1,1,305,0,0.2485684,"\int \frac{d+e x^3}{a+c x^6} \, dx","Int[(d + e*x^3)/(a + c*x^6),x]","-\frac{\left(\sqrt{3} \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{3} \sqrt{c} d\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{3} \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{c} d-\sqrt{3} \sqrt{a} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt[6]{c}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}","-\frac{\left(\sqrt{3} \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{3} \sqrt{c} d\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{3} \sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{c} d-\sqrt{3} \sqrt{a} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt[6]{c}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}",1,"(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) - ((Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/3)) + ((Sqrt[c]*d - Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3]*Sqrt[c]*d - Sqrt[a]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3)) + ((Sqrt[3]*Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))","A",12,8,17,0.4706,1,"{1416, 635, 203, 260, 634, 617, 204, 628}"
2,1,323,0,0.1889832,"\int \frac{d+e x^3}{a-c x^6} \, dx","Int[(d + e*x^3)/(a - c*x^6),x]","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a}-\sqrt[6]{c} x\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}+2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} \sqrt[6]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt[6]{a}+\sqrt[6]{c} x\right)}{6 a^{5/6} \sqrt[6]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}-2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} \sqrt[6]{c}}","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} c^{2/3}}-\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \log \left(\sqrt[6]{a}-\sqrt[6]{c} x\right)}{6 a^{5/6} c^{2/3}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}+2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 a^{5/6} \sqrt[6]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt[6]{a}+\sqrt[6]{c} x\right)}{6 a^{5/6} \sqrt[6]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt[6]{a}-2 \sqrt[6]{c} x}{\sqrt{3} \sqrt[6]{a}}\right)}{2 \sqrt{3} a^{5/6} \sqrt[6]{c}}",1,"-((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[(a^(1/6) - 2*c^(1/6)*x)/(Sqrt[3]*a^(1/6))])/(2*Sqrt[3]*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(a^(1/6) + 2*c^(1/6)*x)/(Sqrt[3]*a^(1/6))])/(2*Sqrt[3]*a^(5/6)*c^(2/3)) - ((Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/6) - c^(1/6)*x])/(6*a^(5/6)*c^(2/3)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/6) + c^(1/6)*x])/(6*a^(5/6)*c^(1/6)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))","A",13,7,18,0.3889,1,"{1417, 200, 31, 634, 617, 204, 628}"
3,1,754,0,1.2468305,"\int \frac{d+e x^4}{a+c x^8} \, dx","Int[(d + e*x^4)/(a + c*x^8),x]","\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\left(-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} \sqrt[8]{c}}","\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} c^{5/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\left(1-\sqrt{2}\right) \sqrt{c} d-\sqrt{a} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{7/8} c^{5/8}}+\frac{\left(-\frac{\sqrt{a} e}{\sqrt{c}}+\sqrt{2} d+d\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{7/8} \sqrt[8]{c}}",1,"-(Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) + (Sqrt[2 + Sqrt[2]]*((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) + (Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) - (Sqrt[2 + Sqrt[2]]*((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(7/8)*c^(5/8)) + (((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*Log[a^(1/4) - Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(7/8)*c^(5/8)) - (((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*Log[a^(1/4) + Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(7/8)*c^(5/8)) - (((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*Log[a^(1/4) - Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(7/8)*c^(5/8)) + ((d + Sqrt[2]*d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(7/8)*c^(1/8))","A",19,6,17,0.3529,1,"{1415, 1169, 634, 618, 204, 628}"
4,1,329,0,0.2092229,"\int \frac{d+e x^4}{a-c x^8} \, dx","Int[(d + e*x^4)/(a - c*x^8),x]","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}+1\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}","\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tan ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}+\frac{\left(\sqrt{a} e+\sqrt{c} d\right) \tanh ^{-1}\left(\frac{\sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 a^{7/8} c^{5/8}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(-\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \log \left(\sqrt{2} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2} a^{7/8} \sqrt[8]{c}}-\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}+\frac{\left(d-\frac{\sqrt{a} e}{\sqrt{c}}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{c} x}{\sqrt[8]{a}}+1\right)}{4 \sqrt{2} a^{7/8} \sqrt[8]{c}}",1,"((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/8)*x)/a^(1/8)])/(4*Sqrt[2]*a^(7/8)*c^(1/8)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTanh[(c^(1/8)*x)/a^(1/8)])/(4*a^(7/8)*c^(5/8)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7/8)*c^(1/8)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a^(7/8)*c^(1/8))","A",13,10,18,0.5556,1,"{1417, 212, 208, 205, 211, 1165, 628, 1162, 617, 204}"
5,1,791,0,0.8627354,"\int \frac{d+e x^4}{d^2+b x^4+e^2 x^8} \, dx","Int[(d + e*x^4)/(d^2 + b*x^4 + e^2*x^8),x]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-b}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-b}}}",1,"-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[-b + 2*d*e]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[-b + 2*d*e]])","A",19,6,26,0.2308,1,"{1419, 1094, 634, 618, 204, 628}"
6,1,791,0,0.8054573,"\int \frac{d+e x^4}{d^2+f x^4+e^2 x^8} \, dx","Int[(d + e*x^4)/(d^2 + f*x^4 + e^2*x^8),x]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e-f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e-f}}}",1,"-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e - f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e - f]])","A",19,6,26,0.2308,1,"{1419, 1094, 634, 618, 204, 628}"
7,1,349,0,0.4215939,"\int \frac{d+e x^4}{d^2-b x^4+e^2 x^8} \, dx","Int[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right)}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}",1,"-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]])","A",7,4,27,0.1481,1,"{1419, 1093, 207, 203}"
8,1,751,0,0.9245036,"\int \frac{d+e x^4}{d^2-f x^4+e^2 x^8} \, dx","Int[(d + e*x^4)/(d^2 - f*x^4 + e^2*x^8),x]","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}","-\frac{\log \left(-x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\log \left(x \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}-\frac{\log \left(-x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\log \left(x \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+\sqrt{d}+\sqrt{e} x^2\right)}{8 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}-2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}-2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}+2 \sqrt{e} x}{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}\right)}{4 \sqrt{d} \sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2 d e+f}+2 \sqrt{d} \sqrt{e}}+2 \sqrt{e} x}{\sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}\right)}{4 \sqrt{d} \sqrt{2 \sqrt{d} \sqrt{e}-\sqrt{2 d e+f}}}",1,"-ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]) - ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]] - 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]) + ArcTan[(Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]] + 2*Sqrt[e]*x)/Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]]/(4*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] - Sqrt[2*d*e + f]]) - Log[Sqrt[d] - Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]) + Log[Sqrt[d] + Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]]*x + Sqrt[e]*x^2]/(8*Sqrt[d]*Sqrt[2*Sqrt[d]*Sqrt[e] + Sqrt[2*d*e + f]])","A",19,6,27,0.2222,1,"{1419, 1094, 634, 618, 204, 628}"
