2 + 2*x*Sqrt[1 + 2*x^2]])/4","A",1
880,1,77,54,0.0570483,"\int \frac{2 x-x^3+x^2 \sqrt{2-x^2}}{-2+2 x^2} \, dx","Integrate[(2*x - x^3 + x^2*Sqrt[2 - x^2])/(-2 + 2*x^2),x]","\frac{1}{4} \left(-x^2+\sqrt{2-x^2} x+\log \left(1-x^2\right)-\log \left(\sqrt{2-x^2}-x+2\right)+\log \left(\sqrt{2-x^2}+x+2\right)+\log (1-x)-\log (x+1)\right)","-\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x+\frac{1}{4} \log \left(1-x^2\right)-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{2-x^2}}\right)",1,"(-x^2 + x*Sqrt[2 - x^2] + Log[1 - x] - Log[1 + x] + Log[1 - x^2] - Log[2 - x + Sqrt[2 - x^2]] + Log[2 + x + Sqrt[2 - x^2]])/4","A",1
881,1,77,60,0.0867738,"\int \frac{x \sqrt{2-x^2}}{x-\sqrt{2-x^2}} \, dx","Integrate[(x*Sqrt[2 - x^2])/(x - Sqrt[2 - x^2]),x]","\frac{1}{4} \left(-x^2+\sqrt{2-x^2} x+\log \left(1-x^2\right)-\log \left(\sqrt{2-x^2}-x+2\right)+\log \left(\sqrt{2-x^2}+x+2\right)+\log (1-x)-\log (x+1)\right)","-\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x-\frac{1}{2} \tanh ^{-1}\left(\frac{x}{\sqrt{2-x^2}}\right)+\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1)",1,"(-x^2 + x*Sqrt[2 - x^2] + Log[1 - x] - Log[1 + x] + Log[1 - x^2] - Log[2 - x + Sqrt[2 - x^2]] + Log[2 + x + Sqrt[2 - x^2]])/4","A",1
882,1,39,51,0.0518392,"\int \frac{x}{-x+\sqrt{2 x-x^2}} \, dx","Integrate[x/(-x + Sqrt[2*x - x^2]),x]","\frac{1}{2} \left(-x-\sqrt{-((x-2) x)}-\log (1-x)+\tanh ^{-1}\left(\sqrt{-((x-2) x)}\right)\right)","-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)-\frac{x}{2}-\frac{1}{2} \log (1-x)",1,"(-x - Sqrt[-((-2 + x)*x)] + ArcTanh[Sqrt[-((-2 + x)*x)]] - Log[1 - x])/2","A",1
883,1,39,51,0.045901,"\int \frac{x+\sqrt{2 x-x^2}}{2-2 x} \, dx","Integrate[(x + Sqrt[2*x - x^2])/(2 - 2*x),x]","\frac{1}{2} \left(-x-\sqrt{-((x-2) x)}-\log (1-x)+\tanh ^{-1}\left(\sqrt{-((x-2) x)}\right)\right)","-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)-\frac{x}{2}-\frac{1}{2} \log (1-x)",1,"(-x - Sqrt[-((-2 + x)*x)] + ArcTanh[Sqrt[-((-2 + x)*x)]] - Log[1 - x])/2","A",1
884,1,39,51,0.022579,"\int \frac{\sqrt{2-x} \sqrt{x}+x}{2-2 x} \, dx","Integrate[(Sqrt[2 - x]*Sqrt[x] + x)/(2 - 2*x),x]","\frac{1}{2} \left(-x-\sqrt{-((x-2) x)}-\log (1-x)+\tanh ^{-1}\left(\sqrt{-((x-2) x)}\right)\right)","-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2 x-x^2}\right)-\frac{x}{2}-\frac{1}{2} \log (1-x)",1,"(-x - Sqrt[-((-2 + x)*x)] + ArcTanh[Sqrt[-((-2 + x)*x)]] - Log[1 - x])/2","A",1
885,1,82,54,0.14229,"\int \frac{\sqrt{x}}{\sqrt{2-x}-\sqrt{x}} \, dx","Integrate[Sqrt[x]/(Sqrt[2 - x] - Sqrt[x]),x]","\frac{1}{2} \left(-x-\sqrt{-((x-2) x)}-\log \left(1-\sqrt{x}\right)-\log \left(\sqrt{x}+1\right)+\tanh ^{-1}\left(\frac{2-\sqrt{x}}{\sqrt{2-x}}\right)-\tanh ^{-1}\left(\frac{\sqrt{x}+2}{\sqrt{2-x}}\right)\right)","-\frac{x}{2}-\frac{1}{2} \sqrt{2-x} \sqrt{x}-\frac{1}{2} \log (1-x)+\frac{1}{2} \tanh ^{-1}\left(\sqrt{2-x} \sqrt{x}\right)",1,"(-x - Sqrt[-((-2 + x)*x)] + ArcTanh[(2 - Sqrt[x])/Sqrt[2 - x]] - ArcTanh[(2 + Sqrt[x])/Sqrt[2 - x]] - Log[1 - Sqrt[x]] - Log[1 + Sqrt[x]])/2","A",0
886,1,23,27,0.0105304,"\int \frac{1}{\left((1+x) \left(-1+x^2\right)\right)^{2/3}} \, dx","Integrate[((1 + x)*(-1 + x^2))^(-2/3),x]","\frac{3 (x-1) (x+1)}{2 \left((x-1) (x+1)^2\right)^{2/3}}","-\frac{3 \left(1-x^2\right)}{2 \left(-\left((x+1) \left(1-x^2\right)\right)\right)^{2/3}}",1,"(3*(-1 + x)*(1 + x))/(2*((-1 + x)*(1 + x)^2)^(2/3))","A",1
887,1,12,14,0.0241415,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x \left(1+x^2\right)}} \, dx","Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x*(1 + x^2)]),x]","-\frac{2 x}{\sqrt{x^3+x}}","-\frac{2 x}{\sqrt{x \left(x^2+1\right)}}",1,"(-2*x)/Sqrt[x + x^3]","A",1
888,1,12,12,0.010741,"\int \frac{-1+x^2}{\left(1+x^2\right) \sqrt{x+x^3}} \, dx","Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]","-\frac{2 x}{\sqrt{x^3+x}}","-\frac{2 x}{\sqrt{x^3+x}}",1,"(-2*x)/Sqrt[x + x^3]","A",1
889,1,29,36,0.0192727,"\int \frac{\sqrt{\frac{\left(-1+x^2\right)^2}{x \left(1+x^2\right)}}}{1+x^2} \, dx","Integrate[Sqrt[(-1 + x^2)^2/(x*(1 + x^2))]/(1 + x^2),x]","-\frac{2 x \sqrt{\frac{\left(x^2-1\right)^2}{x^3+x}}}{x^2-1}","\frac{2 x \sqrt{\frac{\left(1-x^2\right)^2}{x \left(x^2+1\right)}}}{1-x^2}",1,"(-2*x*Sqrt[(-1 + x^2)^2/(x + x^3)])/(-1 + x^2)","A",1
890,1,29,33,0.0122686,"\int \frac{\sqrt{\frac{\left(-1+x^2\right)^2}{x+x^3}}}{1+x^2} \, dx","Integrate[Sqrt[(-1 + x^2)^2/(x + x^3)]/(1 + x^2),x]","-\frac{2 x \sqrt{\frac{\left(x^2-1\right)^2}{x^3+x}}}{x^2-1}","\frac{2 x \sqrt{\frac{\left(1-x^2\right)^2}{x^3+x}}}{1-x^2}",1,"(-2*x*Sqrt[(-1 + x^2)^2/(x + x^3)])/(-1 + x^2)","A",1
891,1,105,70,0.1312723,"\int \frac{1}{\sqrt{a+\frac{b}{x^2}} \sqrt{c+d x^2}} \, dx","Integrate[1/(Sqrt[a + b/x^2]*Sqrt[c + d*x^2]),x]","\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x^2} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a x^2+b}}{\sqrt{a c-b d}}\right)}{\sqrt{d} \sqrt{a x^2+b} \sqrt{a c-b d} \sqrt{\frac{a \left(c+d x^2\right)}{a c-b d}}}","\frac{\sqrt{a x^2+b} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a x^2+b}}{\sqrt{a} \sqrt{c+d x^2}}\right)}{\sqrt{a} \sqrt{d} x \sqrt{a+\frac{b}{x^2}}}",1,"(Sqrt[a + b/x^2]*x*Sqrt[c + d*x^2]*ArcSinh[(Sqrt[d]*Sqrt[b + a*x^2])/Sqrt[a*c - b*d]])/(Sqrt[d]*Sqrt[a*c - b*d]*Sqrt[b + a*x^2]*Sqrt[(a*(c + d*x^2))/(a*c - b*d)])","A",1
892,1,52,83,0.0642256,"\int \frac{\sqrt{-2 x^2+x^4}}{\left(-1+x^2\right) \left(2+x^2\right)} \, dx","Integrate[Sqrt[-2*x^2 + x^4]/((-1 + x^2)*(2 + x^2)),x]","-\frac{x \sqrt{x^2-2} \left(2 \tan ^{-1}\left(\frac{2}{\sqrt{x^2-2}}\right)+\tan ^{-1}\left(\sqrt{x^2-2}\right)\right)}{3 \sqrt{x^2 \left(x^2-2\right)}}","\frac{2 \sqrt{x^4-2 x^2} \tan ^{-1}\left(\frac{\sqrt{x^2-2}}{2}\right)}{3 x \sqrt{x^2-2}}-\frac{\sqrt{x^4-2 x^2} \tan ^{-1}\left(\sqrt{x^2-2}\right)}{3 x \sqrt{x^2-2}}",1,"-1/3*(x*Sqrt[-2 + x^2]*(2*ArcTan[2/Sqrt[-2 + x^2]] + ArcTan[Sqrt[-2 + x^2]]))/Sqrt[x^2*(-2 + x^2)]","A",1
893,1,91,47,0.1182113,"\int \frac{\sqrt{1-\frac{1}{\left(-1+x^2\right)^2}}}{2-x^2} \, dx","Integrate[Sqrt[1 - (-1 + x^2)^(-2)]/(2 - x^2),x]","\frac{1}{2} \tan ^{-1}\left(\frac{(x-1) (x+1) (x+2) \sqrt{\frac{x^2 \left(x^2-2\right)}{\left(x^2-1\right)^2}}}{x \left(x^2-2\right)}\right)-\frac{1}{2} \tan ^{-1}\left(\frac{(x-2) (x-1) (x+1) \sqrt{\frac{x^2 \left(x^2-2\right)}{\left(x^2-1\right)^2}}}{x \left(x^2-2\right)}\right)","\frac{\left(1-x^2\right) \sqrt{1-\frac{1}{\left(1-x^2\right)^2}} \tan ^{-1}\left(\sqrt{x^2-2}\right)}{x \sqrt{x^2-2}}",1,"-1/2*ArcTan[((-2 + x)*(-1 + x)*(1 + x)*Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2])/(x*(-2 + x^2))] + ArcTan[((-1 + x)*(1 + x)*(2 + x)*Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2])/(x*(-2 + x^2))]/2","A",1
894,1,70,123,0.0253943,"\int \frac{\sqrt{\frac{-2 x^2+x^4}{\left(-1+x^2\right)^2}}}{2+x^2} \, dx","Integrate[Sqrt[(-2*x^2 + x^4)/(-1 + x^2)^2]/(2 + x^2),x]","\frac{\sqrt{\frac{x^2 \left(x^2-2\right)}{\left(x^2-1\right)^2}} \left(x^2-1\right) \left(2 \tan ^{-1}\left(\frac{\sqrt{x^2-2}}{2}\right)-\tan ^{-1}\left(\sqrt{x^2-2}\right)\right)}{3 x \sqrt{x^2-2}}","\frac{\left(1-x^2\right) \sqrt{-\frac{2 x^2-x^4}{\left(1-x^2\right)^2}} \tan ^{-1}\left(\sqrt{x^2-2}\right)}{3 x \sqrt{x^2-2}}-\frac{2 \left(1-x^2\right) \sqrt{-\frac{2 x^2-x^4}{\left(1-x^2\right)^2}} \tan ^{-1}\left(\frac{\sqrt{x^2-2}}{2}\right)}{3 x \sqrt{x^2-2}}",1,"(Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2]*(-1 + x^2)*(2*ArcTan[Sqrt[-2 + x^2]/2] - ArcTan[Sqrt[-2 + x^2]]))/(3*x*Sqrt[-2 + x^2])","A",1
895,1,64,133,0.100341,"\int \left(1+\frac{2 x}{1+x^2}\right)^{5/2} \, dx","Integrate[(1 + (2*x)/(1 + x^2))^(5/2),x]","\frac{(x+1) \left(3 x^4-8 x^3-18 x^2+15 \left(x^2+1\right)^{3/2} \sinh ^{-1}(x)-12 x-17\right)}{3 \sqrt{\frac{(x+1)^2}{x^2+1}} \left(x^2+1\right)^2}","-\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)^3}{3 \left(x^2+1\right)}-\frac{4}{3} (1-2 x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)-\frac{(3 x+4) \left(x^2+1\right) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{5 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1}",1,"((1 + x)*(-17 - 12*x - 18*x^2 - 8*x^3 + 3*x^4 + 15*(1 + x^2)^(3/2)*ArcSinh[x]))/(3*Sqrt[(1 + x)^2/(1 + x^2)]*(1 + x^2)^2)","A",1
896,1,44,90,0.038652,"\int \left(1+\frac{2 x}{1+x^2}\right)^{3/2} \, dx","Integrate[(1 + (2*x)/(1 + x^2))^(3/2),x]","\frac{\sqrt{\frac{(x+1)^2}{x^2+1}} \left(x^2+3 \sqrt{x^2+1} \sinh ^{-1}(x)-2 x-1\right)}{x+1}","-\left((1-x) \sqrt{\frac{2 x}{x^2+1}+1} (x+1)\right)-\frac{x \left(x^2+1\right) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}+\frac{3 \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1} \sinh ^{-1}(x)}{x+1}",1,"(Sqrt[(1 + x)^2/(1 + x^2)]*(-1 - 2*x + x^2 + 3*Sqrt[1 + x^2]*ArcSinh[x]))/(1 + x)","A",1
897,1,40,61,0.0177318,"\int \sqrt{1+\frac{2 x}{1+x^2}} \, dx","Integrate[Sqrt[1 + (2*x)/(1 + x^2)],x]","\frac{\sqrt{\frac{(x+1)^2}{x^2+1}} \left(x^2+\sqrt{x^2+1} \sinh ^{-1}(x)+1\right)}{x+1}","\frac{\sqrt{\frac{2 x}{x^2+1}+1} \left(x^2+1\right)}{x+1}+\frac{\sqrt{\frac{2 x}{x^2+1}+1} \sqrt{x^2+1} \sinh ^{-1}(x)}{x+1}",1,"(Sqrt[(1 + x)^2/(1 + x^2)]*(1 + x^2 + Sqrt[1 + x^2]*ArcSinh[x]))/(1 + x)","A",1
898,1,72,109,0.0315714,"\int \frac{1}{\sqrt{1+\frac{2 x}{1+x^2}}} \, dx","Integrate[1/Sqrt[1 + (2*x)/(1 + x^2)],x]","\frac{(x+1) \left(\sqrt{x^2+1}-\sqrt{2} \tanh ^{-1}\left(\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right)-\sinh ^{-1}(x)\right)}{\sqrt{\frac{(x+1)^2}{x^2+1}} \sqrt{x^2+1}}","\frac{x+1}{\sqrt{\frac{2 x}{x^2+1}+1}}-\frac{(x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{\sqrt{2} (x+1) \tanh ^{-1}\left(\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}",1,"((1 + x)*(Sqrt[1 + x^2] - ArcSinh[x] - Sqrt[2]*ArcTanh[(1 - x)/(Sqrt[2]*Sqrt[1 + x^2])]))/(Sqrt[(1 + x)^2/(1 + x^2)]*Sqrt[1 + x^2])","A",1
899,1,95,144,0.1154989,"\int \frac{1}{\left(1+\frac{2 x}{1+x^2}\right)^{3/2}} \, dx","Integrate[(1 + (2*x)/(1 + x^2))^(-3/2),x]","\frac{(x+1) \left(2 \sqrt{x^2+1} \left(2 x^2+9 x+5\right)+9 \sqrt{2} (x+1)^2 \tanh ^{-1}\left(\frac{x-1}{\sqrt{2} \sqrt{x^2+1}}\right)-12 (x+1)^2 \sinh ^{-1}(x)\right)}{4 \left(\frac{(x+1)^2}{x^2+1}\right)^{3/2} \left(x^2+1\right)^{3/2}}","\frac{3 (x+2)}{2 \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{x^2+1}{2 (x+1) \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{3 (x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{9 (x+1) \tanh ^{-1}\left(\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right)}{2 \sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}",1,"((1 + x)*(2*Sqrt[1 + x^2]*(5 + 9*x + 2*x^2) - 12*(1 + x)^2*ArcSinh[x] + 9*Sqrt[2]*(1 + x)^2*ArcTanh[(-1 + x)/(Sqrt[2]*Sqrt[1 + x^2])]))/(4*((1 + x)^2/(1 + x^2))^(3/2)*(1 + x^2)^(3/2))","A",1
900,1,26,28,0.0268524,"\int \frac{\sqrt{1+\frac{2 x}{1+x^2}}}{1+x^2} \, dx","Integrate[Sqrt[1 + (2*x)/(1 + x^2)]/(1 + x^2),x]","\frac{(x-1) \sqrt{\frac{(x+1)^2}{x^2+1}}}{x+1}","-\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1}",1,"((-1 + x)*Sqrt[(1 + x)^2/(1 + x^2)])/(1 + x)","A",1