9,1,411,0,0.2917183,"\int \frac{1+x^4}{1+b x^4+x^8} \, dx","Int[(1 + x^4)/(1 + b*x^4 + x^8),x]","-\frac{\log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}+\frac{\log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}-\frac{\log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}+\frac{\log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}","-\frac{\log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}+\frac{\log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-\sqrt{2-b}}}-\frac{\log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}+\frac{\log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{\sqrt{2-b}+2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{2-\sqrt{2-b}}}",1,"-ArcTan[(Sqrt[2 - Sqrt[2 - b]] - 2*x)/Sqrt[2 + Sqrt[2 - b]]]/(4*Sqrt[2 + Sqrt[2 - b]]) - ArcTan[(Sqrt[2 + Sqrt[2 - b]] - 2*x)/Sqrt[2 - Sqrt[2 - b]]]/(4*Sqrt[2 - Sqrt[2 - b]]) + ArcTan[(Sqrt[2 - Sqrt[2 - b]] + 2*x)/Sqrt[2 + Sqrt[2 - b]]]/(4*Sqrt[2 + Sqrt[2 - b]]) + ArcTan[(Sqrt[2 + Sqrt[2 - b]] + 2*x)/Sqrt[2 - Sqrt[2 - b]]]/(4*Sqrt[2 - Sqrt[2 - b]]) - Log[1 - Sqrt[2 - Sqrt[2 - b]]*x + x^2]/(8*Sqrt[2 - Sqrt[2 - b]]) + Log[1 + Sqrt[2 - Sqrt[2 - b]]*x + x^2]/(8*Sqrt[2 - Sqrt[2 - b]]) - Log[1 - Sqrt[2 + Sqrt[2 - b]]*x + x^2]/(8*Sqrt[2 + Sqrt[2 - b]]) + Log[1 + Sqrt[2 + Sqrt[2 - b]]*x + x^2]/(8*Sqrt[2 + Sqrt[2 - b]])","A",19,6,18,0.3333,1,"{1419, 1094, 634, 618, 204, 628}"
10,1,451,0,0.4065474,"\int \frac{1+x^4}{1+3 x^4+x^8} \, dx","Int[(1 + x^4)/(1 + 3*x^4 + x^8),x]","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4} \sqrt{5}}",1,"-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) - ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) + ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5])","A",19,7,18,0.3889,1,"{1420, 211, 1165, 628, 1162, 617, 204}"
11,1,85,0,0.0454182,"\int \frac{1+x^4}{1+2 x^4+x^8} \, dx","Int[(1 + x^4)/(1 + 2*x^4 + x^8),x]","-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{4 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{4 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{2 \sqrt{2}}","-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{4 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{4 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{2 \sqrt{2}}",1,"-ArcTan[1 - Sqrt[2]*x]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/(2*Sqrt[2]) - Log[1 - Sqrt[2]*x + x^2]/(4*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(4*Sqrt[2])","A",10,7,18,0.3889,1,"{28, 211, 1165, 628, 1162, 617, 204}"
12,1,140,0,0.0954812,"\int \frac{1+x^4}{1+x^4+x^8} \, dx","Int[(1 + x^4)/(1 + x^4 + x^8),x]","-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)","-\frac{1}{8} \log \left(x^2-x+1\right)+\frac{1}{8} \log \left(x^2+x+1\right)-\frac{\log \left(x^2-\sqrt{3} x+1\right)}{8 \sqrt{3}}+\frac{\log \left(x^2+\sqrt{3} x+1\right)}{8 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{\tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"-ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[Sqrt[3] - 2*x]/4 + ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) + ArcTan[Sqrt[3] + 2*x]/4 - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8 - Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3]) + Log[1 + Sqrt[3]*x + x^2]/(8*Sqrt[3])","A",19,6,16,0.3750,1,"{1419, 1094, 634, 618, 204, 628}"
13,1,347,0,0.2468075,"\int \frac{1+x^4}{1+x^8} \, dx","Int[(1 + x^4)/(1 + x^8),x]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}-\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)","-\frac{\log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)}{8 \sqrt{2-\sqrt{2}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)}{8 \sqrt{2+\sqrt{2}}}-\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)-\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\frac{1}{4} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)",1,"-(Sqrt[(2 - Sqrt[2])/2]*ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]])/4 - (Sqrt[(2 + Sqrt[2])/2]*ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]])/4 + (Sqrt[(2 - Sqrt[2])/2]*ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]])/4 + (Sqrt[(2 + Sqrt[2])/2]*ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]])/4 - Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2]/(8*Sqrt[2 - Sqrt[2]]) + Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2]/(8*Sqrt[2 - Sqrt[2]]) - Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2]/(8*Sqrt[2 + Sqrt[2]]) + Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2]/(8*Sqrt[2 + Sqrt[2]])","A",19,6,13,0.4615,1,"{1413, 1094, 634, 618, 204, 628}"
14,1,331,0,0.2345641,"\int \frac{1+x^4}{1-x^4+x^8} \, dx","Int[(1 + x^4)/(1 - x^4 + x^8),x]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"-(Sqrt[2 - Sqrt[3]]*ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]])/4 - (Sqrt[2 + Sqrt[3]]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/4 + (Sqrt[2 - Sqrt[3]]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]])/4 + (Sqrt[2 + Sqrt[3]]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/4 - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[2 - Sqrt[3]]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[2 - Sqrt[3]]) - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqrt[2 + Sqrt[3]]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqrt[2 + Sqrt[3]])","A",19,6,18,0.3333,1,"{1419, 1094, 634, 618, 204, 628}"
15,1,27,0,0.0077244,"\int \frac{1+x^4}{1-2 x^4+x^8} \, dx","Int[(1 + x^4)/(1 - 2*x^4 + x^8),x]","\frac{x}{2 \left(1-x^4\right)}+\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x)","\frac{x}{2 \left(1-x^4\right)}+\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x)",1,"x/(2*(1 - x^4)) + ArcTan[x]/4 + ArcTanh[x]/4","A",5,5,18,0.2778,1,"{28, 385, 212, 206, 203}"
16,1,131,0,0.0862422,"\int \frac{1+x^4}{1-3 x^4+x^8} \, dx","Int[(1 + x^4)/(1 - 3*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{2 \left(\sqrt{5}-1\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{2 \left(1+\sqrt{5}\right)}}",1,"ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])]","A",7,4,18,0.2222,1,"{1419, 1093, 203, 207}"
17,1,157,0,0.0865303,"\int \frac{1+x^4}{1-4 x^4+x^8} \, dx","Int[(1 + x^4)/(1 - 4*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{1+\sqrt{3}}}",1,"ArcTan[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[1 + Sqrt[3]]) + ArcTanh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTanh[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[1 + Sqrt[3]])","A",7,4,18,0.2222,1,"{1419, 1093, 203, 207}"
18,1,171,0,0.1512573,"\int \frac{1+x^4}{1-5 x^4+x^8} \, dx","Int[(1 + x^4)/(1 - 5*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{6 \left(\sqrt{7}-\sqrt{3}\right)}}-\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{6 \left(\sqrt{3}+\sqrt{7}\right)}}",1,"ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTan[Sqrt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTanh[Sqrt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])]","A",7,4,18,0.2222,1,"{1419, 1093, 203, 207}"
19,1,117,0,0.0574979,"\int \frac{1+x^4}{1-6 x^4+x^8} \, dx","Int[(1 + x^4)/(1 - 6*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{1+\sqrt{2}}}",1,"ArcTan[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTan[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]]) + ArcTanh[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTanh[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]])","A",7,4,18,0.2222,1,"{1419, 1093, 203, 207}"
20,1,511,0,0.3585093,"\int \frac{1-x^4}{1+b x^4+x^8} \, dx","Int[(1 - x^4)/(1 + b*x^4 + x^8),x]","\frac{\sqrt{2-\sqrt{2-b}} \log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}","\frac{\sqrt{2-\sqrt{2-b}} \log \left(-\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left(\sqrt{2-\sqrt{2-b}} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left(-\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left(\sqrt{\sqrt{2-b}+2} x+x^2+1\right)}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right)}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right)}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}",1,"-(Sqrt[2 + b]*ArcTan[(Sqrt[2 - Sqrt[2 - b]] - 2*x)/Sqrt[2 + Sqrt[2 - b]]])/(4*Sqrt[2 - Sqrt[2 - b]]*Sqrt[2 - b]) + (Sqrt[2 + b]*ArcTan[(Sqrt[2 + Sqrt[2 - b]] - 2*x)/Sqrt[2 - Sqrt[2 - b]]])/(4*Sqrt[2 + Sqrt[2 - b]]*Sqrt[2 - b]) + (Sqrt[2 + b]*ArcTan[(Sqrt[2 - Sqrt[2 - b]] + 2*x)/Sqrt[2 + Sqrt[2 - b]]])/(4*Sqrt[2 - Sqrt[2 - b]]*Sqrt[2 - b]) - (Sqrt[2 + b]*ArcTan[(Sqrt[2 + Sqrt[2 - b]] + 2*x)/Sqrt[2 - Sqrt[2 - b]]])/(4*Sqrt[2 + Sqrt[2 - b]]*Sqrt[2 - b]) + (Sqrt[2 - Sqrt[2 - b]]*Log[1 - Sqrt[2 - Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) - (Sqrt[2 - Sqrt[2 - b]]*Log[1 + Sqrt[2 - Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) - (Sqrt[2 + Sqrt[2 - b]]*Log[1 - Sqrt[2 + Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) + (Sqrt[2 + Sqrt[2 - b]]*Log[1 + Sqrt[2 + Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b])","A",19,6,20,0.3000,1,"{1421, 1169, 634, 618, 204, 628}"
21,1,411,0,0.3212872,"\int \frac{1-x^4}{1+3 x^4+x^8} \, dx","Int[(1 - x^4)/(1 + 3*x^4 + x^8),x]","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4}}","-\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3-\sqrt{5}\right)} x+\sqrt{2 \left(3-\sqrt{5}\right)}\right)}{4\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2-2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \log \left(2 x^2+2 \sqrt[4]{2 \left(3+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)}\right)}{4\ 2^{3/4}}-\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3+\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3-\sqrt{5}}}+1\right)}{2\ 2^{3/4}}+\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(1-\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}\right)}{2\ 2^{3/4}}-\frac{\sqrt[4]{3-\sqrt{5}} \tan ^{-1}\left(\frac{2^{3/4} x}{\sqrt[4]{3+\sqrt{5}}}+1\right)}{2\ 2^{3/4}}",1,"-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) - ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) - ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4))","A",19,7,20,0.3500,1,"{1420, 211, 1165, 628, 1162, 617, 204}"
22,1,97,0,0.0523075,"\int \frac{1-x^4}{1+2 x^4+x^8} \, dx","Int[(1 - x^4)/(1 + 2*x^4 + x^8),x]","\frac{x}{2 \left(x^4+1\right)}-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{8 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{8 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{4 \sqrt{2}}","\frac{x}{2 \left(x^4+1\right)}-\frac{\log \left(x^2-\sqrt{2} x+1\right)}{8 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2} x+1\right)}{8 \sqrt{2}}-\frac{\tan ^{-1}\left(1-\sqrt{2} x\right)}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{2} x+1\right)}{4 \sqrt{2}}",1,"x/(2*(1 + x^4)) - ArcTan[1 - Sqrt[2]*x]/(4*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/(4*Sqrt[2]) - Log[1 - Sqrt[2]*x + x^2]/(8*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(8*Sqrt[2])","A",11,8,20,0.4000,1,"{28, 385, 211, 1165, 628, 1162, 617, 204}"
23,1,140,0,0.0987737,"\int \frac{1-x^4}{1+x^4+x^8} \, dx","Int[(1 - x^4)/(1 + x^4 + x^8),x]","\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)-\frac{1}{8} \sqrt{3} \log \left(x^2-\sqrt{3} x+1\right)+\frac{1}{8} \sqrt{3} \log \left(x^2+\sqrt{3} x+1\right)-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)","\frac{1}{8} \log \left(x^2-x+1\right)-\frac{1}{8} \log \left(x^2+x+1\right)-\frac{1}{8} \sqrt{3} \log \left(x^2-\sqrt{3} x+1\right)+\frac{1}{8} \sqrt{3} \log \left(x^2+\sqrt{3} x+1\right)-\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{1-2 x}{\sqrt{3}}\right)+\frac{1}{4} \tan ^{-1}\left(\sqrt{3}-2 x\right)+\frac{1}{4} \sqrt{3} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)-\frac{1}{4} \tan ^{-1}\left(2 x+\sqrt{3}\right)",1,"-(Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[3]])/4 + ArcTan[Sqrt[3] - 2*x]/4 + (Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]])/4 - ArcTan[Sqrt[3] + 2*x]/4 + Log[1 - x + x^2]/8 - Log[1 + x + x^2]/8 - (Sqrt[3]*Log[1 - Sqrt[3]*x + x^2])/8 + (Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/8","A",19,6,18,0.3333,1,"{1421, 1169, 634, 618, 204, 628}"
24,1,347,0,0.2702869,"\int \frac{1-x^4}{1+x^8} \, dx","Int[(1 - x^4)/(1 + x^8),x]","\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}","\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2-\sqrt{2}\right)} \log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{2} \left(2+\sqrt{2}\right)} \log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)}{4 \sqrt{2-\sqrt{2}}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)}{4 \sqrt{2+\sqrt{2}}}",1,"-ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqrt[2]]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqrt[2]]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + (Sqrt[(2 - Sqrt[2])/2]*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[2])/2]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2 + Sqrt[2])/2]*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[2])/2]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2])/8","A",19,6,15,0.4000,1,"{1414, 1169, 634, 618, 204, 628}"
25,1,355,0,0.2772977,"\int \frac{1-x^4}{1-x^4+x^8} \, dx","Int[(1 - x^4)/(1 - x^4 + x^8),x]","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}","\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2-\sqrt{2+\sqrt{3}} x+1\right)+\frac{1}{8} \sqrt{\frac{1}{3} \left(2+\sqrt{3}\right)} \log \left(x^2+\sqrt{2+\sqrt{3}} x+1\right)-\frac{\tan ^{-1}\left(\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right)}{4 \sqrt{3 \left(2-\sqrt{3}\right)}}-\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{4 \sqrt{3 \left(2+\sqrt{3}\right)}}",1,"-ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) + ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) + ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) + (Sqrt[(2 - Sqrt[3])/3]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[3])/3]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 + Sqrt[3])/3]*Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[3])/3]*Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2])/8","A",19,6,20,0.3000,1,"{1421, 1169, 634, 618, 204, 628}"
26,1,13,0,0.0046864,"\int \frac{1-x^4}{1-2 x^4+x^8} \, dx","Int[(1 - x^4)/(1 - 2*x^4 + x^8),x]","\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)","\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)",1,"ArcTan[x]/2 + ArcTanh[x]/2","A",5,5,20,0.2500,1,"{28, 21, 212, 206, 203}"
27,1,129,0,0.1178341,"\int \frac{1-x^4}{1-3 x^4+x^8} \, dx","Int[(1 - x^4)/(1 - 3*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)}{\sqrt{10 \left(\sqrt{5}-1\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)}{\sqrt{10 \left(1+\sqrt{5}\right)}}",1,"ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]","A",7,4,20,0.2000,1,"{1419, 1093, 207, 203}"
28,1,165,0,0.1040541,"\int \frac{1-x^4}{1-4 x^4+x^8} \, dx","Int[(1 - x^4)/(1 - 4*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{\sqrt{3}-1}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(\sqrt{3}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{2} x}{\sqrt{1+\sqrt{3}}}\right)}{2 \sqrt[4]{2} \sqrt{3 \left(1+\sqrt{3}\right)}}",1,"ArcTan[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(-1 + Sqrt[3])]) + ArcTan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(1 + Sqrt[3])]) + ArcTanh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(-1 + Sqrt[3])]) + ArcTanh[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(1 + Sqrt[3])])","A",7,4,20,0.2000,1,"{1419, 1093, 207, 203}"
29,1,169,0,0.1424469,"\int \frac{1-x^4}{1-5 x^4+x^8} \, dx","Int[(1 - x^4)/(1 - 5*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}","\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tan ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right)}{\sqrt{14 \left(\sqrt{7}-\sqrt{3}\right)}}+\frac{\tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right)}{\sqrt{14 \left(\sqrt{3}+\sqrt{7}\right)}}",1,"ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTan[Sqrt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])]","A",7,4,20,0.2000,1,"{1419, 1093, 207, 203}"
30,1,125,0,0.0674187,"\int \frac{1-x^4}{1-6 x^4+x^8} \, dx","Int[(1 - x^4)/(1 - 6*x^4 + x^8),x]","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}","\frac{\tan ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tan ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{\sqrt{2}-1}}\right)}{4 \sqrt{2 \left(\sqrt{2}-1\right)}}+\frac{\tanh ^{-1}\left(\frac{x}{\sqrt{1+\sqrt{2}}}\right)}{4 \sqrt{2 \left(1+\sqrt{2}\right)}}",1,"ArcTan[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 + Sqrt[2])]) + ArcTan[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + ArcTanh[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 + Sqrt[2])]) + ArcTanh[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[2*(1 + Sqrt[2])])","A",7,4,20,0.2000,1,"{1419, 1093, 207, 203}"