901,0,0,17,0.1013283,"\int \sqrt{x-x^2} F(x) \, dx","Integrate[Sqrt[x - x^2]*F[x],x]","\int \sqrt{x-x^2} F(x) \, dx","\text{Int}\left(\sqrt{x-x^2} F(x),x\right)",0,"Integrate[Sqrt[x - x^2]*F[x], x]","A",-1
902,0,0,17,0.1794497,"\int \frac{F(x)}{\sqrt{x-x^2}} \, dx","Integrate[F[x]/Sqrt[x - x^2],x]","\int \frac{F(x)}{\sqrt{x-x^2}} \, dx","\text{Int}\left(\frac{F(x)}{\sqrt{x-x^2}},x\right)",0,"Integrate[F[x]/Sqrt[x - x^2], x]","A",-1
903,0,0,17,0.0272157,"\int \sqrt{1-x} \sqrt{x} F(x) \, dx","Integrate[Sqrt[1 - x]*Sqrt[x]*F[x],x]","\int \sqrt{1-x} \sqrt{x} F(x) \, dx","\text{Int}\left(\sqrt{x-x^2} F(x),x\right)",0,"Integrate[Sqrt[1 - x]*Sqrt[x]*F[x], x]","A",-1
904,0,0,17,0.021922,"\int \frac{F(x)}{\sqrt{1-x} \sqrt{x}} \, dx","Integrate[F[x]/(Sqrt[1 - x]*Sqrt[x]),x]","\int \frac{F(x)}{\sqrt{1-x} \sqrt{x}} \, dx","\text{Int}\left(\frac{F(x)}{\sqrt{x-x^2}},x\right)",0,"Integrate[F[x]/(Sqrt[1 - x]*Sqrt[x]), x]","A",-1
905,0,0,11,0.0055542,"\int F\left(\frac{a+b x}{x}\right) \, dx","Integrate[F[(a + b*x)/x],x]","\int F\left(\frac{a+b x}{x}\right) \, dx","\text{Int}\left(F\left(\frac{a}{x}+b\right),x\right)",0,"Integrate[F[(a + b*x)/x], x]","A",-1
906,0,0,11,0.0064167,"\int F\left(\frac{a+b x^2}{x^2}\right) \, dx","Integrate[F[(a + b*x^2)/x^2],x]","\int F\left(\frac{a+b x^2}{x^2}\right) \, dx","\text{Int}\left(F\left(\frac{a}{x^2}+b\right),x\right)",0,"Integrate[F[(a + b*x^2)/x^2], x]","A",-1
907,0,0,13,0.0072089,"\int F\left(\frac{x}{a+b x}\right) \, dx","Integrate[F[x/(a + b*x)],x]","\int F\left(\frac{x}{a+b x}\right) \, dx","\text{Int}\left(F\left(\frac{x}{a+b x}\right),x\right)",0,"Integrate[F[x/(a + b*x)], x]","A",-1
908,0,0,17,0.0089897,"\int F\left(\frac{x^2}{a+b x^2}\right) \, dx","Integrate[F[x^2/(a + b*x^2)],x]","\int F\left(\frac{x^2}{a+b x^2}\right) \, dx","\text{Int}\left(F\left(\frac{x^2}{a+b x^2}\right),x\right)",0,"Integrate[F[x^2/(a + b*x^2)], x]","A",-1
909,0,0,15,0.0118108,"\int F\left(\frac{x^2}{(a+b x)^2}\right) \, dx","Integrate[F[x^2/(a + b*x)^2],x]","\int F\left(\frac{x^2}{(a+b x)^2}\right) \, dx","\text{Int}\left(F\left(\frac{x^2}{(a+b x)^2}\right),x\right)",0,"Integrate[F[x^2/(a + b*x)^2], x]","A",-1
910,0,0,17,0.0117559,"\int F\left(\frac{x^4}{\left(a+b x^2\right)^2}\right) \, dx","Integrate[F[x^4/(a + b*x^2)^2],x]","\int F\left(\frac{x^4}{\left(a+b x^2\right)^2}\right) \, dx","\text{Int}\left(F\left(\frac{x^4}{\left(a+b x^2\right)^2}\right),x\right)",0,"Integrate[F[x^4/(a + b*x^2)^2], x]","A",-1
911,1,47,47,0.0227063,"\int \frac{\sqrt{b x^2+\sqrt{a+b^2 x^4}}}{\sqrt{a+b^2 x^4}} \, dx","Integrate[Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right)}{\sqrt{2} \sqrt{b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}+b x^2}}\right)}{\sqrt{2} \sqrt{b}}",1,"ArcTanh[(Sqrt[2]*Sqrt[b]*x)/Sqrt[b*x^2 + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])","A",1
912,1,48,48,0.0197307,"\int \frac{\sqrt{-b x^2+\sqrt{a+b^2 x^4}}}{\sqrt{a+b^2 x^4}} \, dx","Integrate[Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]/Sqrt[a + b^2*x^4],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}-b x^2}}\right)}{\sqrt{2} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\sqrt{a+b^2 x^4}-b x^2}}\right)}{\sqrt{2} \sqrt{b}}",1,"ArcTan[(Sqrt[2]*Sqrt[b]*x)/Sqrt[-(b*x^2) + Sqrt[a + b^2*x^4]]]/(Sqrt[2]*Sqrt[b])","A",1
913,0,0,169,0.1136447,"\int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x) \sqrt{3+4 x^4}} \, dx","Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)*Sqrt[3 + 4*x^4]),x]","\int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x) \sqrt{3+4 x^4}} \, dx","\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \tan ^{-1}\left(\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right)}{\sqrt{-\sqrt{3} d^2+2 i c^2}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \tanh ^{-1}\left(\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right)}{\sqrt{\sqrt{3} d^2+2 i c^2}}",1,"Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)*Sqrt[3 + 4*x^4]), x]","F",-1
914,0,0,268,0.0998964,"\int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x)^2 \sqrt{3+4 x^4}} \, dx","Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]),x]","\int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x)^2 \sqrt{3+4 x^4}} \, dx","\frac{\left(\frac{1}{2}-\frac{i}{2}\right) d \sqrt{\sqrt{3}-2 i x^2}}{\left(-\sqrt{3} d^2+2 i c^2\right) (c+d x)}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) d \sqrt{\sqrt{3}+2 i x^2}}{\left(\sqrt{3} d^2+2 i c^2\right) (c+d x)}+\frac{(1+i) c \tan ^{-1}\left(\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right)}{\left(-\sqrt{3} d^2+2 i c^2\right)^{3/2}}+\frac{(1-i) c \tanh ^{-1}\left(\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right)}{\left(\sqrt{3} d^2+2 i c^2\right)^{3/2}}",1,"Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]), x]","F",-1
915,1,41,41,0.0263191,"\int \frac{-4+x}{\left(1+\sqrt[3]{x}\right) \sqrt{x}} \, dx","Integrate[(-4 + x)/((1 + x^(1/3))*Sqrt[x]),x]","\frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-30 \sqrt[6]{x}+30 \tan ^{-1}\left(\sqrt[6]{x}\right)","\frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-30 \sqrt[6]{x}+30 \tan ^{-1}\left(\sqrt[6]{x}\right)",1,"-30*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 30*ArcTan[x^(1/6)]","A",1
916,1,38,26,0.0245896,"\int \frac{1+\sqrt{x}}{x^{5/6}+x^{7/6}} \, dx","Integrate[(1 + Sqrt[x])/(x^(5/6) + x^(7/6)),x]","3 \sqrt[3]{x}+(-3-3 i) \log \left(-\sqrt[6]{x}+i\right)-(3-3 i) \log \left(\sqrt[6]{x}+i\right)","3 \sqrt[3]{x}-3 \log \left(\sqrt[3]{x}+1\right)+6 \tan ^{-1}\left(\sqrt[6]{x}\right)",1,"3*x^(1/3) - (3 + 3*I)*Log[I - x^(1/6)] - (3 - 3*I)*Log[I + x^(1/6)]","C",1
917,1,54,42,0.0225494,"\int \frac{1+\sqrt{x}}{\left(1+\sqrt[3]{x}\right) \sqrt{x}} \, dx","Integrate[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]","\frac{3 x^{2/3}}{2}-3 \sqrt[3]{x}+6 \sqrt[6]{x}+(3+3 i) \log \left(-\sqrt[6]{x}+i\right)+(3-3 i) \log \left(\sqrt[6]{x}+i\right)","\frac{3 x^{2/3}}{2}-3 \sqrt[3]{x}+6 \sqrt[6]{x}+3 \log \left(\sqrt[3]{x}+1\right)-6 \tan ^{-1}\left(\sqrt[6]{x}\right)",1,"6*x^(1/6) - 3*x^(1/3) + (3*x^(2/3))/2 + (3 + 3*I)*Log[I - x^(1/6)] + (3 - 3*I)*Log[I + x^(1/6)]","C",1
918,1,48,20,0.0135536,"\int \frac{\sqrt{2+\frac{b}{x^2}}}{b+2 x^2} \, dx","Integrate[Sqrt[2 + b/x^2]/(b + 2*x^2),x]","-\frac{x \sqrt{\frac{b}{x^2}+2} \tanh ^{-1}\left(\frac{\sqrt{b+2 x^2}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{b+2 x^2}}","-\frac{\text{csch}^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b}}\right)}{\sqrt{b}}",1,"-((Sqrt[2 + b/x^2]*x*ArcTanh[Sqrt[b + 2*x^2]/Sqrt[b]])/(Sqrt[b]*Sqrt[b + 2*x^2]))","B",1
919,1,52,20,0.0136877,"\int \frac{\sqrt{2-\frac{b}{x^2}}}{-b+2 x^2} \, dx","Integrate[Sqrt[2 - b/x^2]/(-b + 2*x^2),x]","\frac{x \sqrt{2-\frac{b}{x^2}} \tan ^{-1}\left(\frac{\sqrt{2 x^2-b}}{\sqrt{b}}\right)}{\sqrt{b} \sqrt{2 x^2-b}}","-\frac{\csc ^{-1}\left(\frac{\sqrt{2} x}{\sqrt{b}}\right)}{\sqrt{b}}",1,"(Sqrt[2 - b/x^2]*x*ArcTan[Sqrt[-b + 2*x^2]/Sqrt[b]])/(Sqrt[b]*Sqrt[-b + 2*x^2])","B",1
920,1,136,121,0.1023651,"\int \frac{\sqrt{a+\frac{c}{x^2}}}{d+e x} \, dx","Integrate[Sqrt[a + c/x^2]/(d + e*x),x]","\frac{x \sqrt{a+\frac{c}{x^2}} \left(\sqrt{a d^2+c e^2} \tanh ^{-1}\left(\frac{c e-a d x}{\sqrt{a x^2+c} \sqrt{a d^2+c e^2}}\right)+\sqrt{a} d \tanh ^{-1}\left(\frac{\sqrt{a} x}{\sqrt{a x^2+c}}\right)-\sqrt{c} e \tanh ^{-1}\left(\frac{\sqrt{a x^2+c}}{\sqrt{c}}\right)\right)}{d e \sqrt{a x^2+c}}","-\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left(\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}} \sqrt{a d^2+c e^2}}\right)}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c}}{x \sqrt{a+\frac{c}{x^2}}}\right)}{d}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right)}{e}",1,"(Sqrt[a + c/x^2]*x*(Sqrt[a]*d*ArcTanh[(Sqrt[a]*x)/Sqrt[c + a*x^2]] + Sqrt[a*d^2 + c*e^2]*ArcTanh[(c*e - a*d*x)/(Sqrt[a*d^2 + c*e^2]*Sqrt[c + a*x^2])] - Sqrt[c]*e*ArcTanh[Sqrt[c + a*x^2]/Sqrt[c]]))/(d*e*Sqrt[c + a*x^2])","A",1
921,1,189,181,0.2806201,"\int \frac{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}{d+e x} \, dx","Integrate[Sqrt[a + c/x^2 + b/x]/(d + e*x),x]","-\frac{x \sqrt{a+\frac{b x+c}{x^2}} \left(\sqrt{a d^2-b d e+c e^2} \tanh ^{-1}\left(\frac{2 a d x+b d-b e x-2 c e}{2 \sqrt{x (a x+b)+c} \sqrt{a d^2-b d e+c e^2}}\right)-\sqrt{a} d \tanh ^{-1}\left(\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right)+\sqrt{c} e \tanh ^{-1}\left(\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right)\right)}{d e \sqrt{x (a x+b)+c}}","-\frac{\sqrt{a d^2-e (b d-c e)} \tanh ^{-1}\left(\frac{2 a d+\frac{b d-2 c e}{x}-b e}{2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{a d^2-e (b d-c e)}}\right)}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{d}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right)}{e}",1,"-((x*Sqrt[a + (c + b*x)/x^2]*(-(Sqrt[a]*d*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x*(b + a*x)])]) + Sqrt[c]*e*ArcTanh[(2*c + b*x)/(2*Sqrt[c]*Sqrt[c + x*(b + a*x)])] + Sqrt[a*d^2 - b*d*e + c*e^2]*ArcTanh[(b*d - 2*c*e + 2*a*d*x - b*e*x)/(2*Sqrt[a*d^2 - b*d*e + c*e^2]*Sqrt[c + x*(b + a*x)])]))/(d*e*Sqrt[c + x*(b + a*x)]))","A",1
922,1,26,26,0.0125477,"\int \frac{\sqrt[6]{x}+\sqrt[5]{x^3}}{\sqrt{x}} \, dx","Integrate[(x^(1/6) + (x^3)^(1/5))/Sqrt[x],x]","\frac{3 x^{2/3}}{2}+\frac{10}{11} \sqrt[5]{x^3} \sqrt{x}","\frac{3 x^{2/3}}{2}+\frac{10}{11} \sqrt[5]{x^3} \sqrt{x}",1,"(3*x^(2/3))/2 + (10*Sqrt[x]*(x^3)^(1/5))/11","A",1
923,1,27,26,0.0308802,"\int \frac{2+x}{\sqrt{4 x-x^2}} \, dx","Integrate[(2 + x)/Sqrt[4*x - x^2],x]","-\sqrt{-((x-4) x)}-8 \sin ^{-1}\left(\sqrt{1-\frac{x}{4}}\right)","-\sqrt{4 x-x^2}-4 \sin ^{-1}\left(1-\frac{x}{2}\right)",1,"-Sqrt[-((-4 + x)*x)] - 8*ArcSin[Sqrt[1 - x/4]]","A",1
924,1,13,15,0.0119268,"\int \frac{3+x}{\sqrt[3]{6 x+x^2}} \, dx","Integrate[(3 + x)/(6*x + x^2)^(1/3),x]","\frac{3}{4} (x (x+6))^{2/3}","\frac{3}{4} \left(x^2+6 x\right)^{2/3}",1,"(3*(x*(6 + x))^(2/3))/4","A",1
925,1,19,22,0.0131036,"\int \frac{4+x}{\left(6 x-x^2\right)^{3/2}} \, dx","Integrate[(4 + x)/(6*x - x^2)^(3/2),x]","\frac{7 x-12}{9 \sqrt{-((x-6) x)}}","-\frac{12-7 x}{9 \sqrt{6 x-x^2}}",1,"(-12 + 7*x)/(9*Sqrt[-((-6 + x)*x)])","A",1
926,1,37,12,0.0149302,"\int \frac{1}{(1+x) \sqrt{2 x+x^2}} \, dx","Integrate[1/((1 + x)*Sqrt[2*x + x^2]),x]","\frac{2 \sqrt{x} \sqrt{x+2} \tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+2}}\right)}{\sqrt{x (x+2)}}","\tan ^{-1}\left(\sqrt{x^2+2 x}\right)",1,"(2*Sqrt[x]*Sqrt[2 + x]*ArcTan[Sqrt[x]/Sqrt[2 + x]])/Sqrt[x*(2 + x)]","B",1
927,1,37,12,0.017268,"\int \frac{1}{(1+2 x) \sqrt{x+x^2}} \, dx","Integrate[1/((1 + 2*x)*Sqrt[x + x^2]),x]","\frac{2 \sqrt{x} \sqrt{x+1} \tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+1}}\right)}{\sqrt{x (x+1)}}","\tan ^{-1}\left(2 \sqrt{x^2+x}\right)",1,"(2*Sqrt[x]*Sqrt[1 + x]*ArcTan[Sqrt[x]/Sqrt[1 + x]])/Sqrt[x*(1 + x)]","B",1