31,1,135,0,0.1235384,"\int \frac{-1+\sqrt{3}+2 x^4}{1-x^4+x^8} \, dx","Int[(-1 + Sqrt[3] + 2*x^4)/(1 - x^4 + x^8),x]","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}","-\frac{\log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}+\frac{\log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)}{2 \sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2]) + ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2] - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2])","A",9,6,25,0.2400,1,"{1423, 1161, 618, 204, 1164, 628}"
32,1,164,0,0.0947788,"\int \frac{1+\left(1+\sqrt{3}\right) x^4}{1-x^4+x^8} \, dx","Int[(1 + (1 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]","-\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)","-\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"-(Sqrt[2 + Sqrt[3]]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/2 + (Sqrt[2 + Sqrt[3]]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/2 - (Sqrt[2 + Sqrt[3]]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/4 + (Sqrt[2 + Sqrt[3]]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/4","A",9,6,26,0.2308,1,"{1423, 1161, 618, 204, 1164, 628}"
33,1,180,0,0.1223599,"\int \frac{3-2 \sqrt{3}+\left(-3+\sqrt{3}\right) x^4}{1-x^4+x^8} \, dx","Int[(3 - 2*Sqrt[3] + (-3 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]","\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)-\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)","\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2-\sqrt{2-\sqrt{3}} x+1\right)-\frac{1}{4} \sqrt{3 \left(2-\sqrt{3}\right)} \log \left(x^2+\sqrt{2-\sqrt{3}} x+1\right)+\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right)-\frac{1}{2} \sqrt{3 \left(2-\sqrt{3}\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right)",1,"(Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/2 - (Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/2 + (Sqrt[3*(2 - Sqrt[3])]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/4 - (Sqrt[3*(2 - Sqrt[3])]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/4","A",9,6,33,0.1818,1,"{1423, 1161, 618, 204, 1164, 628}"
34,1,49,0,0.0300692,"\int \frac{d+\frac{e}{x}}{c+\frac{a}{x^2}} \, dx","Int[(d + e/x)/(c + a/x^2),x]","-\frac{\sqrt{a} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2}}+\frac{e \log \left(a+c x^2\right)}{2 c}+\frac{d x}{c}","-\frac{\sqrt{a} d \tan ^{-1}\left(\frac{\sqrt{c} x}{\sqrt{a}}\right)}{c^{3/2}}+\frac{e \log \left(a+c x^2\right)}{2 c}+\frac{d x}{c}",1,"(d*x)/c - (Sqrt[a]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/c^(3/2) + (e*Log[a + c*x^2])/(2*c)","A",5,5,17,0.2941,1,"{1394, 774, 635, 205, 260}"
35,1,86,0,0.0812762,"\int \frac{d+\frac{e}{x}}{c+\frac{a}{x^2}+\frac{b}{x}} \, dx","Int[(d + e/x)/(c + a/x^2 + b/x),x]","-\frac{\left(-2 a c d+b^2 d-b c e\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{(b d-c e) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{d x}{c}","-\frac{\left(-2 a c d+b^2 d-b c e\right) \tanh ^{-1}\left(\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right)}{c^2 \sqrt{b^2-4 a c}}-\frac{(b d-c e) \log \left(a+b x+c x^2\right)}{2 c^2}+\frac{d x}{c}",1,"(d*x)/c - ((b^2*d - 2*a*c*d - b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) - ((b*d - c*e)*Log[a + b*x + c*x^2])/(2*c^2)","A",6,6,22,0.2727,1,"{1393, 773, 634, 618, 206, 628}"
36,1,253,0,0.2109309,"\int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}} \, dx","Int[(d + e/x^2)/(c + a/x^4),x]","\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{d x}{c}","\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{c} x+\sqrt{a}+\sqrt{c} x^2\right)}{4 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}-\frac{\left(\sqrt{a} d-\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{c} x}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} \sqrt[4]{a} c^{5/4}}+\frac{d x}{c}",1,"(d*x)/c + ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) + ((Sqrt[a]*d + Sqrt[c]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[a]*d + Sqrt[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4))","A",11,8,17,0.4706,1,"{1394, 1280, 1168, 1162, 617, 204, 1165, 628}"
37,1,208,0,0.5429669,"\int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}+\frac{b}{x^2}} \, dx","Int[(d + e/x^2)/(c + a/x^4 + b/x^2),x]","-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{d x}{c}","-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right)}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right)}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{d x}{c}",1,"(d*x)/c - ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])","A",5,4,22,0.1818,1,"{1393, 1279, 1166, 205}"
38,1,311,0,0.289657,"\int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}} \, dx","Int[(d + e/x^3)/(c + a/x^6),x]","\frac{\left(\sqrt{3} \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{3} \sqrt{a} d-\sqrt{c} e\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}+\frac{\left(\sqrt{a} d-\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{a} d+\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\sqrt[6]{a} d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 c^{7/6}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}+\frac{d x}{c}","\frac{\left(\sqrt{3} \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{3} \sqrt{a} d-\sqrt{c} e\right) \log \left(\sqrt{3} \sqrt[6]{a} \sqrt[6]{c} x+\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{12 \sqrt[3]{a} c^{7/6}}+\frac{\left(\sqrt{a} d-\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\left(\sqrt{a} d+\sqrt{3} \sqrt{c} e\right) \tan ^{-1}\left(\frac{2 \sqrt[6]{c} x}{\sqrt[6]{a}}+\sqrt{3}\right)}{6 \sqrt[3]{a} c^{7/6}}-\frac{\sqrt[6]{a} d \tan ^{-1}\left(\frac{\sqrt[6]{c} x}{\sqrt[6]{a}}\right)}{3 c^{7/6}}-\frac{e \log \left(\sqrt[3]{a}+\sqrt[3]{c} x^2\right)}{6 \sqrt[3]{a} c^{2/3}}+\frac{d x}{c}",1,"(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((Sqrt[a]*d - Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) - ((Sqrt[a]*d + Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) + ((Sqrt[3]*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6)) - ((Sqrt[3]*Sqrt[a]*d - Sqrt[c]*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6))","A",14,10,17,0.5882,1,"{1394, 1503, 1416, 635, 203, 260, 634, 617, 204, 628}"
39,1,716,0,1.6335118,"\int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}+\frac{b}{x^3}} \, dx","Int[(d + e/x^3)/(c + a/x^6 + b/x^3),x]","\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{d x}{c}","\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{b-\sqrt{b^2-4 a c}}+\left(b-\sqrt{b^2-4 a c}\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(-\sqrt[3]{2} \sqrt[3]{c} x \sqrt[3]{\sqrt{b^2-4 a c}+b}+\left(\sqrt{b^2-4 a c}+b\right)^{2/3}+2^{2/3} c^{2/3} x^2\right)}{6 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}-\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{b-\sqrt{b^2-4 a c}}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}-\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \log \left(\sqrt[3]{\sqrt{b^2-4 a c}+b}+\sqrt[3]{2} \sqrt[3]{c} x\right)}{3 \sqrt[3]{2} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(b-\sqrt{b^2-4 a c}\right)^{2/3}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[3]{2} \sqrt[3]{c} x}{\sqrt[3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right)}{\sqrt[3]{2} \sqrt{3} c^{4/3} \left(\sqrt{b^2-4 a c}+b\right)^{2/3}}+\frac{d x}{c}",1,"(d*x)/c + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) - ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3))","A",15,9,22,0.4091,1,"{1393, 1502, 1422, 200, 31, 634, 617, 204, 628}"