928,1,12,15,0.0084394,"\int \frac{-1+x}{\sqrt{2 x-x^2}} \, dx","Integrate[(-1 + x)/Sqrt[2*x - x^2],x]","-\sqrt{-((x-2) x)}","-\sqrt{2 x-x^2}",1,"-Sqrt[-((-2 + x)*x)]","A",1
929,1,95,54,0.0779172,"\int \frac{\sqrt{x-x^2}}{1+x} \, dx","Integrate[Sqrt[x - x^2]/(1 + x),x]","\sqrt{-((x-1) x)}-\frac{3 \sqrt{-((x-1) x)} \sin ^{-1}\left(\sqrt{1-x}\right)}{\sqrt{1-x} \sqrt{x}}+\frac{2 \sqrt{2} \sqrt{-((x-1) x)} \tanh ^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{2} \sqrt{x}}\right)}{\sqrt{x-1} \sqrt{x}}","\sqrt{x-x^2}+\sqrt{2} \tan ^{-1}\left(\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right)-\frac{3}{2} \sin ^{-1}(1-2 x)",1,"Sqrt[-((-1 + x)*x)] - (3*Sqrt[-((-1 + x)*x)]*ArcSin[Sqrt[1 - x]])/(Sqrt[1 - x]*Sqrt[x]) + (2*Sqrt[2]*Sqrt[-((-1 + x)*x)]*ArcTanh[Sqrt[-1 + x]/(Sqrt[2]*Sqrt[x])])/(Sqrt[-1 + x]*Sqrt[x])","A",1
930,1,57,59,0.0357243,"\int \sqrt{\sqrt[4]{x}+x} \, dx","Integrate[Sqrt[x^(1/4) + x],x]","\frac{3 x^{5/4}-\sqrt{x^{3/4}+1} \sqrt[8]{x} \sinh ^{-1}\left(x^{3/8}\right)+2 x^2+\sqrt{x}}{3 \sqrt{x+\sqrt[4]{x}}}","\frac{2}{3} \sqrt{x+\sqrt[4]{x}} x+\frac{1}{3} \sqrt{x+\sqrt[4]{x}} \sqrt[4]{x}-\frac{1}{3} \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+\sqrt[4]{x}}}\right)",1,"(Sqrt[x] + 3*x^(5/4) + 2*x^2 - Sqrt[1 + x^(3/4)]*x^(1/8)*ArcSinh[x^(3/8)])/(3*Sqrt[x^(1/4) + x])","A",1
931,1,39,59,0.0195342,"\int \sqrt{x+x^{3/2}} \, dx","Integrate[Sqrt[x + x^(3/2)],x]","\frac{4 \left(\sqrt{x}+1\right) \left(15 x-12 \sqrt{x}+8\right) \sqrt{x^{3/2}+x}}{105 \sqrt{x}}","\frac{4 \left(x^{3/2}+x\right)^{3/2}}{7 \sqrt{x}}-\frac{16 \left(x^{3/2}+x\right)^{3/2}}{35 x}+\frac{32 \left(x^{3/2}+x\right)^{3/2}}{105 x^{3/2}}",1,"(4*(1 + Sqrt[x])*(8 - 12*Sqrt[x] + 15*x)*Sqrt[x + x^(3/2)])/(105*Sqrt[x])","A",1
932,1,51,94,0.0298856,"\int x \sqrt{x+x^{3/2}} \, dx","Integrate[x*Sqrt[x + x^(3/2)],x]","\frac{4 \left(\sqrt{x}+1\right) \sqrt{x^{3/2}+x} \left(-280 x^{3/2}+315 x^2+240 x-192 \sqrt{x}+128\right)}{3465 \sqrt{x}}","\frac{4}{11} \sqrt{x} \left(x^{3/2}+x\right)^{3/2}+\frac{64 \left(x^{3/2}+x\right)^{3/2}}{231 \sqrt{x}}-\frac{256 \left(x^{3/2}+x\right)^{3/2}}{1155 x}+\frac{512 \left(x^{3/2}+x\right)^{3/2}}{3465 x^{3/2}}-\frac{32}{99} \left(x^{3/2}+x\right)^{3/2}",1,"(4*(1 + Sqrt[x])*Sqrt[x + x^(3/2)]*(128 - 192*Sqrt[x] + 240*x - 280*x^(3/2) + 315*x^2))/(3465*Sqrt[x])","A",1
933,1,18,18,0.0097945,"\int \left(1-x^2\right) \sqrt{\frac{1}{2-x^2}} \, dx","Integrate[(1 - x^2)*Sqrt[(2 - x^2)^(-1)],x]","\frac{x}{2 \sqrt{\frac{1}{2-x^2}}}","\frac{x}{2 \sqrt{\frac{1}{2-x^2}}}",1,"x/(2*Sqrt[(2 - x^2)^(-1)])","A",1
934,1,84,107,0.0276627,"\int \sqrt{x^2+x^3-x^4} \, dx","Integrate[Sqrt[x^2 + x^3 - x^4],x]","\frac{\sqrt{-x^4+x^3+x^2} \left(2 \sqrt{x^2-x-1} \left(8 x^2-2 x-11\right)-15 \tanh ^{-1}\left(\frac{2 x-1}{2 \sqrt{x^2-x-1}}\right)\right)}{48 x \sqrt{x^2-x-1}}","-\frac{\sqrt{-x^4+x^3+x^2} (1-2 x)}{8 x}-\frac{\left(-x^2+x+1\right) \sqrt{-x^4+x^3+x^2}}{3 x}-\frac{5 \sqrt{-x^4+x^3+x^2} \sin ^{-1}\left(\frac{1-2 x}{\sqrt{5}}\right)}{16 x \sqrt{-x^2+x+1}}",1,"(Sqrt[x^2 + x^3 - x^4]*(2*Sqrt[-1 - x + x^2]*(-11 - 2*x + 8*x^2) - 15*ArcTanh[(-1 + 2*x)/(2*Sqrt[-1 - x + x^2])]))/(48*x*Sqrt[-1 - x + x^2])","A",1
935,1,25,25,0.0192203,"\int \frac{1}{\sqrt{\left(a^2+x^2\right)^3}} \, dx","Integrate[1/Sqrt[(a^2 + x^2)^3],x]","\frac{x \left(a^2+x^2\right)}{a^2 \sqrt{\left(a^2+x^2\right)^3}}","\frac{x \left(a^2+x^2\right)}{a^2 \sqrt{\left(a^2+x^2\right)^3}}",1,"(x*(a^2 + x^2))/(a^2*Sqrt[(a^2 + x^2)^3])","A",1
936,1,42,42,0.0113573,"\int \frac{\sqrt{x}}{1+\sqrt{x}+x} \, dx","Integrate[Sqrt[x]/(1 + Sqrt[x] + x),x]","2 \sqrt{x}-\log \left(x+\sqrt{x}+1\right)-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)}{\sqrt{3}}","2 \sqrt{x}-\log \left(x+\sqrt{x}+1\right)-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"2*Sqrt[x] - (2*ArcTan[(1 + 2*Sqrt[x])/Sqrt[3]])/Sqrt[3] - Log[1 + Sqrt[x] + x]","A",1
937,1,32,32,0.0099336,"\int \frac{x}{1+\sqrt{x}+x} \, dx","Integrate[x/(1 + Sqrt[x] + x),x]","x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)}{\sqrt{3}}","x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-2*Sqrt[x] + x + (4*ArcTan[(1 + 2*Sqrt[x])/Sqrt[3]])/Sqrt[3]","A",1
938,1,49,76,0.016427,"\int \frac{1}{\sqrt{x} \left(1+\sqrt{x}+x\right)^{7/2}} \, dx","Integrate[1/(Sqrt[x]*(1 + Sqrt[x] + x)^(7/2)),x]","\frac{4 \left(2 \sqrt{x}+1\right) \left(256 x^{3/2}+128 x^2+432 x+304 \sqrt{x}+203\right)}{405 \left(x+\sqrt{x}+1\right)^{5/2}}","\frac{512 \left(2 \sqrt{x}+1\right)}{405 \sqrt{x+\sqrt{x}+1}}+\frac{64 \left(2 \sqrt{x}+1\right)}{135 \left(x+\sqrt{x}+1\right)^{3/2}}+\frac{4 \left(2 \sqrt{x}+1\right)}{15 \left(x+\sqrt{x}+1\right)^{5/2}}",1,"(4*(1 + 2*Sqrt[x])*(203 + 304*Sqrt[x] + 432*x + 256*x^(3/2) + 128*x^2))/(405*(1 + Sqrt[x] + x)^(5/2))","A",1
939,1,46,46,0.0343815,"\int \frac{-1+x}{1+\sqrt{1+x^2}} \, dx","Integrate[(-1 + x)/(1 + Sqrt[1 + x^2]),x]","\frac{\sqrt{x^2+1}}{x}+\sqrt{x^2+1}-\log \left(\sqrt{x^2+1}+1\right)-\frac{1}{x}-\sinh ^{-1}(x)","\frac{\sqrt{x^2+1}}{x}+\sqrt{x^2+1}-\log \left(\sqrt{x^2+1}+1\right)-\frac{1}{x}-\sinh ^{-1}(x)",1,"-x^(-1) + Sqrt[1 + x^2] + Sqrt[1 + x^2]/x - ArcSinh[x] - Log[1 + Sqrt[1 + x^2]]","A",1
940,1,23,20,0.027167,"\int \frac{1}{(1+x)^{2/3} \left(-1+x^2\right)^{2/3}} \, dx","Integrate[1/((1 + x)^(2/3)*(-1 + x^2)^(2/3)),x]","\frac{3 (x-1) \sqrt[3]{x+1}}{2 \left(x^2-1\right)^{2/3}}","\frac{3 \sqrt[3]{x^2-1}}{2 (x+1)^{2/3}}",1,"(3*(-1 + x)*(1 + x)^(1/3))/(2*(-1 + x^2)^(2/3))","A",1
941,1,18,35,0.0074674,"\int \left(\left(1-x^6\right)^{2/3}+\frac{\left(1-x^6\right)^{2/3}}{x^6}\right) \, dx","Integrate[(1 - x^6)^(2/3) + (1 - x^6)^(2/3)/x^6,x]","-\frac{\left(1-x^6\right)^{5/3}}{5 x^5}","\frac{1}{5} x \left(1-x^6\right)^{2/3}-\frac{\left(1-x^6\right)^{2/3}}{5 x^5}",1,"-1/5*(1 - x^6)^(5/3)/x^5","A",1
942,1,111,15,0.1794655,"\int \frac{x^{-1+m} \left(2 a m+b (2 m-n) x^n\right)}{2 \left(a+b x^n\right)^{3/2}} \, dx","Integrate[(x^(-1 + m)*(2*a*m + b*(2*m - n)*x^n))/(2*(a + b*x^n)^(3/2)),x]","\frac{x^m \sqrt{\frac{b x^n}{a}+1} \left(b (2 m-n) x^n \, _2F_1\left(\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right)+2 a (m+n) \, _2F_1\left(\frac{3}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right)\right)}{2 a (m+n) \sqrt{a+b x^n}}","\frac{x^m}{\sqrt{a+b x^n}}",1,"(x^m*Sqrt[1 + (b*x^n)/a]*(2*a*(m + n)*Hypergeometric2F1[3/2, m/n, (m + n)/n, -((b*x^n)/a)] + b*(2*m - n)*x^n*Hypergeometric2F1[3/2, (m + n)/n, 2 + m/n, -((b*x^n)/a)]))/(2*a*(m + n)*Sqrt[a + b*x^n])","C",1
943,1,28,53,0.0258057,"\int \frac{x-2 x^3}{\sqrt{2+3 x}} \, dx","Integrate[(x - 2*x^3)/Sqrt[2 + 3*x],x]","-\frac{2 \sqrt{3 x+2} \left(270 x^3-216 x^2-123 x+164\right)}{2835}","-\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2}",1,"(-2*Sqrt[2 + 3*x]*(164 - 123*x - 216*x^2 + 270*x^3))/2835","A",1
944,1,31,31,0.0115314,"\int \frac{1}{\sqrt[4]{1+x}+\sqrt{1+x}} \, dx","Integrate[((1 + x)^(1/4) + Sqrt[1 + x])^(-1),x]","2 \sqrt{x+1}-4 \sqrt[4]{x+1}+4 \log \left(\sqrt[4]{x+1}+1\right)","2 \sqrt{x+1}-4 \sqrt[4]{x+1}+4 \log \left(\sqrt[4]{x+1}+1\right)",1,"-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]","A",1
945,1,11,11,0.0059058,"\int \frac{1+2 x}{\sqrt{x+x^2}} \, dx","Integrate[(1 + 2*x)/Sqrt[x + x^2],x]","2 \sqrt{x (x+1)}","2 \sqrt{x^2+x}",1,"2*Sqrt[x*(1 + x)]","A",1
946,1,6,6,0.0031455,"\int \frac{1}{2 \sqrt{x} (1+x)} \, dx","Integrate[1/(2*Sqrt[x]*(1 + x)),x]","\tan ^{-1}\left(\sqrt{x}\right)","\tan ^{-1}\left(\sqrt{x}\right)",1,"ArcTan[Sqrt[x]]","A",1
947,1,17,20,0.0062305,"\int \frac{1}{x \sqrt{6 x-x^2}} \, dx","Integrate[1/(x*Sqrt[6*x - x^2]),x]","\frac{x-6}{3 \sqrt{-((x-6) x)}}","-\frac{\sqrt{6 x-x^2}}{3 x}",1,"(-6 + x)/(3*Sqrt[-((-6 + x)*x)])","A",1
948,1,17,17,0.0022798,"\int \left(1+\sqrt{x}\right) \sqrt{x} \, dx","Integrate[(1 + Sqrt[x])*Sqrt[x],x]","\frac{2 x^{3/2}}{3}+\frac{x^2}{2}","\frac{2 x^{3/2}}{3}+\frac{x^2}{2}",1,"(2*x^(3/2))/3 + x^2/2","A",1
949,1,19,19,0.0038622,"\int \frac{1-\sqrt{x}}{\sqrt[3]{x}} \, dx","Integrate[(1 - Sqrt[x])/x^(1/3),x]","\frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7}","\frac{3 x^{2/3}}{2}-\frac{6 x^{7/6}}{7}",1,"(3*x^(2/3))/2 - (6*x^(7/6))/7","A",1
950,1,41,41,0.0079448,"\int \frac{\sqrt{x}}{1+\sqrt[3]{x}} \, dx","Integrate[Sqrt[x]/(1 + x^(1/3)),x]","\frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt[6]{x}+6 \tan ^{-1}\left(\sqrt[6]{x}\right)","\frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt[6]{x}+6 \tan ^{-1}\left(\sqrt[6]{x}\right)",1,"-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]","A",1
951,1,88,67,0.0195228,"\int \frac{\sqrt[3]{1+\sqrt{x}}}{x} \, dx","Integrate[(1 + Sqrt[x])^(1/3)/x,x]","6 \sqrt[3]{\sqrt{x}+1}+2 \log \left(1-\sqrt[3]{\sqrt{x}+1}\right)-\log \left(\left(\sqrt{x}+1\right)^{2/3}+\sqrt[3]{\sqrt{x}+1}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{\sqrt{x}+1}+1}{\sqrt{3}}\right)","6 \sqrt[3]{\sqrt{x}+1}+3 \log \left(1-\sqrt[3]{\sqrt{x}+1}\right)-\frac{\log (x)}{2}-2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{\sqrt{x}+1}+1}{\sqrt{3}}\right)",1,"6*(1 + Sqrt[x])^(1/3) - 2*Sqrt[3]*ArcTan[(1 + 2*(1 + Sqrt[x])^(1/3))/Sqrt[3]] + 2*Log[1 - (1 + Sqrt[x])^(1/3)] - Log[1 + (1 + Sqrt[x])^(1/3) + (1 + Sqrt[x])^(2/3)]","A",1
952,1,11,11,0.0009802,"\int \left(1-\sqrt{x}\right) \, dx","Integrate[1 - Sqrt[x],x]","x-\frac{2 x^{3/2}}{3}","x-\frac{2 x^{3/2}}{3}",1,"x - (2*x^(3/2))/3","A",1
953,1,11,11,0.0017614,"\int \left(1-\sqrt[4]{x}\right) \, dx","Integrate[1 - x^(1/4),x]","x-\frac{4 x^{5/4}}{5}","x-\frac{4 x^{5/4}}{5}",1,"x - (4*x^(5/4))/5","A",1
954,1,11,11,0.0005146,"\int \frac{1-\sqrt{x}}{1+\sqrt[4]{x}} \, dx","Integrate[(1 - Sqrt[x])/(1 + x^(1/4)),x]","x-\frac{4 x^{5/4}}{5}","x-\frac{4 x^{5/4}}{5}",1,"x - (4*x^(5/4))/5","A",1
955,1,95,61,0.0645483,"\int \frac{1}{\sqrt{(a+b x) (c+d x)}} \, dx","Integrate[1/Sqrt[(a + b*x)*(c + d*x)],x]","\frac{2 \sqrt{a+b x} \sqrt{b c-a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right)}{b \sqrt{d} \sqrt{(a+b x) (c+d x)}}","\frac{\tanh ^{-1}\left(\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (a d+b c)+a c+b d x^2}}\right)}{\sqrt{b} \sqrt{d}}",1,"(2*Sqrt[b*c - a*d]*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(b*Sqrt[d]*Sqrt[(a + b*x)*(c + d*x)])","A",1
956,1,94,65,0.076547,"\int \frac{1}{\sqrt{(a+b x) (c-d x)}} \, dx","Integrate[1/Sqrt[(a + b*x)*(c - d*x)],x]","\frac{2 \sqrt{a+b x} \sqrt{a d+b c} \sqrt{\frac{b (c-d x)}{a d+b c}} \sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d+b c}}\right)}{b \sqrt{d} \sqrt{(a+b x) (c-d x)}}","-\frac{\tan ^{-1}\left(\frac{-a d+b c-2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (b c-a d)+a c-b d x^2}}\right)}{\sqrt{b} \sqrt{d}}",1,"(2*Sqrt[b*c + a*d]*Sqrt[a + b*x]*Sqrt[(b*(c - d*x))/(b*c + a*d)]*ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c + a*d]])/(b*Sqrt[d]*Sqrt[(a + b*x)*(c - d*x)])","A",1
957,1,13,13,0.0039556,"\int \frac{1}{\sqrt{x} \left(1-x^2\right)} \, dx","Integrate[1/(Sqrt[x]*(1 - x^2)),x]","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)",1,"ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]","A",1
958,1,13,13,0.0034153,"\int \frac{\sqrt{x}}{x-x^3} \, dx","Integrate[Sqrt[x]/(x - x^3),x]","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)","\tan ^{-1}\left(\sqrt{x}\right)+\tanh ^{-1}\left(\sqrt{x}\right)",1,"ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]","A",1
959,1,72,72,0.0960427,"\int \frac{x}{2-\sqrt{3}+\left(1+\sqrt{3}\right) x+x^2} \, dx","Integrate[x/(2 - Sqrt[3] + (1 + Sqrt[3])*x + x^2),x]","\frac{1}{2} \log \left(x^2+\sqrt{3} x+x-\sqrt{3}+2\right)+\frac{\left(1+\sqrt{3}\right) \tanh ^{-1}\left(\frac{2 x+\sqrt{3}+1}{\sqrt{6 \sqrt{3}-4}}\right)}{\sqrt{6 \sqrt{3}-4}}","\frac{1}{2} \log \left(x^2+\left(1+\sqrt{3}\right) x-\sqrt{3}+2\right)+\sqrt{\frac{1}{23} \left(13+8 \sqrt{3}\right)} \tanh ^{-1}\left(\frac{2 x+\sqrt{3}+1}{\sqrt{2 \left(3 \sqrt{3}-2\right)}}\right)",1,"((1 + Sqrt[3])*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[-4 + 6*Sqrt[3]]])/Sqrt[-4 + 6*Sqrt[3]] + Log[2 - Sqrt[3] + x + Sqrt[3]*x + x^2]/2","A",1
960,1,23,37,0.008557,"\int \sqrt{x^2+x^3} \, dx","Integrate[Sqrt[x^2 + x^3],x]","\frac{2 \left(x^2 (x+1)\right)^{3/2} (3 x-2)}{15 x^3}","\frac{2 \left(x^3+x^2\right)^{3/2}}{5 x^2}-\frac{4 \left(x^3+x^2\right)^{3/2}}{15 x^3}",1,"(2*(x^2*(1 + x))^(3/2)*(-2 + 3*x))/(15*x^3)","A",1
961,1,37,12,0.0082882,"\int \frac{1}{(1+x) \sqrt{2 x+x^2}} \, dx","Integrate[1/((1 + x)*Sqrt[2*x + x^2]),x]","\frac{2 \sqrt{x} \sqrt{x+2} \tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+2}}\right)}{\sqrt{x (x+2)}}","\tan ^{-1}\left(\sqrt{x^2+2 x}\right)",1,"(2*Sqrt[x]*Sqrt[2 + x]*ArcTan[Sqrt[x]/Sqrt[2 + x]])/Sqrt[x*(2 + x)]","B",1
962,1,60,95,0.0366535,"\int \sqrt{1-\sqrt{x}-x} \sqrt{x} \, dx","Integrate[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]","\frac{1}{96} \sqrt{-x-\sqrt{x}+1} \left(48 x^{3/2}+8 x-34 \sqrt{x}+67\right)-\frac{45}{64} \sin ^{-1}\left(\frac{-2 \sqrt{x}-1}{\sqrt{5}}\right)","-\frac{1}{2} \sqrt{x} \left(-x-\sqrt{x}+1\right)^{3/2}+\frac{5}{12} \left(-x-\sqrt{x}+1\right)^{3/2}+\frac{9}{32} \left(2 \sqrt{x}+1\right) \sqrt{-x-\sqrt{x}+1}+\frac{45}{64} \sin ^{-1}\left(\frac{2 \sqrt{x}+1}{\sqrt{5}}\right)",1,"(Sqrt[1 - Sqrt[x] - x]*(67 - 34*Sqrt[x] + 8*x + 48*x^(3/2)))/96 - (45*ArcSin[(-1 - 2*Sqrt[x])/Sqrt[5]])/64","A",1
963,1,28,35,0.0106803,"\int \sqrt[3]{1+\sqrt{-3+x}} \, dx","Integrate[(1 + Sqrt[-3 + x])^(1/3),x]","\frac{3}{14} \left(\sqrt{x-3}+1\right)^{4/3} \left(4 \sqrt{x-3}-3\right)","\frac{6}{7} \left(\sqrt{x-3}+1\right)^{7/3}-\frac{3}{2} \left(\sqrt{x-3}+1\right)^{4/3}",1,"(3*(1 + Sqrt[-3 + x])^(4/3)*(-3 + 4*Sqrt[-3 + x]))/14","A",1
964,1,30,37,0.0110258,"\int \frac{1}{\sqrt{3+\sqrt{-1+2 x}}} \, dx","Integrate[1/Sqrt[3 + Sqrt[-1 + 2*x]],x]","\frac{2}{3} \left(\sqrt{2 x-1}-6\right) \sqrt{\sqrt{2 x-1}+3}","\frac{2}{3} \left(\sqrt{2 x-1}+3\right)^{3/2}-6 \sqrt{\sqrt{2 x-1}+3}",1,"(2*(-6 + Sqrt[-1 + 2*x])*Sqrt[3 + Sqrt[-1 + 2*x]])/3","A",1
965,1,26,29,0.0279727,"\int \frac{\sqrt{1-x}}{1+\sqrt{x}} \, dx","Integrate[Sqrt[1 - x]/(1 + Sqrt[x]),x]","\left(\sqrt{x}-2\right) \sqrt{1-x}-\sin ^{-1}\left(\sqrt{x}\right)","-\sqrt{1-x} \left(2-\sqrt{x}\right)-\sin ^{-1}\left(\sqrt{x}\right)",1,"(-2 + Sqrt[x])*Sqrt[1 - x] - ArcSin[Sqrt[x]]","A",1
966,1,26,25,0.0306541,"\int \frac{\sqrt{1-x}}{1-\sqrt{x}} \, dx","Integrate[Sqrt[1 - x]/(1 - Sqrt[x]),x]","\sqrt{1-x} \left(-\sqrt{x}-2\right)+\sin ^{-1}\left(\sqrt{x}\right)","\sin ^{-1}\left(\sqrt{x}\right)-\left(\sqrt{x}+2\right) \sqrt{1-x}",1,"(-2 - Sqrt[x])*Sqrt[1 - x] + ArcSin[Sqrt[x]]","A",1
967,1,21,21,0.0250443,"\int \frac{x}{x-\sqrt{1+x^2}} \, dx","Integrate[x/(x - Sqrt[1 + x^2]),x]","-\frac{x^3}{3}-\frac{1}{3} \left(x^2+1\right)^{3/2}","-\frac{x^3}{3}-\frac{1}{3} \left(x^2+1\right)^{3/2}",1,"-1/3*x^3 - (1 + x^2)^(3/2)/3","A",1
968,1,54,65,0.0542019,"\int \frac{x}{x-\sqrt{1-x^2}} \, dx","Integrate[x/(x - Sqrt[1 - x^2]),x]","\frac{1}{4} \left(2 \left(\sqrt{1-x^2}+x\right)-\sqrt{2} \tanh ^{-1}\left(\sqrt{2-2 x^2}\right)-\sqrt{2} \tanh ^{-1}\left(\sqrt{2} x\right)\right)","\frac{\sqrt{1-x^2}}{2}-\frac{\tanh ^{-1}\left(\sqrt{2} \sqrt{1-x^2}\right)}{2 \sqrt{2}}+\frac{x}{2}-\frac{\tanh ^{-1}\left(\sqrt{2} x\right)}{2 \sqrt{2}}",1,"(2*(x + Sqrt[1 - x^2]) - Sqrt[2]*ArcTanh[Sqrt[2]*x] - Sqrt[2]*ArcTanh[Sqrt[2 - 2*x^2]])/4","A",1
969,1,31,31,0.0360596,"\int \frac{x}{x-\sqrt{1+2 x^2}} \, dx","Integrate[x/(x - Sqrt[1 + 2*x^2]),x]","-\sqrt{2 x^2+1}+\tan ^{-1}\left(\sqrt{2 x^2+1}\right)-x+\tan ^{-1}(x)","-\sqrt{2 x^2+1}+\tan ^{-1}\left(\sqrt{2 x^2+1}\right)-x+\tan ^{-1}(x)",1,"-x - Sqrt[1 + 2*x^2] + ArcTan[x] + ArcTan[Sqrt[1 + 2*x^2]]","A",1
970,1,58,82,0.0665508,"\int \sqrt{x} \sqrt{\sqrt{x}+x} \, dx","Integrate[Sqrt[x]*Sqrt[Sqrt[x] + x],x]","\frac{1}{96} \sqrt{x+\sqrt{x}} \left(48 x^{3/2}+8 x-10 \sqrt{x}-\frac{15 \sinh ^{-1}\left(\sqrt[4]{x}\right)}{\sqrt{\sqrt{x}+1} \sqrt[4]{x}}+15\right)","\frac{1}{2} \sqrt{x} \left(x+\sqrt{x}\right)^{3/2}-\frac{5}{12} \left(x+\sqrt{x}\right)^{3/2}+\frac{5}{32} \left(2 \sqrt{x}+1\right) \sqrt{x+\sqrt{x}}-\frac{5}{32} \tanh ^{-1}\left(\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right)",1,"(Sqrt[Sqrt[x] + x]*(15 - 10*Sqrt[x] + 8*x + 48*x^(3/2) - (15*ArcSinh[x^(1/4)])/(Sqrt[1 + Sqrt[x]]*x^(1/4))))/96","A",1
971,1,74,74,0.0378098,"\int \frac{1+\sqrt[3]{x}}{1+\sqrt{x}} \, dx","Integrate[(1 + x^(1/3))/(1 + Sqrt[x]),x]","\frac{6 x^{5/6}}{5}+2 \sqrt{x}-3 \sqrt[3]{x}-4 \log \left(\sqrt[6]{x}+1\right)-\log \left(\sqrt[3]{x}-\sqrt[6]{x}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \sqrt[6]{x}}{\sqrt{3}}\right)","\frac{6 x^{5/6}}{5}+2 \sqrt{x}-3 \sqrt[3]{x}-4 \log \left(\sqrt[6]{x}+1\right)-\log \left(\sqrt[3]{x}-\sqrt[6]{x}+1\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \sqrt[6]{x}}{\sqrt{3}}\right)",1,"-3*x^(1/3) + 2*Sqrt[x] + (6*x^(5/6))/5 - 2*Sqrt[3]*ArcTan[(1 - 2*x^(1/6))/Sqrt[3]] - 4*Log[1 + x^(1/6)] - Log[1 - x^(1/6) + x^(1/3)]","A",1
972,1,123,115,0.0866564,"\int \frac{1+\sqrt[3]{x}}{1+\sqrt[4]{x}} \, dx","Integrate[(1 + x^(1/3))/(1 + x^(1/4)),x]","\frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt[3]{x}+4 \sqrt[4]{x}+12 \sqrt[12]{x}+4 \left(\sqrt[3]{-1}-1\right) \log \left(\sqrt[3]{-1}-\sqrt[12]{x}\right)-4 \left(1+(-1)^{2/3}\right) \log \left(-\sqrt[12]{x}-(-1)^{2/3}\right)-8 \log \left(\sqrt[12]{x}+1\right)","\frac{12 x^{13/12}}{13}-\frac{6 x^{5/6}}{5}+\frac{4 x^{3/4}}{3}+\frac{12 x^{7/12}}{7}-2 \sqrt{x}-3 \sqrt[3]{x}+4 \sqrt[4]{x}+12 \sqrt[12]{x}-8 \log \left(\sqrt[12]{x}+1\right)-2 \log \left(\sqrt[6]{x}-\sqrt[12]{x}+1\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \sqrt[12]{x}}{\sqrt{3}}\right)",1,"12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*Sqrt[x] + (12*x^(7/12))/7 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(13/12))/13 + 4*(-1 + (-1)^(1/3))*Log[(-1)^(1/3) - x^(1/12)] - 4*(1 + (-1)^(2/3))*Log[-(-1)^(2/3) - x^(1/12)] - 8*Log[1 + x^(1/12)]","A",1
973,1,4,4,0.0625892,"\int \frac{x^2}{-1+x^2+\sqrt{1-x^2}} \, dx","Integrate[x^2/(-1 + x^2 + Sqrt[1 - x^2]),x]","x+\sin ^{-1}(x)","x+\sin ^{-1}(x)",1,"x + ArcSin[x]","A",1
974,1,22,22,0.009768,"\int \sqrt{\frac{1+x}{x}} \, dx","Integrate[Sqrt[(1 + x)/x],x]","\sqrt{\frac{1}{x}+1} x+\tanh ^{-1}\left(\sqrt{\frac{1}{x}+1}\right)","\sqrt{\frac{1}{x}+1} x+\tanh ^{-1}\left(\sqrt{\frac{1}{x}+1}\right)",1,"Sqrt[1 + x^(-1)]*x + ArcTanh[Sqrt[1 + x^(-1)]]","A",1
975,1,24,24,0.0082373,"\int \sqrt{\frac{1-x}{x}} \, dx","Integrate[Sqrt[(1 - x)/x],x]","\sqrt{\frac{1}{x}-1} x-\tan ^{-1}\left(\sqrt{\frac{1}{x}-1}\right)","\sqrt{\frac{1}{x}-1} x-\tan ^{-1}\left(\sqrt{\frac{1}{x}-1}\right)",1,"Sqrt[-1 + x^(-1)]*x - ArcTan[Sqrt[-1 + x^(-1)]]","A",1
976,1,38,24,0.0163485,"\int \sqrt{\frac{-1+x}{x}} \, dx","Integrate[Sqrt[(-1 + x)/x],x]","\frac{\sqrt{x} (x-1)+\sqrt{1-x} \sin ^{-1}\left(\sqrt{1-x}\right)}{\sqrt{x-1}}","\sqrt{x-1} \sqrt{x}-\sinh ^{-1}\left(\sqrt{x-1}\right)",1,"((-1 + x)*Sqrt[x] + Sqrt[1 - x]*ArcSin[Sqrt[1 - x]])/Sqrt[-1 + x]","A",1
977,1,24,24,0.0067672,"\int \frac{\sqrt{\frac{1+x}{x}}}{x} \, dx","Integrate[Sqrt[(1 + x)/x]/x,x]","2 \tanh ^{-1}\left(\sqrt{\frac{1}{x}+1}\right)-2 \sqrt{\frac{1}{x}+1}","2 \tanh ^{-1}\left(\sqrt{\frac{1}{x}+1}\right)-2 \sqrt{\frac{1}{x}+1}",1,"-2*Sqrt[1 + x^(-1)] + 2*ArcTanh[Sqrt[1 + x^(-1)]]","A",1
978,1,42,22,0.0199979,"\int \sqrt{\frac{x}{1+x}} \, dx","Integrate[Sqrt[x/(1 + x)],x]","\frac{\sqrt{\frac{x}{x+1}} \left(\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)\right)}{\sqrt{x}}","\sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]","A",1
979,1,43,29,0.0132226,"\int \frac{1}{\sqrt{\frac{-1-x}{x}}} \, dx","Integrate[1/Sqrt[(-1 - x)/x],x]","\frac{\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left(\sqrt{x}\right)}{\sqrt{x} \sqrt{-\frac{x+1}{x}}}","\tan ^{-1}\left(\sqrt{-\frac{x+1}{x}}\right)-x \sqrt{-\frac{x+1}{x}}",1,"(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]])/(Sqrt[x]*Sqrt[-((1 + x)/x)])","A",1
980,1,32,33,0.051265,"\int \sqrt{(4-x) x} \, dx","Integrate[Sqrt[(4 - x)*x],x]","\frac{1}{2} (x-2) \sqrt{-((x-4) x)}-4 \sin ^{-1}\left(\sqrt{1-\frac{x}{4}}\right)","-\frac{1}{2} \sqrt{4 x-x^2} (2-x)-2 \sin ^{-1}\left(1-\frac{x}{2}\right)",1,"((-2 + x)*Sqrt[-((-4 + x)*x)])/2 - 4*ArcSin[Sqrt[1 - x/4]]","A",1
981,1,12,8,0.0083161,"\int \frac{1}{\sqrt{(1-x) x}} \, dx","Integrate[1/Sqrt[(1 - x)*x],x]","-2 \sin ^{-1}\left(\sqrt{1-x}\right)","-\sin ^{-1}(1-2 x)",1,"-2*ArcSin[Sqrt[1 - x]]","A",1
982,1,11,13,0.0035452,"\int \frac{x}{(x (2+x))^{3/2}} \, dx","Integrate[x/(x*(2 + x))^(3/2),x]","\frac{x}{\sqrt{x (x+2)}}","\frac{x}{\sqrt{x^2+2 x}}",1,"x/Sqrt[x*(2 + x)]","A",1
983,1,22,22,0.0275399,"\int \frac{\sqrt{1+\frac{1}{x}}}{1-x^2} \, dx","Integrate[Sqrt[1 + x^(-1)]/(1 - x^2),x]","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right)","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{x}+1}}{\sqrt{2}}\right)",1,"Sqrt[2]*ArcTanh[Sqrt[1 + x^(-1)]/Sqrt[2]]","A",1
984,1,39,24,0.0235145,"\int \frac{1}{1+\sqrt{5}-x^2+\sqrt{5} x^2} \, dx","Integrate[(1 + Sqrt[5] - x^2 + Sqrt[5]*x^2)^(-1),x]","\frac{1}{4} i \log \left(-2 i x+\sqrt{5}+1\right)-\frac{1}{4} i \log \left(2 i x+\sqrt{5}+1\right)","\frac{1}{2} \tan ^{-1}\left(\sqrt{\frac{1}{2} \left(3-\sqrt{5}\right)} x\right)",1,"(I/4)*Log[1 + Sqrt[5] - (2*I)*x] - (I/4)*Log[1 + Sqrt[5] + (2*I)*x]","C",1
985,1,57,28,0.0224352,"\int \frac{1}{\sqrt{a x+b x^2}} \, dx","Integrate[1/Sqrt[a*x + b*x^2],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
986,1,57,28,0.0037269,"\int \frac{1}{\sqrt{x (a+b x)}} \, dx","Integrate[1/Sqrt[x*(a + b*x)],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
987,1,57,28,0.004072,"\int \frac{1}{\sqrt{\left(b+\frac{a}{x}\right) x^2}} \, dx","Integrate[1/Sqrt[(b + a/x)*x^2],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