40,1,753,0,1.4363195,"\int \frac{d+\frac{e}{x^4}}{c+\frac{a}{x^8}} \, dx","Int[(d + e/x^4)/(c + a/x^8),x]","-\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{d x}{c}","-\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(-\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \log \left(\sqrt{2-\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2-\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(-\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}-\frac{\left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \log \left(\sqrt{2+\sqrt{2}} \sqrt[8]{a} \sqrt[8]{c} x+\sqrt[4]{a}+\sqrt[4]{c} x^2\right)}{8 \sqrt{2 \left(2+\sqrt{2}\right)} a^{3/8} c^{9/8}}+\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}-2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}-\frac{\sqrt{2-\sqrt{2}} \left(\left(1+\sqrt{2}\right) \sqrt{a} d+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2+\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{\sqrt{2+\sqrt{2}} \left(\sqrt{a} \left(d-\sqrt{2} d\right)+\sqrt{c} e\right) \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{a}+2 \sqrt[8]{c} x}{\sqrt{2-\sqrt{2}} \sqrt[8]{a}}\right)}{8 a^{3/8} c^{9/8}}+\frac{d x}{c}",1,"(d*x)/c + (Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) - 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - (Sqrt[2 - Sqrt[2]]*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 + Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) + (Sqrt[2 + Sqrt[2]]*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/(8*a^(3/8)*c^(9/8)) - ((Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*Log[a^(1/4) - Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(3/8)*c^(9/8)) + ((Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 - Sqrt[2])]*a^(3/8)*c^(9/8)) + (((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4) - Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8)) - (((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2*(2 + Sqrt[2])]*a^(3/8)*c^(9/8))","A",21,8,17,0.4706,1,"{1394, 1503, 1415, 1169, 634, 618, 204, 628}"
41,1,433,0,0.9886401,"\int \frac{d+\frac{e}{x^4}}{c+\frac{a}{x^8}+\frac{b}{x^4}} \, dx","Int[(d + e/x^4)/(c + a/x^8 + b/x^4),x]","\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{d x}{c}","\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tan ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{-\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(-\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{\left(-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right) \tanh ^{-1}\left(\frac{\sqrt[4]{2} \sqrt[4]{c} x}{\sqrt[4]{\sqrt{b^2-4 a c}-b}}\right)}{2 \sqrt[4]{2} c^{5/4} \left(\sqrt{b^2-4 a c}-b\right)^{3/4}}+\frac{d x}{c}",1,"(d*x)/c + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))","A",9,6,22,0.2727,1,"{1393, 1502, 1422, 212, 208, 205}"
42,1,141,0,0.1452321,"\int \frac{\left(d+e x^n\right)^3}{a+c x^{2 n}} \, dx","Int[(d + e*x^n)^3/(a + c*x^(2*n)),x]","\frac{e x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}+\frac{d x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)}","\frac{e x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}+\frac{d x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{3 d e^2 x}{c}+\frac{e^3 x^{n+1}}{c (n+1)}",1,"(3*d*e^2*x)/c + (e^3*x^(1 + n))/(c*(1 + n)) + (d*(c*d^2 - 3*a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (e*(3*c*d^2 - a*e^2)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c*(1 + n))","A",5,4,21,0.1905,1,"{1425, 1418, 245, 364}"
43,1,107,0,0.097461,"\int \frac{\left(d+e x^n\right)^2}{a+c x^{2 n}} \, dx","Int[(d + e*x^n)^2/(a + c*x^(2*n)),x]","\frac{x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^2 x}{c}","\frac{x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}+\frac{e^2 x}{c}",1,"(e^2*x)/c + ((c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) + (2*d*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))","A",5,4,21,0.1905,1,"{1425, 1418, 245, 364}"
44,1,83,0,0.0274489,"\int \frac{d+e x^n}{a+c x^{2 n}} \, dx","Int[(d + e*x^n)/(a + c*x^(2*n)),x]","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1)}",1,"(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n))","A",3,3,19,0.1579,1,"{1418, 245, 364}"
45,1,152,0,0.1105922,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)} \, dx","Int[1/((d + e*x^n)*(a + c*x^(2*n))),x]","-\frac{c e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)}+\frac{c d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)}","-\frac{c e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)}+\frac{c d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)}",1,"(c*d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)) - (c*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)*(1 + n))","A",6,4,21,0.1905,1,"{1425, 245, 1418, 364}"
46,1,205,0,0.1738187,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)} \, dx","Int[1/((d + e*x^n)^2*(a + c*x^(2*n))),x]","-\frac{2 c^2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}+\frac{c x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{2 c e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)}","-\frac{2 c^2 d e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}+\frac{c x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{2 c e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)}",1,"(c*(c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2) + (2*c*e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^2 - (2*c^2*d*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2*(1 + n)) + (e^2*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2))","A",7,4,21,0.1905,1,"{1425, 245, 1418, 364}"
47,1,81,0,0.0283905,"\int \frac{d+e x^n}{a-c x^{2 n}} \, dx","Int[(d + e*x^n)/(a - c*x^(2*n)),x]","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a (n+1)}","\frac{d x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a}+\frac{e x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);\frac{c x^{2 n}}{a}\right)}{a (n+1)}",1,"(d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, (c*x^(2*n))/a])/a + (e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, (c*x^(2*n))/a])/(a*(1 + n))","A",3,3,20,0.1500,1,"{1418, 245, 364}"
48,1,288,0,0.2521692,"\int \frac{\left(d+e x^n\right)^3}{\left(a+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)^3/(a + c*x^(2*n))^2,x]","-\frac{e (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{3 d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}","-\frac{e (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{3 d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c (n+1)}",1,"(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(2*a*c*n*(a + c*x^(2*n))) + (3*d*e^2*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) - (d*(c*d^2 - 3*a*e^2)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n) + (e^3*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c*(1 + n)) - (e*(3*c*d^2 - a*e^2)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n*(1 + n))","A",9,5,21,0.2381,1,"{1437, 1431, 1418, 245, 364}"
49,1,203,0,0.1673524,"\int \frac{\left(d+e x^n\right)^2}{\left(a+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)^2/(a + c*x^(2*n))^2,x]","-\frac{(1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1)}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}","-\frac{(1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1)}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a c}",1,"(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(2*a*c*n*(a + c*x^(2*n))) + (e^2*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*c) - ((c*d^2 - a*e^2)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n) - (d*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*n*(1 + n))","A",7,5,21,0.2381,1,"{1437, 1431, 1418, 245, 364}"
50,1,134,0,0.0543761,"\int \frac{d+e x^n}{\left(a+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)/(a + c*x^(2*n))^2,x]","-\frac{d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n}-\frac{e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1)}+\frac{x \left(d+e x^n\right)}{2 a n \left(a+c x^{2 n}\right)}","-\frac{d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n}-\frac{e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1)}+\frac{x \left(d+e x^n\right)}{2 a n \left(a+c x^{2 n}\right)}",1,"(x*(d + e*x^n))/(2*a*n*(a + c*x^(2*n))) - (d*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n) - (e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n*(1 + n))","A",4,4,19,0.2105,1,"{1431, 1418, 245, 364}"
51,1,333,0,0.2243944,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)^2} \, dx","Int[1/((d + e*x^n)*(a + c*x^(2*n))^2),x]","\frac{c e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)}-\frac{c d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)}-\frac{c e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}+\frac{c d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^2}+\frac{c x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}","\frac{c e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)}-\frac{c d (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)}-\frac{c e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^2}+\frac{c d e^2 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^2}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^2}+\frac{c x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}",1,"(c*x*(d - e*x^n))/(2*a*(c*d^2 + a*e^2)*n*(a + c*x^(2*n))) + (c*d*e^2*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2) - (c*d*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)*n) + (e^4*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^2) - (c*e^3*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^2*(1 + n)) + (c*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)*n*(1 + n))","A",10,5,21,0.2381,1,"{1437, 245, 1431, 1418, 364}"