988,1,57,28,0.00427,"\int \frac{1}{\sqrt{\left(\frac{a}{x^2}+\frac{b}{x}\right) x^3}} \, dx","Integrate[1/Sqrt[(a/x^2 + b/x)*x^3],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
989,1,57,28,0.0044126,"\int \frac{1}{\sqrt{\frac{a x^2+b x^3}{x}}} \, dx","Integrate[1/Sqrt[(a*x^2 + b*x^3)/x],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
990,1,57,28,0.004488,"\int \frac{1}{\sqrt{\frac{a x^3+b x^4}{x^2}}} \, dx","Integrate[1/Sqrt[(a*x^3 + b*x^4)/x^2],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right)}{\sqrt{b}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[x*(a + b*x)])","B",1
991,1,58,40,0.0173618,"\int \frac{1}{\sqrt{a c x+b c x^2}} \, dx","Integrate[1/Sqrt[a*c*x + b*c*x^2],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{c x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right)}{\sqrt{b} \sqrt{c}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])","A",1
992,1,58,40,0.0043749,"\int \frac{1}{\sqrt{c \left(a x+b x^2\right)}} \, dx","Integrate[1/Sqrt[c*(a*x + b*x^2)],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{c x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right)}{\sqrt{b} \sqrt{c}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])","A",1
993,1,58,40,0.0041039,"\int \frac{1}{\sqrt{c x (a+b x)}} \, dx","Integrate[1/Sqrt[c*x*(a + b*x)],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{c x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right)}{\sqrt{b} \sqrt{c}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])","A",1
994,1,58,40,0.0042636,"\int \frac{1}{\sqrt{c \left(b+\frac{a}{x}\right) x^2}} \, dx","Integrate[1/Sqrt[c*(b + a/x)*x^2],x]","\frac{2 \sqrt{a} \sqrt{x} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{c x (a+b x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a c x+b c x^2}}\right)}{\sqrt{b} \sqrt{c}}",1,"(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])","A",1
995,0,0,63,0.026816,"\int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx","Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]","\int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx","\frac{1}{4} \sqrt{-x^2+\sqrt{x^2-1} x+1} \left(\sqrt{x^2-1}+3 x\right)+\frac{3 \sin ^{-1}\left(x-\sqrt{x^2-1}\right)}{4 \sqrt{2}}",1,"Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x]","F",-1
996,1,180,66,0.5250197,"\int \frac{\sqrt{-x+\sqrt{x} \sqrt{1+x}}}{\sqrt{1+x}} \, dx","Integrate[Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]/Sqrt[1 + x],x]","-\frac{(x+1) \left(2 x-2 \sqrt{x+1} \sqrt{x}+1\right)^2 \left(2 \sqrt{\sqrt{x} \sqrt{x+1}-x} \left(-2 x+2 \sqrt{x+1} \sqrt{x}-3\right)+3 \sqrt{-4 x+4 \sqrt{x+1} \sqrt{x}-2} \log \left(2 \sqrt{\sqrt{x} \sqrt{x+1}-x}+\sqrt{-4 x+4 \sqrt{x+1} \sqrt{x}-2}\right)\right)}{4 \left(\sqrt{x+1}-\sqrt{x}\right)^3 \left(x-\sqrt{x+1} \sqrt{x}+1\right)^2}","\frac{1}{2} \left(\sqrt{x}+3 \sqrt{x+1}\right) \sqrt{\sqrt{x} \sqrt{x+1}-x}-\frac{3 \sin ^{-1}\left(\sqrt{x}-\sqrt{x+1}\right)}{2 \sqrt{2}}",1,"-1/4*((1 + x)*(1 + 2*x - 2*Sqrt[x]*Sqrt[1 + x])^2*(2*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]*(-3 - 2*x + 2*Sqrt[x]*Sqrt[1 + x]) + 3*Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]*Log[2*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]] + Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]]))/((-Sqrt[x] + Sqrt[1 + x])^3*(1 + x - Sqrt[x]*Sqrt[1 + x])^2)","B",1
997,1,34,78,0.4269873,"\int -\frac{x+2 \sqrt{1+x^2}}{x+x^3+\sqrt{1+x^2}} \, dx","Integrate[-((x + 2*Sqrt[1 + x^2])/(x + x^3 + Sqrt[1 + x^2])),x]","-\int \frac{2 \sqrt{x^2+1}+x}{x^3+\sqrt{x^2+1}+x} \, dx","\sqrt{2 \left(\sqrt{5}-1\right)} \tanh ^{-1}\left(\sqrt{2+\sqrt{5}} \left(\sqrt{x^2+1}+x\right)\right)-\sqrt{2 \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\sqrt{5}-2} \left(\sqrt{x^2+1}+x\right)\right)",1,"-Integrate[(x + 2*Sqrt[1 + x^2])/(x + x^3 + Sqrt[1 + x^2]), x]","F",1
998,1,87,126,0.0358011,"\int \frac{1+2 x}{\left(1+x^2\right) \sqrt{2+2 x+x^2}} \, dx","Integrate[(1 + 2*x)/((1 + x^2)*Sqrt[2 + 2*x + x^2]),x]","\frac{1}{2} i \left(\sqrt{1+2 i} \tanh ^{-1}\left(\frac{(1+i) x+(2+i)}{\sqrt{1+2 i} \sqrt{x^2+2 x+2}}\right)-\sqrt{1-2 i} \tanh ^{-1}\left(\frac{(2-2 i) x+(4-2 i)}{2 \sqrt{1-2 i} \sqrt{x^2+2 x+2}}\right)\right)","-\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{2 \sqrt{5}-\left(5+\sqrt{5}\right) x}{\sqrt{10 \left(1+\sqrt{5}\right)} \sqrt{x^2+2 x+2}}\right)-\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)} \tanh ^{-1}\left(\frac{\left(5-\sqrt{5}\right) x+2 \sqrt{5}}{\sqrt{10 \left(\sqrt{5}-1\right)} \sqrt{x^2+2 x+2}}\right)",1,"(I/2)*(Sqrt[1 + 2*I]*ArcTanh[((2 + I) + (1 + I)*x)/(Sqrt[1 + 2*I]*Sqrt[2 + 2*x + x^2])] - Sqrt[1 - 2*I]*ArcTanh[((4 - 2*I) + (2 - 2*I)*x)/(2*Sqrt[1 - 2*I]*Sqrt[2 + 2*x + x^2])])","C",1
999,1,24,22,1.0582417,"\int \frac{1}{\left(1+x^4\right) \sqrt{-x^2+\sqrt{1+x^4}}} \, dx","Integrate[1/((1 + x^4)*Sqrt[-x^2 + Sqrt[1 + x^4]]),x]","\cot ^{-1}\left(\frac{\sqrt{\sqrt{x^4+1}-x^2}}{x}\right)","\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right)",1,"ArcCot[Sqrt[-x^2 + Sqrt[1 + x^4]]/x]","A",1
1000,1,50,40,0.4885676,"\int \frac{1}{\left(a+b x^4\right) \sqrt{c x^2+d \sqrt{a+b x^4}}} \, dx","Integrate[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]","\frac{\sqrt{-\frac{1}{c}} \cot ^{-1}\left(\frac{\sqrt{-\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}+c x^2}}{x}\right)}{a}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right)}{a \sqrt{c}}",1,"(Sqrt[-c^(-1)]*ArcCot[(Sqrt[-c^(-1)]*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]])/x])/a","A",1
1001,1,47,41,0.5637058,"\int \frac{1}{\left(a+b x^4\right) \sqrt{-c x^2+d \sqrt{a+b x^4}}} \, dx","Integrate[1/((a + b*x^4)*Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]]),x]","\frac{\sqrt{\frac{1}{c}} \cot ^{-1}\left(\frac{\sqrt{\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}-c x^2}}{x}\right)}{a}","\frac{\tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}-c x^2}}\right)}{a \sqrt{c}}",1,"(Sqrt[c^(-1)]*ArcCot[(Sqrt[c^(-1)]*Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]])/x])/a","A",1
1002,1,330,184,0.5832441,"\int \frac{x}{\sqrt{a+b c^4+4 b c^3 d x+6 b c^2 d^2 x^2+4 b c d^3 x^3+b d^4 x^4}} \, dx","Integrate[x/Sqrt[a + b*c^4 + 4*b*c^3*d*x + 6*b*c^2*d^2*x^2 + 4*b*c*d^3*x^3 + b*d^4*x^4],x]","\frac{\sqrt[4]{-1} \sqrt{2} \sqrt{-\frac{i \left(\sqrt[4]{-1} \sqrt[4]{a}+\sqrt[4]{b} (c+d x)\right)}{\sqrt[4]{-1} \sqrt[4]{a}-\sqrt[4]{b} (c+d x)}} \left(\sqrt{b} (c+d x)^2+i \sqrt{a}\right) \left(\left(\sqrt[4]{-1} \sqrt[4]{a}-\sqrt[4]{b} c\right) F\left(\left.\sin ^{-1}\left(\sqrt{-\frac{i \left(\sqrt[4]{b} (c+d x)+\sqrt[4]{-1} \sqrt[4]{a}\right)}{\sqrt[4]{-1} \sqrt[4]{a}-\sqrt[4]{b} (c+d x)}}\right)\right|-1\right)-2 \sqrt[4]{-1} \sqrt[4]{a} \Pi \left(-i;\left.\sin ^{-1}\left(\sqrt{-\frac{i \left(\sqrt[4]{b} (c+d x)+\sqrt[4]{-1} \sqrt[4]{a}\right)}{\sqrt[4]{-1} \sqrt[4]{a}-\sqrt[4]{b} (c+d x)}}\right)\right|-1\right)\right)}{\sqrt[4]{a} \sqrt{b} d^2 \sqrt{\frac{\sqrt{b} (c+d x)^2+i \sqrt{a}}{\left(\sqrt[4]{-1} \sqrt[4]{a}-\sqrt[4]{b} (c+d x)\right)^2}} \sqrt{a+b (c+d x)^4}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} d^2 \left(\frac{c}{d}+x\right)^2}{\sqrt{a+b d^4 \left(\frac{c}{d}+x\right)^4}}\right)}{2 \sqrt{b} d^2}-\frac{c \left(\sqrt{a}+\sqrt{b} d^2 \left(\frac{c}{d}+x\right)^2\right) \sqrt{\frac{a+b d^4 \left(\frac{c}{d}+x\right)^4}{\left(\sqrt{a}+\sqrt{b} d^2 \left(\frac{c}{d}+x\right)^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} d^2 \sqrt{a+b d^4 \left(\frac{c}{d}+x\right)^4}}",1,"((-1)^(1/4)*Sqrt[2]*Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*(I*Sqrt[a] + Sqrt[b]*(c + d*x)^2)*(((-1)^(1/4)*a^(1/4) - b^(1/4)*c)*EllipticF[ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]], -1] - 2*(-1)^(1/4)*a^(1/4)*EllipticPi[-I, ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]], -1]))/(a^(1/4)*Sqrt[b]*d^2*Sqrt[(I*Sqrt[a] + Sqrt[b]*(c + d*x)^2)/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))^2]*Sqrt[a + b*(c + d*x)^4])","C",1
1003,1,90,131,0.0589525,"\int \frac{1}{\sqrt{a+b c^4+4 b c^3 d x+6 b c^2 d^2 x^2+4 b c d^3 x^3+b d^4 x^4}} \, dx","Integrate[1/Sqrt[a + b*c^4 + 4*b*c^3*d*x + 6*b*c^2*d^2*x^2 + 4*b*c*d^3*x^3 + b*d^4*x^4],x]","-\frac{i \sqrt{\frac{a+b (c+d x)^4}{a}} F\left(\left.i \sinh ^{-1}\left(\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} (c+d x)\right)\right|-1\right)}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b (c+d x)^4}}","\frac{\left(\sqrt{a}+\sqrt{b} d^2 \left(\frac{c}{d}+x\right)^2\right) \sqrt{\frac{a+b d^4 \left(\frac{c}{d}+x\right)^4}{\left(\sqrt{a}+\sqrt{b} d^2 \left(\frac{c}{d}+x\right)^2\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt[4]{b} d \sqrt{a+b d^4 \left(\frac{c}{d}+x\right)^4}}",1,"((-I)*Sqrt[(a + b*(c + d*x)^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*(c + d*x)], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*d*Sqrt[a + b*(c + d*x)^4])","C",1
1004,1,419,54,1.9638744,"\int \frac{a-c x^4}{\sqrt{a+b x^2+c x^4} \left(a d+a e x^2+c d x^4\right)} \, dx","Integrate[(a - c*x^4)/(Sqrt[a + b*x^2 + c*x^4]*(a*d + a*e*x^2 + c*d*x^4)),x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(-\Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) d}{a e-\sqrt{a} \sqrt{a e^2-4 c d^2}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-\Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) d}{a e+\sqrt{a} \sqrt{a e^2-4 c d^2}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} d \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\tanh ^{-1}\left(\frac{x \sqrt{b d-a e}}{\sqrt{d} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{d} \sqrt{b d-a e}}",1,"(I*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(a*e - Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(a*e + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a + b*x^2 + c*x^4])","C",1
1005,1,416,53,1.450497,"\int \frac{a-c x^4}{\sqrt{a-b x^2+c x^4} \left(a d+a e x^2+c d x^4\right)} \, dx","Integrate[(a - c*x^4)/(Sqrt[a - b*x^2 + c*x^4]*(a*d + a*e*x^2 + c*d*x^4)),x]","\frac{i \sqrt{\frac{4 c x^2}{\sqrt{b^2-4 a c}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left(-\Pi \left(\frac{\left(b-\sqrt{b^2-4 a c}\right) d}{\sqrt{a} \sqrt{a e^2-4 c d^2}-a e};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)-\Pi \left(\frac{\left(\sqrt{b^2-4 a c}-b\right) d}{a e+\sqrt{a} \sqrt{a e^2-4 c d^2}};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)+F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right)|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right)\right)}{2 d \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} \sqrt{a-b x^2+c x^4}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{a e+b d}}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right)}{\sqrt{d} \sqrt{a e+b d}}",1,"((I/2)*Sqrt[2 + (4*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])] - EllipticPi[((b - Sqrt[b^2 - 4*a*c])*d)/(-(a*e) + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])] - EllipticPi[((-b + Sqrt[b^2 - 4*a*c])*d)/(a*e + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]))/(Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a - b*x^2 + c*x^4])","C",1