52,1,410,0,0.3787252,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^2} \, dx","Int[1/((d + e*x^n)^2*(a + c*x^(2*n))^2),x]","\frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^2}-\frac{c (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}+\frac{4 c e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^3}+\frac{e^4 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^2}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}","\frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^2}-\frac{c (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}+\frac{4 c e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^3}+\frac{e^4 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^2}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}",1,"(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) + (c*e^2*(3*c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3) - (c*(c*d^2 - a*e^2)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n) + (4*c*e^4*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^3 - (4*c^2*d*e^3*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3*(1 + n)) + (c^2*d*e*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(c*d^2 + a*e^2)^2*n*(1 + n)) + (e^4*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^2)","A",11,5,21,0.2381,1,"{1437, 245, 1431, 1418, 364}"
53,1,424,0,0.3825564,"\int \frac{\left(d+e x^n\right)^3}{\left(a+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)^3/(a + c*x^(2*n))^3,x]","\frac{e (1-3 n) (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2 (n+1)}+\frac{d (1-4 n) (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}-\frac{x \left(e (1-3 n) x^n \left(3 c d^2-a e^2\right)+d (1-4 n) \left(c d^2-3 a e^2\right)\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}-\frac{3 d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{4 a c n \left(a+c x^{2 n}\right)^2}+\frac{e^2 x \left(3 d+e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}","\frac{e (1-3 n) (1-n) x^{n+1} \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2 (n+1)}+\frac{d (1-4 n) (1-2 n) x \left(c d^2-3 a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}-\frac{x \left(e (1-3 n) x^n \left(3 c d^2-a e^2\right)+d (1-4 n) \left(c d^2-3 a e^2\right)\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}-\frac{3 d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n}-\frac{e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 c n (n+1)}+\frac{x \left(e x^n \left(3 c d^2-a e^2\right)+d \left(c d^2-3 a e^2\right)\right)}{4 a c n \left(a+c x^{2 n}\right)^2}+\frac{e^2 x \left(3 d+e x^n\right)}{2 a c n \left(a+c x^{2 n}\right)}",1,"(x*(d*(c*d^2 - 3*a*e^2) + e*(3*c*d^2 - a*e^2)*x^n))/(4*a*c*n*(a + c*x^(2*n))^2) + (e^2*x*(3*d + e*x^n))/(2*a*c*n*(a + c*x^(2*n))) - (x*(d*(c*d^2 - 3*a*e^2)*(1 - 4*n) + e*(3*c*d^2 - a*e^2)*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + (d*(c*d^2 - 3*a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*c*n^2) - (3*d*e^2*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n) + (e*(3*c*d^2 - a*e^2)*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*c*n^2*(1 + n)) - (e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*c*n*(1 + n))","A",11,5,21,0.2381,1,"{1437, 1431, 1418, 245, 364}"
54,1,272,0,0.2469664,"\int \frac{\left(d+e x^n\right)^2}{\left(a+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)^2/(a + c*x^(2*n))^3,x]","\frac{(1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}-\frac{x \left((1-4 n) \left(c d^2-a e^2\right)+2 c d e (1-3 n) x^n\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}+\frac{d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1)}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 c}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{4 a c n \left(a+c x^{2 n}\right)^2}","\frac{(1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 c n^2}-\frac{x \left((1-4 n) \left(c d^2-a e^2\right)+2 c d e (1-3 n) x^n\right)}{8 a^2 c n^2 \left(a+c x^{2 n}\right)}+\frac{d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1)}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 c}+\frac{x \left(-a e^2+c d^2+2 c d e x^n\right)}{4 a c n \left(a+c x^{2 n}\right)^2}",1,"(x*(c*d^2 - a*e^2 + 2*c*d*e*x^n))/(4*a*c*n*(a + c*x^(2*n))^2) - (x*((c*d^2 - a*e^2)*(1 - 4*n) + 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*c*n^2*(a + c*x^(2*n))) + ((c*d^2 - a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*c*n^2) + (d*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(4*a^3*n^2*(1 + n)) + (e^2*x*Hypergeometric2F1[2, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*c)","A",8,5,21,0.2381,1,"{1437, 1431, 1418, 245, 364}"
55,1,184,0,0.1007402,"\int \frac{d+e x^n}{\left(a+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)/(a + c*x^(2*n))^3,x]","-\frac{x \left(d (1-4 n)+e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a+c x^{2 n}\right)}+\frac{d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2}+\frac{e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1)}+\frac{x \left(d+e x^n\right)}{4 a n \left(a+c x^{2 n}\right)^2}","-\frac{x \left(d (1-4 n)+e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a+c x^{2 n}\right)}+\frac{d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2}+\frac{e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1)}+\frac{x \left(d+e x^n\right)}{4 a n \left(a+c x^{2 n}\right)^2}",1,"(x*(d + e*x^n))/(4*a*n*(a + c*x^(2*n))^2) - (x*(d*(1 - 4*n) + e*(1 - 3*n)*x^n))/(8*a^2*n^2*(a + c*x^(2*n))) + (d*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*n^2) + (e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*n^2*(1 + n))","A",5,4,19,0.2105,1,"{1431, 1418, 245, 364}"
56,1,582,0,0.4160654,"\int \frac{1}{\left(d+e x^n\right) \left(a+c x^{2 n}\right)^3} \, dx","Int[1/((d + e*x^n)*(a + c*x^(2*n))^3),x]","-\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1) \left(a e^2+c d^2\right)}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)^2}-\frac{c d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{c x \left(d (1-4 n)-e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}-\frac{c e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c d e^4 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}+\frac{e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}+\frac{c x \left(d-e x^n\right)}{4 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)^2}","-\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 (n+1) \left(a e^2+c d^2\right)}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n (n+1) \left(a e^2+c d^2\right)^2}-\frac{c d e^2 (1-2 n) x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^2}-\frac{c x \left(d (1-4 n)-e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)}-\frac{c e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^3}+\frac{c d e^4 x \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^3}+\frac{e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(d-e x^n\right)}{2 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}+\frac{c x \left(d-e x^n\right)}{4 a n \left(a e^2+c d^2\right) \left(a+c x^{2 n}\right)^2}",1,"(c*x*(d - e*x^n))/(4*a*(c*d^2 + a*e^2)*n*(a + c*x^(2*n))^2) + (c*e^2*x*(d - e*x^n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) - (c*x*(d*(1 - 4*n) - e*(1 - 3*n)*x^n))/(8*a^2*(c*d^2 + a*e^2)*n^2*(a + c*x^(2*n))) + (c*d*e^4*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3) + (c*d*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)*n^2) - (c*d*e^2*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n) + (e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^3) - (c*e^5*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3*(1 + n)) - (c*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)*n^2*(1 + n)) + (c*e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n*(1 + n))","A",15,5,21,0.2381,1,"{1437, 245, 1431, 1418, 364}"