1006,1,160,84,0.3317122,"\int \frac{1}{\sqrt{5-2 x+x^2} \left(8+x^3\right)} \, dx","Integrate[1/(Sqrt[5 - 2*x + x^2]*(8 + x^3)),x]","\frac{1}{312} \left(-2 \sqrt{13} \tanh ^{-1}\left(\frac{7-3 x}{\sqrt{13} \sqrt{x^2-2 x+5}}\right)-13 \left(\left(\sqrt{3}+i\right) \tan ^{-1}\left(\frac{-2 \sqrt[3]{-1} x+4 x+5 i \sqrt{3}+1}{\sqrt{2-2 i \sqrt{3}} \sqrt{x^2-2 x+5}}\right)+\left(\sqrt{3}-i\right) \tan ^{-1}\left(\frac{2 \left(2+(-1)^{2/3}\right) x-5 i \sqrt{3}+1}{\sqrt{2+2 i \sqrt{3}} \sqrt{x^2-2 x+5}}\right)\right)\right)","-\frac{\tan ^{-1}\left(\frac{1-x}{\sqrt{3} \sqrt{x^2-2 x+5}}\right)}{4 \sqrt{3}}-\frac{\tanh ^{-1}\left(\frac{7-3 x}{\sqrt{13} \sqrt{x^2-2 x+5}}\right)}{12 \sqrt{13}}+\frac{1}{12} \tanh ^{-1}\left(\sqrt{x^2-2 x+5}\right)",1,"(-13*((I + Sqrt[3])*ArcTan[(1 + (5*I)*Sqrt[3] + 4*x - 2*(-1)^(1/3)*x)/(Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[5 - 2*x + x^2])] + (-I + Sqrt[3])*ArcTan[(1 - (5*I)*Sqrt[3] + 2*(2 + (-1)^(2/3))*x)/(Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[5 - 2*x + x^2])]) - 2*Sqrt[13]*ArcTanh[(7 - 3*x)/(Sqrt[13]*Sqrt[5 - 2*x + x^2])])/312","C",1
1007,1,17,20,0.005467,"\int \sqrt{\frac{x^2}{1+x^2}} \, dx","Integrate[Sqrt[x^2/(1 + x^2)],x]","\frac{x}{\sqrt{\frac{x^2}{x^2+1}}}","\frac{\sqrt{x^2} \sqrt{x^2+1}}{x}",1,"x/Sqrt[x^2/(1 + x^2)]","A",1
1008,1,38,46,0.0118705,"\int \sqrt{\frac{x^n}{1+x^n}} \, dx","Integrate[Sqrt[x^n/(1 + x^n)],x]","\frac{2 x \sqrt{x^n} \, _2F_1\left(\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-x^n\right)}{n+2}","\frac{2 x \sqrt{x^n} \, _2F_1\left(\frac{1}{2},\frac{1}{2} \left(1+\frac{2}{n}\right);\frac{1}{2} \left(3+\frac{2}{n}\right);-x^n\right)}{n+2}",1,"(2*x*Sqrt[x^n]*Hypergeometric2F1[1/2, 1/2 + n^(-1), 3/2 + n^(-1), -x^n])/(2 + n)","A",1
1009,1,13884,88,6.5177092,"\int \frac{e f-e f x^2}{\left(a d+b d x+a d x^2\right) \sqrt{a+b x+c x^2+b x^3+a x^4}} \, dx","Integrate[(e*f - e*f*x^2)/((a*d + b*d*x + a*d*x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]","\text{Result too large to show}","\frac{e f \tan ^{-1}\left(\frac{x \left(4 a^2-2 a c+b^2\right)+a b x^2+a b}{2 a \sqrt{2 a-c} \sqrt{a x^4+a+b x^3+b x+c x^2}}\right)}{a d \sqrt{2 a-c}}",1,"Result too large to show","C",0
1010,1,15147,88,6.5461748,"\int \frac{e f-e f x^2}{\left(-a d+b d x-a d x^2\right) \sqrt{-a+b x+c x^2+b x^3-a x^4}} \, dx","Integrate[(e*f - e*f*x^2)/((-(a*d) + b*d*x - a*d*x^2)*Sqrt[-a + b*x + c*x^2 + b*x^3 - a*x^4]),x]","\text{Result too large to show}","\frac{e f \tanh ^{-1}\left(\frac{-x \left(4 a^2+2 a c+b^2\right)+a b x^2+a b}{2 a \sqrt{2 a+c} \sqrt{-a x^4-a+b x^3+b x+c x^2}}\right)}{a d \sqrt{2 a+c}}",1,"Result too large to show","C",0
1011,1,148,46,1.1182121,"\int \frac{\sqrt{a x^2+b x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","Integrate[Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]),x]","\frac{\sqrt{2} x \sqrt{a x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)} \left(b x \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x^2-1\right) \tanh ^{-1}\left(\frac{\sqrt{a x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)}}{\sqrt{2} a x}\right)}{\sqrt{\frac{a \left(a x^2-1\right)}{b^2}} \left(x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)\right)^{3/2}}","\frac{\sqrt{2} b \sinh ^{-1}\left(\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*x*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*ArcTanh[Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)])/(Sqrt[(a*(-1 + a*x^2))/b^2]*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(3/2))","B",1
1012,1,161,46,1.185825,"\int \frac{\sqrt{-a x^2+b x \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}}}{x \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","Integrate[Sqrt[-(a*x^2) + b*x*Sqrt[a/b^2 + (a^2*x^2)/b^2]]/(x*Sqrt[a/b^2 + (a^2*x^2)/b^2]),x]","\frac{\sqrt{2} b^2 \sqrt{\frac{a \left(a x^2+1\right)}{b^2}} \sqrt{a x \left(a x-b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}\right)} \sqrt{x \left(b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}-a x\right)} \tanh ^{-1}\left(\frac{\sqrt{a x \left(a x-b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}\right)}}{\sqrt{2} a x}\right)}{a^2 \left(-b x^2 \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}+a x^3+x\right)}","\frac{\sqrt{2} b \sin ^{-1}\left(\frac{a x-b \sqrt{\frac{a^2 x^2}{b^2}+\frac{a}{b^2}}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*b^2*Sqrt[(a*(1 + a*x^2))/b^2]*Sqrt[a*x*(a*x - b*Sqrt[(a*(1 + a*x^2))/b^2])]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(1 + a*x^2))/b^2])]*ArcTanh[Sqrt[a*x*(a*x - b*Sqrt[(a*(1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)])/(a^2*(x + a*x^3 - b*x^2*Sqrt[(a*(1 + a*x^2))/b^2]))","B",1
1013,1,148,46,0.1394244,"\int \frac{\sqrt{x \left(a x+b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)}}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","Integrate[Sqrt[x*(a*x + b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])]/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]),x]","\frac{\sqrt{2} x \sqrt{a x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)} \left(b x \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x^2-1\right) \tanh ^{-1}\left(\frac{\sqrt{a x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)}}{\sqrt{2} a x}\right)}{\sqrt{\frac{a \left(a x^2-1\right)}{b^2}} \left(x \left(b \sqrt{\frac{a \left(a x^2-1\right)}{b^2}}+a x\right)\right)^{3/2}}","\frac{\sqrt{2} b \sinh ^{-1}\left(\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*x*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*ArcTanh[Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)])/(Sqrt[(a*(-1 + a*x^2))/b^2]*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(3/2))","B",1
1014,1,161,46,0.1881542,"\int \frac{\sqrt{x \left(-a x+b \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}\right)}}{x \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx","Integrate[Sqrt[x*(-(a*x) + b*Sqrt[a/b^2 + (a^2*x^2)/b^2])]/(x*Sqrt[a/b^2 + (a^2*x^2)/b^2]),x]","\frac{\sqrt{2} b^2 \sqrt{\frac{a \left(a x^2+1\right)}{b^2}} \sqrt{a x \left(a x-b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}\right)} \sqrt{x \left(b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}-a x\right)} \tanh ^{-1}\left(\frac{\sqrt{a x \left(a x-b \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}\right)}}{\sqrt{2} a x}\right)}{a^2 \left(-b x^2 \sqrt{\frac{a \left(a x^2+1\right)}{b^2}}+a x^3+x\right)}","\frac{\sqrt{2} b \sin ^{-1}\left(\frac{a x-b \sqrt{\frac{a^2 x^2}{b^2}+\frac{a}{b^2}}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*b^2*Sqrt[(a*(1 + a*x^2))/b^2]*Sqrt[a*x*(a*x - b*Sqrt[(a*(1 + a*x^2))/b^2])]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(1 + a*x^2))/b^2])]*ArcTanh[Sqrt[a*x*(a*x - b*Sqrt[(a*(1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)])/(a^2*(x + a*x^3 - b*x^2*Sqrt[(a*(1 + a*x^2))/b^2]))","B",1
1015,1,75,19,1.3762964,"\int \frac{-\sqrt{-4+x}-4 \sqrt{-1+x}+\sqrt{-4+x} x+\sqrt{-1+x} x}{\left(1+\sqrt{-4+x}+\sqrt{-1+x}\right) \left(4-5 x+x^2\right)} \, dx","Integrate[(-Sqrt[-4 + x] - 4*Sqrt[-1 + x] + Sqrt[-4 + x]*x + Sqrt[-1 + x]*x)/((1 + Sqrt[-4 + x] + Sqrt[-1 + x])*(4 - 5*x + x^2)),x]","\frac{1}{2} \log \left(-5 x-4 \sqrt{x-4} \sqrt{x-1}+17\right)+\frac{1}{2} \log \left(-2 x-2 \sqrt{x-4} \sqrt{x-1}+5\right)-\tanh ^{-1}\left(\sqrt{x-4}\right)+\tanh ^{-1}\left(\frac{\sqrt{x-1}}{2}\right)","2 \log \left(\sqrt{x-4}+\sqrt{x-1}+1\right)",1,"-ArcTanh[Sqrt[-4 + x]] + ArcTanh[Sqrt[-1 + x]/2] + Log[17 - 4*Sqrt[-4 + x]*Sqrt[-1 + x] - 5*x]/2 + Log[5 - 2*Sqrt[-4 + x]*Sqrt[-1 + x] - 2*x]/2","B",1
1016,1,120,90,0.1586826,"\int \frac{1}{x \left(3+3 x+x^2\right) \sqrt[3]{3+3 x+3 x^2+x^3}} \, dx","Integrate[1/(x*(3 + 3*x + x^2)*(3 + 3*x + 3*x^2 + x^3)^(1/3)),x]","\frac{\sqrt{3} \left(2 \log \left(1-\frac{\sqrt[3]{3} (x+1)}{\sqrt[3]{(x+1)^3+2}}\right)-\log \left(\frac{3^{2/3} (x+1)^2}{\left((x+1)^3+2\right)^{2/3}}+\frac{\sqrt[3]{3} (x+1)}{\sqrt[3]{(x+1)^3+2}}+1\right)\right)-6 \tan ^{-1}\left(\frac{2 (x+1)}{\sqrt[6]{3} \sqrt[3]{(x+1)^3+2}}+\frac{1}{\sqrt{3}}\right)}{6\ 3^{5/6}}","-\frac{\log \left(1-(x+1)^3\right)}{6 \sqrt[3]{3}}+\frac{\log \left(\sqrt[3]{3} (x+1)-\sqrt[3]{(x+1)^3+2}\right)}{2 \sqrt[3]{3}}-\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{3} (x+1)}{\sqrt[3]{(x+1)^3+2}}+1}{\sqrt{3}}\right)}{3^{5/6}}",1,"(-6*ArcTan[1/Sqrt[3] + (2*(1 + x))/(3^(1/6)*(2 + (1 + x)^3)^(1/3))] + Sqrt[3]*(2*Log[1 - (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)] - Log[1 + (3^(2/3)*(1 + x)^2)/(2 + (1 + x)^3)^(2/3) + (3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3)]))/(6*3^(5/6))","A",1
1017,0,0,103,0.1859008,"\int \frac{1-x^2}{\left(1-x+x^2\right) \left(1-x^3\right)^{2/3}} \, dx","Integrate[(1 - x^2)/((1 - x + x^2)*(1 - x^3)^(2/3)),x]","\int \frac{1-x^2}{\left(1-x+x^2\right) \left(1-x^3\right)^{2/3}} \, dx","-\frac{\log \left(-x^3+2 (1-x)^3+1\right)}{2\ 2^{2/3}}+\frac{3 \log \left(\sqrt[3]{1-x^3}+\sqrt[3]{2} (1-x)\right)}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} (1-x)}{\sqrt[3]{1-x^3}}}{\sqrt{3}}\right)}{2^{2/3}}",1,"Integrate[(1 - x^2)/((1 - x + x^2)*(1 - x^3)^(2/3)), x]","F",-1
1018,1,46,49,0.0209582,"\int \frac{x^2}{\sqrt{-1+x^4} \left(1+x^4\right)} \, dx","Integrate[x^2/(Sqrt[-1 + x^4]*(1 + x^4)),x]","\frac{x^3 \sqrt{1-x^4} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};x^4,-x^4\right)}{3 \sqrt{x^4-1}}","-\frac{1}{4} \tan ^{-1}\left(\frac{x^2+1}{x \sqrt{x^4-1}}\right)-\frac{1}{4} \tanh ^{-1}\left(\frac{1-x^2}{x \sqrt{x^4-1}}\right)",1,"(x^3*Sqrt[1 - x^4]*AppellF1[3/4, 1/2, 1, 7/4, x^4, -x^4])/(3*Sqrt[-1 + x^4])","C",0
1019,1,383,80,0.9142941,"\int \frac{a-c x^4}{\left(a e+c d x^2\right) \left(d+e x^2\right) \sqrt{a+b x^2+c x^4}} \, dx","Integrate[(a - c*x^4)/((a*e + c*d*x^2)*(d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]","\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left(-\Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) d}{2 a e};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)-\Pi \left(\frac{\left(b+\sqrt{b^2-4 a c}\right) e}{2 c d};i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)+F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)\right)}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}","\frac{\tan ^{-1}\left(\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right)}{\sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}}",1,"(I*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(2*a*e), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])","C",1
1020,1,1,1,0.0002543,"\int \left(x+\frac{1-x^2}{1+x}\right) \, dx","Integrate[x + (1 - x^2)/(1 + x),x]","x","x",1,"x","A",1