57,1,701,0,0.6923857,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^3} \, dx","Int[1/((d + e*x^n)^2*(a + c*x^(2*n))^3),x]","-\frac{c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1) \left(a e^2+c d^2\right)^2}+\frac{2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^3}+\frac{c (1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)^2}-\frac{c e^2 (1-2 n) x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^3}-\frac{c x \left((1-4 n) \left(c d^2-a e^2\right)-2 c d e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}-\frac{6 c^2 d e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^4}+\frac{c e^4 x \left(5 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^4}+\frac{6 c e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^4}+\frac{e^6 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(-a e^2+3 c d^2-4 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^3 \left(a+c x^{2 n}\right)}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{4 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)^2}","-\frac{c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{4 a^3 n^2 (n+1) \left(a e^2+c d^2\right)^2}+\frac{2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a^2 n (n+1) \left(a e^2+c d^2\right)^3}+\frac{c (1-4 n) (1-2 n) x \left(c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{8 a^3 n^2 \left(a e^2+c d^2\right)^2}-\frac{c e^2 (1-2 n) x \left(3 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 a^2 n \left(a e^2+c d^2\right)^3}-\frac{c x \left((1-4 n) \left(c d^2-a e^2\right)-2 c d e (1-3 n) x^n\right)}{8 a^2 n^2 \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)}-\frac{6 c^2 d e^5 x^{n+1} \, _2F_1\left(1,\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a (n+1) \left(a e^2+c d^2\right)^4}+\frac{c e^4 x \left(5 c d^2-a e^2\right) \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{a \left(a e^2+c d^2\right)^4}+\frac{6 c e^6 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{\left(a e^2+c d^2\right)^4}+\frac{e^6 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2+c d^2\right)^3}+\frac{c e^2 x \left(-a e^2+3 c d^2-4 c d e x^n\right)}{2 a n \left(a e^2+c d^2\right)^3 \left(a+c x^{2 n}\right)}+\frac{c x \left(-a e^2+c d^2-2 c d e x^n\right)}{4 a n \left(a e^2+c d^2\right)^2 \left(a+c x^{2 n}\right)^2}",1,"(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(4*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))^2) + (c*e^2*x*(3*c*d^2 - a*e^2 - 4*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^3*n*(a + c*x^(2*n))) - (c*x*((c*d^2 - a*e^2)*(1 - 4*n) - 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*(c*d^2 + a*e^2)^2*n^2*(a + c*x^(2*n))) + (c*e^4*(5*c*d^2 - a*e^2)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4) + (c*(c*d^2 - a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)^2*n^2) - (c*e^2*(3*c*d^2 - a*e^2)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^3*n) + (6*c*e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*e^2)^4 - (6*c^2*d*e^5*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4*(1 + n)) - (c^2*d*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(4*a^3*(c*d^2 + a*e^2)^2*n^2*(1 + n)) + (2*c^2*d*e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(c*d^2 + a*e^2)^3*n*(1 + n)) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^3)","A",16,5,21,0.2381,1,"{1437, 245, 1431, 1418, 364}"
58,1,171,0,0.1662208,"\int \frac{1}{\left(d+e x^n\right) \sqrt{a+c x^{2 n}}} \, dx","Int[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]),x]","\frac{x \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d \sqrt{a+c x^{2 n}}}-\frac{e x^{n+1} \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{n+1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1) \sqrt{a+c x^{2 n}}}","\frac{x \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d \sqrt{a+c x^{2 n}}}-\frac{e x^{n+1} \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left(\frac{n+1}{2 n};\frac{1}{2},1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1) \sqrt{a+c x^{2 n}}}",1,"(x*Sqrt[1 + (c*x^(2*n))/a]*AppellF1[1/(2*n), 1/2, 1, (2 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d*Sqrt[a + c*x^(2*n)]) - (e*x^(1 + n)*Sqrt[1 + (c*x^(2*n))/a]*AppellF1[(1 + n)/(2*n), 1/2, 1, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + n)*Sqrt[a + c*x^(2*n)])","A",6,5,23,0.2174,1,"{1438, 430, 429, 511, 510}"
59,0,0,0,0.0080332,"\int \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^q*(a + c*x^(2*n))^p,x]","\int \left(d+e x^n\right)^q \left(a+c x^{2 n}\right)^p \, dx","\text{Int}\left(\left(a+c x^{2 n}\right)^p \left(d+e x^n\right)^q,x\right)",0,"Defer[Int][(d + e*x^n)^q*(a + c*x^(2*n))^p, x]","A",0,0,0,0,-1,"{}"
60,1,299,0,0.1602869,"\int \left(d+e x^n\right)^3 \left(a+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^3*(a + c*x^(2*n))^p,x]","\frac{3 d^2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+d^3 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{3 d e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right),-p;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{3 n+1}","\frac{3 d^2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+d^3 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{3 d e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right),-p;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{3 n+1}",1,"(3*d*e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (e^3*x^(1 + 3*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(3 + n^(-1))/2, -p, (5 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 3*n)*(1 + (c*x^(2*n))/a)^p) + (d^3*x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (3*d^2*e*x^(1 + n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + n)*(1 + (c*x^(2*n))/a)^p)","A",10,5,21,0.2381,1,"{1437, 246, 245, 365, 364}"
61,1,217,0,0.1008809,"\int \left(d+e x^n\right)^2 \left(a+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]","d^2 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{2 d e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}","d^2 x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{2 d e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right),-p;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{2 n+1}",1,"(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, -p, (4 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (d^2*x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (2*d*e*x^(1 + n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + n)*(1 + (c*x^(2*n))/a)^p)","A",8,5,21,0.2381,1,"{1437, 246, 245, 365, 364}"
62,1,135,0,0.0604615,"\int \left(d+e x^n\right) \left(a+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)*(a + c*x^(2*n))^p,x]","d x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}","d x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2 n},-p;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)+\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} \, _2F_1\left(\frac{n+1}{2 n},-p;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a}\right)}{n+1}",1,"(d*x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (e*x^(1 + n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + n)*(1 + (c*x^(2*n))/a)^p)","A",6,5,19,0.2632,1,"{1433, 246, 245, 365, 364}"
63,1,167,0,0.1415182,"\int \frac{\left(a+c x^{2 n}\right)^p}{d+e x^n} \, dx","Int[(a + c*x^(2*n))^p/(d + e*x^n),x]","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d}-\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1)}","\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,1;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d}-\frac{e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,1;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2 (n+1)}",1,"(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, (2 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d*(1 + (c*x^(2*n))/a)^p) - (e*x^(1 + n)*(a + c*x^(2*n))^p*AppellF1[(1 + n)/(2*n), -p, 1, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + n)*(1 + (c*x^(2*n))/a)^p)","A",6,5,21,0.2381,1,"{1438, 430, 429, 511, 510}"