1021,1,1932,42,3.9614024,"\int \frac{1}{\frac{1}{x}+\sqrt{1-x^2}} \, dx","Integrate[(x^(-1) + Sqrt[1 - x^2])^(-1),x]","\frac{1}{24} \left(24 \sin ^{-1}(x)-\frac{2 \left(-i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(-7-i \sqrt{3}+8 \sqrt{3} x+i \left(7 i+\sqrt{3}\right) x^2\right)}{-6-2 i \sqrt{3}+3 \left(-i+\sqrt{3}\right) x^3-2 \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+2 i x^2 \left(9 i+\sqrt{3}+i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3 i+11 \sqrt{3}+2 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}+\frac{2 \left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(7-i \sqrt{3}-8 \sqrt{3} x+\left(7+i \sqrt{3}\right) x^2\right)}{-6+2 i \sqrt{3}+3 \left(i+\sqrt{3}\right) x^3-2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}+x^2 \left(-18-2 i \sqrt{3}-2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(-3 i+11 \sqrt{3}+2 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}-\frac{2 \left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(7-i \sqrt{3}+8 \sqrt{3} x+\left(7+i \sqrt{3}\right) x^2\right)}{6-2 i \sqrt{3}+3 \left(i+\sqrt{3}\right) x^3+2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}+2 x^2 \left(9+i \sqrt{3}+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(-3 i+11 \sqrt{3}+2 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}+\frac{2 \left(1+i \sqrt{3}\right) \tanh ^{-1}\left(\frac{x \left(7 i-\sqrt{3}+8 i \sqrt{3} x+\left(7 i+\sqrt{3}\right) x^2\right)}{6+2 i \sqrt{3}+3 \left(-i+\sqrt{3}\right) x^3+2 \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+2 x^2 \left(9-i \sqrt{3}+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3 i+11 \sqrt{3}+2 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-4 i \sqrt{3} \log \left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^2\right)+4 i \sqrt{3} \log \left(\frac{1}{2} i \left(i+\sqrt{3}\right)+x^2\right)-\frac{i \left(-i+\sqrt{3}\right) \log \left(16 \left(1+\sqrt{3} x+x^2\right)^2\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}+\frac{i \left(i+\sqrt{3}\right) \log \left(16 \left(1+\sqrt{3} x+x^2\right)^2\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}+\frac{\left(1-i \sqrt{3}\right) \log \left(\left(4-4 \sqrt{3} x+4 x^2\right)^2\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}+\frac{\left(1+i \sqrt{3}\right) \log \left(\left(4-4 \sqrt{3} x+4 x^2\right)^2\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}+\frac{\left(1+i \sqrt{3}\right) \log \left(3 i+\sqrt{3}-\left(-i+\sqrt{3}\right) x^4+2 i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+5 i x^2 \left(2+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3+5 i \sqrt{3}+3 i \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)+i x^3 \left(3 i+3 \sqrt{3}+\sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(3 i+\sqrt{3}-\left(-i+\sqrt{3}\right) x^4+2 i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+5 i x^2 \left(2+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x^3 \left(3-3 i \sqrt{3}-i \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)-i x \left(-3 i+5 \sqrt{3}+3 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{\sqrt{\frac{1}{6} \left(1-i \sqrt{3}\right)}}+\frac{\left(1-i \sqrt{3}\right) \log \left(-3 i+\sqrt{3}-\left(i+\sqrt{3}\right) x^4-2 i \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}-5 i x^2 \left(2+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3-5 i \sqrt{3}-3 i \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)-i x^3 \left(-3 i+3 \sqrt{3}+\sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}+\frac{i \left(i+\sqrt{3}\right) \log \left(-3 i+\sqrt{3}-\left(i+\sqrt{3}\right) x^4-2 i \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}-5 i x^2 \left(2+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x^3 \left(3+3 i \sqrt{3}+i \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)+i x \left(3 i+5 \sqrt{3}+3 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{\sqrt{\frac{1}{6} \left(1+i \sqrt{3}\right)}}\right)","\sin ^{-1}(x)-\frac{\tan ^{-1}\left(\frac{1+4 x \sqrt{1-x^2}}{\sqrt{3} \left(1-2 x^2\right)}\right)}{\sqrt{3}}",1,"(24*ArcSin[x] - (2*(-I + Sqrt[3])*ArcTan[(x*(-7 - I*Sqrt[3] + 8*Sqrt[3]*x + I*(7*I + Sqrt[3])*x^2))/(-6 - (2*I)*Sqrt[3] + 3*(-I + Sqrt[3])*x^3 - 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (2*I)*x^2*(9*I + Sqrt[3] + I*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3*I + 11*Sqrt[3] + 2*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/Sqrt[(1 - I*Sqrt[3])/6] + (2*(I + Sqrt[3])*ArcTan[(x*(7 - I*Sqrt[3] - 8*Sqrt[3]*x + (7 + I*Sqrt[3])*x^2))/(-6 + (2*I)*Sqrt[3] + 3*(I + Sqrt[3])*x^3 - 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + x^2*(-18 - (2*I)*Sqrt[3] - 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(-3*I + 11*Sqrt[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/Sqrt[(1 + I*Sqrt[3])/6] - (2*(I + Sqrt[3])*ArcTan[(x*(7 - I*Sqrt[3] + 8*Sqrt[3]*x + (7 + I*Sqrt[3])*x^2))/(6 - (2*I)*Sqrt[3] + 3*(I + Sqrt[3])*x^3 + 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + 2*x^2*(9 + I*Sqrt[3] + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(-3*I + 11*Sqrt[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/Sqrt[(1 + I*Sqrt[3])/6] + (2*(1 + I*Sqrt[3])*ArcTanh[(x*(7*I - Sqrt[3] + (8*I)*Sqrt[3]*x + (7*I + Sqrt[3])*x^2))/(6 + (2*I)*Sqrt[3] + 3*(-I + Sqrt[3])*x^3 + 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + 2*x^2*(9 - I*Sqrt[3] + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3*I + 11*Sqrt[3] + 2*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/Sqrt[(1 - I*Sqrt[3])/6] - (4*I)*Sqrt[3]*Log[-1/2 - (I/2)*Sqrt[3] + x^2] + (4*I)*Sqrt[3]*Log[(I/2)*(I + Sqrt[3]) + x^2] - (I*(-I + Sqrt[3])*Log[16*(1 + Sqrt[3]*x + x^2)^2])/Sqrt[(1 - I*Sqrt[3])/6] + (I*(I + Sqrt[3])*Log[16*(1 + Sqrt[3]*x + x^2)^2])/Sqrt[(1 + I*Sqrt[3])/6] + ((1 - I*Sqrt[3])*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2])/Sqrt[(1 + I*Sqrt[3])/6] + ((1 + I*Sqrt[3])*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2])/Sqrt[(1 - I*Sqrt[3])/6] + ((1 + I*Sqrt[3])*Log[3*I + Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 + (5*I)*Sqrt[3] + (3*I)*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) + I*x^3*(3*I + 3*Sqrt[3] + Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[(1 - I*Sqrt[3])/6] - (I*(-I + Sqrt[3])*Log[3*I + Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 - (3*I)*Sqrt[3] - I*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x*(-3*I + 5*Sqrt[3] + 3*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[(1 - I*Sqrt[3])/6] + ((1 - I*Sqrt[3])*Log[-3*I + Sqrt[3] - (I + Sqrt[3])*x^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 - (5*I)*Sqrt[3] - (3*I)*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x^3*(-3*I + 3*Sqrt[3] + Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[(1 + I*Sqrt[3])/6] + (I*(I + Sqrt[3])*Log[-3*I + Sqrt[3] - (I + Sqrt[3])*x^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 + (3*I)*Sqrt[3] + I*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) + I*x*(3*I + 5*Sqrt[3] + 3*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[(1 + I*Sqrt[3])/6])/24","C",1
1022,1,1910,42,0.9254888,"\int \frac{x \sqrt{1-x^2}}{x-x^3+\sqrt{1-x^2}} \, dx","Integrate[(x*Sqrt[1 - x^2])/(x - x^3 + Sqrt[1 - x^2]),x]","\sin ^{-1}(x)-\frac{\left(-i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(-7-i \sqrt{3}+8 \sqrt{3} x+i \left(7 i+\sqrt{3}\right) x^2\right)}{-6-2 i \sqrt{3}+3 \left(-i+\sqrt{3}\right) x^3-2 \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+2 i x^2 \left(9 i+\sqrt{3}+i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3 i+11 \sqrt{3}+2 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{2 \sqrt{6-6 i \sqrt{3}}}+\frac{\left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(7-i \sqrt{3}-8 \sqrt{3} x+\left(7+i \sqrt{3}\right) x^2\right)}{-6+2 i \sqrt{3}+3 \left(i+\sqrt{3}\right) x^3-2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}+x^2 \left(-18-2 i \sqrt{3}-2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(-3 i+11 \sqrt{3}+2 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{2 \sqrt{6+6 i \sqrt{3}}}-\frac{\left(i+\sqrt{3}\right) \tan ^{-1}\left(\frac{x \left(7-i \sqrt{3}+8 \sqrt{3} x+\left(7+i \sqrt{3}\right) x^2\right)}{6-2 i \sqrt{3}+3 \left(i+\sqrt{3}\right) x^3+2 \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}+2 x^2 \left(9+i \sqrt{3}+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(-3 i+11 \sqrt{3}+2 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{2 \sqrt{6+6 i \sqrt{3}}}+\frac{\left(1+i \sqrt{3}\right) \tanh ^{-1}\left(\frac{x \left(7 i-\sqrt{3}+8 i \sqrt{3} x+\left(7 i+\sqrt{3}\right) x^2\right)}{6+2 i \sqrt{3}+3 \left(-i+\sqrt{3}\right) x^3+2 \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+2 x^2 \left(9-i \sqrt{3}+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3 i+11 \sqrt{3}+2 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)}\right)}{2 \sqrt{6-6 i \sqrt{3}}}-\frac{i \log \left(-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^2\right)}{2 \sqrt{3}}+\frac{i \log \left(\frac{1}{2} i \left(i+\sqrt{3}\right)+x^2\right)}{2 \sqrt{3}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(16 \left(1+\sqrt{3} x+x^2\right)^2\right)}{4 \sqrt{6-6 i \sqrt{3}}}+\frac{i \left(i+\sqrt{3}\right) \log \left(16 \left(1+\sqrt{3} x+x^2\right)^2\right)}{4 \sqrt{6+6 i \sqrt{3}}}+\frac{\left(1+i \sqrt{3}\right) \log \left(\left(4-4 \sqrt{3} x+4 x^2\right)^2\right)}{4 \sqrt{6-6 i \sqrt{3}}}+\frac{\left(1-i \sqrt{3}\right) \log \left(\left(4-4 \sqrt{3} x+4 x^2\right)^2\right)}{4 \sqrt{6+6 i \sqrt{3}}}+\frac{\left(1+i \sqrt{3}\right) \log \left(3 i+\sqrt{3}-\left(-i+\sqrt{3}\right) x^4+2 i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+5 i x^2 \left(2+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3+5 i \sqrt{3}+3 i \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)+i x^3 \left(3 i+3 \sqrt{3}+\sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{4 \sqrt{6-6 i \sqrt{3}}}-\frac{i \left(-i+\sqrt{3}\right) \log \left(3 i+\sqrt{3}-\left(-i+\sqrt{3}\right) x^4+2 i \sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}+5 i x^2 \left(2+\sqrt{2-2 i \sqrt{3}} \sqrt{1-x^2}\right)+x^3 \left(3-3 i \sqrt{3}-i \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)-i x \left(-3 i+5 \sqrt{3}+3 \sqrt{6-6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{4 \sqrt{6-6 i \sqrt{3}}}+\frac{\left(1-i \sqrt{3}\right) \log \left(-3 i+\sqrt{3}-\left(i+\sqrt{3}\right) x^4-2 i \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}-5 i x^2 \left(2+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x \left(3-5 i \sqrt{3}-3 i \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)-i x^3 \left(-3 i+3 \sqrt{3}+\sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{4 \sqrt{6+6 i \sqrt{3}}}+\frac{i \left(i+\sqrt{3}\right) \log \left(-3 i+\sqrt{3}-\left(i+\sqrt{3}\right) x^4-2 i \sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}-5 i x^2 \left(2+\sqrt{2+2 i \sqrt{3}} \sqrt{1-x^2}\right)+x^3 \left(3+3 i \sqrt{3}+i \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)+i x \left(3 i+5 \sqrt{3}+3 \sqrt{6+6 i \sqrt{3}} \sqrt{1-x^2}\right)\right)}{4 \sqrt{6+6 i \sqrt{3}}}","\sin ^{-1}(x)-\frac{\tan ^{-1}\left(\frac{1+4 x \sqrt{1-x^2}}{\sqrt{3} \left(1-2 x^2\right)}\right)}{\sqrt{3}}",1,"ArcSin[x] - ((-I + Sqrt[3])*ArcTan[(x*(-7 - I*Sqrt[3] + 8*Sqrt[3]*x + I*(7*I + Sqrt[3])*x^2))/(-6 - (2*I)*Sqrt[3] + 3*(-I + Sqrt[3])*x^3 - 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (2*I)*x^2*(9*I + Sqrt[3] + I*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3*I + 11*Sqrt[3] + 2*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/(2*Sqrt[6 - (6*I)*Sqrt[3]]) + ((I + Sqrt[3])*ArcTan[(x*(7 - I*Sqrt[3] - 8*Sqrt[3]*x + (7 + I*Sqrt[3])*x^2))/(-6 + (2*I)*Sqrt[3] + 3*(I + Sqrt[3])*x^3 - 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + x^2*(-18 - (2*I)*Sqrt[3] - 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(-3*I + 11*Sqrt[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/(2*Sqrt[6 + (6*I)*Sqrt[3]]) - ((I + Sqrt[3])*ArcTan[(x*(7 - I*Sqrt[3] + 8*Sqrt[3]*x + (7 + I*Sqrt[3])*x^2))/(6 - (2*I)*Sqrt[3] + 3*(I + Sqrt[3])*x^3 + 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + 2*x^2*(9 + I*Sqrt[3] + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(-3*I + 11*Sqrt[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/(2*Sqrt[6 + (6*I)*Sqrt[3]]) + ((1 + I*Sqrt[3])*ArcTanh[(x*(7*I - Sqrt[3] + (8*I)*Sqrt[3]*x + (7*I + Sqrt[3])*x^2))/(6 + (2*I)*Sqrt[3] + 3*(-I + Sqrt[3])*x^3 + 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + 2*x^2*(9 - I*Sqrt[3] + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3*I + 11*Sqrt[3] + 2*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))])/(2*Sqrt[6 - (6*I)*Sqrt[3]]) - ((I/2)*Log[-1/2 - (I/2)*Sqrt[3] + x^2])/Sqrt[3] + ((I/2)*Log[(I/2)*(I + Sqrt[3]) + x^2])/Sqrt[3] - ((I/4)*(-I + Sqrt[3])*Log[16*(1 + Sqrt[3]*x + x^2)^2])/Sqrt[6 - (6*I)*Sqrt[3]] + ((I/4)*(I + Sqrt[3])*Log[16*(1 + Sqrt[3]*x + x^2)^2])/Sqrt[6 + (6*I)*Sqrt[3]] + ((1 + I*Sqrt[3])*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2])/(4*Sqrt[6 - (6*I)*Sqrt[3]]) + ((1 - I*Sqrt[3])*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2])/(4*Sqrt[6 + (6*I)*Sqrt[3]]) + ((1 + I*Sqrt[3])*Log[3*I + Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 + (5*I)*Sqrt[3] + (3*I)*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) + I*x^3*(3*I + 3*Sqrt[3] + Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/(4*Sqrt[6 - (6*I)*Sqrt[3]]) - ((I/4)*(-I + Sqrt[3])*Log[3*I + Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 - (3*I)*Sqrt[3] - I*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x*(-3*I + 5*Sqrt[3] + 3*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[6 - (6*I)*Sqrt[3]] + ((1 - I*Sqrt[3])*Log[-3*I + Sqrt[3] - (I + Sqrt[3])*x^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 - (5*I)*Sqrt[3] - (3*I)*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x^3*(-3*I + 3*Sqrt[3] + Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/(4*Sqrt[6 + (6*I)*Sqrt[3]]) + ((I/4)*(I + Sqrt[3])*Log[-3*I + Sqrt[3] - (I + Sqrt[3])*x^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 + (3*I)*Sqrt[3] + I*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) + I*x*(3*I + 5*Sqrt[3] + 3*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])/Sqrt[6 + (6*I)*Sqrt[3]]","C",1
1023,1,31,34,0.0403725,"\int \left(1+x+x^2+x^3\right)^{-n} \left(1-x^4\right)^n \, dx","Integrate[(1 - x^4)^n/(1 + x + x^2 + x^3)^n,x]","\frac{(x-1) \left(x^3+x^2+x+1\right)^{-n} \left(1-x^4\right)^n}{n+1}","-\frac{(1-x) \left(x^3+x^2+x+1\right)^{-n} \left(1-x^4\right)^n}{n+1}",1,"((-1 + x)*(1 - x^4)^n)/((1 + n)*(1 + x + x^2 + x^3)^n)","A",1
1024,1,1671,177,6.130339,"\int \frac{x}{\sqrt{-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4}} \, dx","Integrate[x/Sqrt[-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4],x]","\frac{2 \left(x-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}\right)^2 \left(\frac{\Pi \left(\frac{\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]}{c}}{\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}};\sin ^{-1}\left(\sqrt{\frac{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}}\right)|-\frac{\left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}{\left(-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]+\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}\right) \left(b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right)}{c}-\frac{b F\left(\sin ^{-1}\left(\sqrt{\frac{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}}\right)|-\frac{\left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}{\left(-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]+\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}\right) \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}\right) \sqrt{\frac{\left(b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right) \left(x-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]}{c}\right)}{c \left(x-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}\right) \left(\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,3\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]}{c}\right)}} \sqrt{\frac{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}{\left(c x-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]\right) \left(\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]-\text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]\right)}} \left(\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}\right) \sqrt{\frac{\left(b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]-b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]\right) \left(x-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}\right)}{c \left(x-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}\right) \left(\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]}{c}\right)}}}{\sqrt{-44375 b^4+576000 c x b^3+576000 c^2 x^2 b^2+5308416 c^4 x^4} \left(\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,1\right]}{c}\right) \left(\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,2\right]}{c}-\frac{b \text{Root}\left[5308416 \text{$\#$1}^4+576000 \text{$\#$1}^2+576000 \text{$\#$1}-44375\&,4\right]}{c}\right)}","\frac{\log \left(20738073600000000 b^8 c^4+597005697024000000 b^6 c^6 x^2+2583100705996800000 b^5 c^7 x^3+951050714480640000 b^4 c^8 x^4+21641687369515008000 b^3 c^9 x^5+32462531054272512000 b^2 c^{10} x^6+5308416 \sqrt{-44375 b^4+576000 b^3 c x+576000 b^2 c^2 x^2+5308416 c^4 x^4} \left(12203125 b^6 c^4+79200000 b^5 c^5 x+38880000 b^4 c^6 x^2+1105920000 b^3 c^7 x^3+1990656000 b^2 c^8 x^4+12230590464 c^{10} x^6\right)+149587343098087735296 c^{12} x^8\right)}{18432 c^2}",1,"(2*(x - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c)^2*(-((b*EllipticF[ArcSin[Sqrt[((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))/((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))]], -(((Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))/((-Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] + Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])))]*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c) + (EllipticPi[(-((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])/c) + (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c)/(-((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c) + (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c), ArcSin[Sqrt[((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))/((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))]], -(((Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))/((-Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] + Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])))]*(-(b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0]) + b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0]))/c)*Sqrt[((-(b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0]) + b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])*(x - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])/c))/(c*(x - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c)*(-((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])/c) + (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 3, 0])/c))]*Sqrt[((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))/((c*x - b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])*(Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0] - Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0]))]*((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])/c - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c)*Sqrt[((-(b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0]) + b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])*(x - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c))/(c*(x - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c)*(-((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])/c) + (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c))])/(Sqrt[-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4]*(-((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 1, 0])/c) + (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c)*((b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 2, 0])/c - (b*Root[-44375 + 576000*#1 + 576000*#1^2 + 5308416*#1^4 & , 4, 0])/c))","C",0
1025,1,2787,100,6.1016805,"\int \frac{1+4 x}{\sqrt{9+120 x+64 x^2+64 x^3+64 x^4}} \, dx","Integrate[(1 + 4*x)/Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4],x]","\text{Result too large to show}","\frac{1}{16} \log \left(4096 x^8+8192 x^7+12288 x^6+19456 x^5+17024 x^4+13440 x^3+9280 x^2+\sqrt{64 x^4+64 x^3+64 x^2+120 x+9} \left(512 x^6+768 x^5+960 x^4+1280 x^3+744 x^2+444 x+179\right)+2864 x+921\right)",1,"(8*(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]], -(((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])))]*Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0]) + EllipticPi[(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])/(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]], -(((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])))]*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0]))*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0]))]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])*Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))])/(Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])) + (2*EllipticF[ArcSin[Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]], ((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]*(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0]))]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]*Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))])/(Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4]*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))","C",0