64,1,261,0,0.2354681,"\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Int[(a + c*x^(2*n))^p/(d + e*x^n)^2,x]","-\frac{2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,2;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3 (n+1)}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,2;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (2 n+1)}+\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,2;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2}","-\frac{2 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,2;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3 (n+1)}+\frac{e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,2;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (2 n+1)}+\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,2;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^2}",1,"(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 2, (4 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 2, (2 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + (c*x^(2*n))/a)^p) - (2*e*x^(1 + n)*(a + c*x^(2*n))^p*AppellF1[(1 + n)/(2*n), -p, 2, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + n)*(1 + (c*x^(2*n))/a)^p)","A",8,5,21,0.2381,1,"{1438, 430, 429, 511, 510}"
65,1,357,0,0.3380363,"\int \frac{\left(a+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","Int[(a + c*x^(2*n))^p/(d + e*x^n)^3,x]","-\frac{3 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,3;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (n+1)}+\frac{3 e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^5 (2 n+1)}-\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^6 (3 n+1)}+\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,3;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3}","-\frac{3 e x^{n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2 n};-p,3;\frac{1}{2} \left(3+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^4 (n+1)}+\frac{3 e^2 x^{2 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(2+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(4+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^5 (2 n+1)}-\frac{e^3 x^{3 n+1} \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2} \left(3+\frac{1}{n}\right);-p,3;\frac{1}{2} \left(5+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^6 (3 n+1)}+\frac{x \left(a+c x^{2 n}\right)^p \left(\frac{c x^{2 n}}{a}+1\right)^{-p} F_1\left(\frac{1}{2 n};-p,3;\frac{1}{2} \left(2+\frac{1}{n}\right);-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right)}{d^3}",1,"(3*e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 3, (4 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^5*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) - (e^3*x^(1 + 3*n)*(a + c*x^(2*n))^p*AppellF1[(3 + n^(-1))/2, -p, 3, (5 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^6*(1 + 3*n)*(1 + (c*x^(2*n))/a)^p) + (x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 3, (2 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + (c*x^(2*n))/a)^p) - (3*e*x^(1 + n)*(a + c*x^(2*n))^p*AppellF1[(1 + n)/(2*n), -p, 3, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + n)*(1 + (c*x^(2*n))/a)^p)","A",10,5,21,0.2381,1,"{1438, 430, 429, 511, 510}"
66,1,62,0,0.038973,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right) \, dx","Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n)),x]","\frac{x^{n+1} (a e+b d)}{n+1}+a d x+\frac{x^{2 n+1} (b e+c d)}{2 n+1}+\frac{c e x^{3 n+1}}{3 n+1}","\frac{x^{n+1} (a e+b d)}{n+1}+a d x+\frac{x^{2 n+1} (b e+c d)}{2 n+1}+\frac{c e x^{3 n+1}}{3 n+1}",1,"a*d*x + ((b*d + a*e)*x^(1 + n))/(1 + n) + ((c*d + b*e)*x^(1 + 2*n))/(1 + 2*n) + (c*e*x^(1 + 3*n))/(1 + 3*n)","A",2,1,22,0.04545,1,"{1407}"
67,1,132,0,0.1016745,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^2 \, dx","Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^2,x]","a^2 d x+\frac{x^{2 n+1} \left(2 a b e+2 a c d+b^2 d\right)}{2 n+1}+\frac{x^{3 n+1} \left(2 a c e+b^2 e+2 b c d\right)}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1}","a^2 d x+\frac{x^{2 n+1} \left(2 a b e+2 a c d+b^2 d\right)}{2 n+1}+\frac{x^{3 n+1} \left(2 a c e+b^2 e+2 b c d\right)}{3 n+1}+\frac{a x^{n+1} (a e+2 b d)}{n+1}+\frac{c x^{4 n+1} (2 b e+c d)}{4 n+1}+\frac{c^2 e x^{5 n+1}}{5 n+1}",1,"a^2*d*x + (a*(2*b*d + a*e)*x^(1 + n))/(1 + n) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^(1 + 2*n))/(1 + 2*n) + ((2*b*c*d + b^2*e + 2*a*c*e)*x^(1 + 3*n))/(1 + 3*n) + (c*(c*d + 2*b*e)*x^(1 + 4*n))/(1 + 4*n) + (c^2*e*x^(1 + 5*n))/(1 + 5*n)","A",2,1,24,0.04167,1,"{1432}"
68,1,218,0,0.20067,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^3 \, dx","Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^3,x]","\frac{x^{3 n+1} \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{3 n+1}+\frac{a^2 x^{n+1} (a e+3 b d)}{n+1}+a^3 d x+\frac{x^{4 n+1} \left(6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right)}{4 n+1}+\frac{3 a x^{2 n+1} \left(a b e+a c d+b^2 d\right)}{2 n+1}+\frac{3 c x^{5 n+1} \left(a c e+b^2 e+b c d\right)}{5 n+1}+\frac{c^2 x^{6 n+1} (3 b e+c d)}{6 n+1}+\frac{c^3 e x^{7 n+1}}{7 n+1}","\frac{x^{3 n+1} \left(3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right)}{3 n+1}+\frac{a^2 x^{n+1} (a e+3 b d)}{n+1}+a^3 d x+\frac{x^{4 n+1} \left(6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right)}{4 n+1}+\frac{3 a x^{2 n+1} \left(a b e+a c d+b^2 d\right)}{2 n+1}+\frac{3 c x^{5 n+1} \left(a c e+b^2 e+b c d\right)}{5 n+1}+\frac{c^2 x^{6 n+1} (3 b e+c d)}{6 n+1}+\frac{c^3 e x^{7 n+1}}{7 n+1}",1,"a^3*d*x + (a^2*(3*b*d + a*e)*x^(1 + n))/(1 + n) + (3*a*(b^2*d + a*c*d + a*b*e)*x^(1 + 2*n))/(1 + 2*n) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^(1 + 3*n))/(1 + 3*n) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^(1 + 4*n))/(1 + 4*n) + (3*c*(b*c*d + b^2*e + a*c*e)*x^(1 + 5*n))/(1 + 5*n) + (c^2*(c*d + 3*b*e)*x^(1 + 6*n))/(1 + 6*n) + (c^3*e*x^(1 + 7*n))/(1 + 7*n)","A",2,1,24,0.04167,1,"{1432}"
69,1,308,0,0.6990901,"\int \frac{\left(d+e x^n\right)^3}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c^2 \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c^2 \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)}","\frac{x \left(\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c^2 \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{(2 c d-b e) \left(-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right)}{\sqrt{b^2-4 a c}}-a c e^3+b^2 e^3-3 b c d e^2+3 c^2 d^2 e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c^2 \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^{n+1}}{c (n+1)}",1,"(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^(1 + n))/(c*(1 + n)) + ((3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3 + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(c^2*(b - Sqrt[b^2 - 4*a*c])) + ((3*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3 - a*c*e^3 - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(c^2*(b + Sqrt[b^2 - 4*a*c]))","A",5,3,26,0.1154,1,"{1424, 1422, 245}"
70,1,224,0,0.475921,"\int \frac{\left(d+e x^n\right)^2}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x}{c}","\frac{x \left(\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left(-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}-b e^2+2 c d e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c \left(\sqrt{b^2-4 a c}+b\right)}+\frac{e^2 x}{c}",1,"(e^2*x)/c + ((2*c*d*e - b*e^2 + (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(c*(b - Sqrt[b^2 - 4*a*c])) + ((2*c*d*e - b*e^2 - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(c*(b + Sqrt[b^2 - 4*a*c]))","A",5,3,26,0.1154,1,"{1424, 1422, 245}"
71,1,154,0,0.1202342,"\int \frac{d+e x^n}{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n)),x]","\frac{x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{b-\sqrt{b^2-4 a c}}+\frac{x \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}+b}","\frac{x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{b-\sqrt{b^2-4 a c}}+\frac{x \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{b^2-4 a c}+b}",1,"((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c]) + ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c])","A",3,2,24,0.08333,1,"{1422, 245}"
72,1,243,0,0.4693909,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)} \, dx","Int[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]","-\frac{c x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{c x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(\sqrt{b^2-4 a c}+b\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)}","-\frac{c x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)}-\frac{c x \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(\sqrt{b^2-4 a c}+b\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)}",1,"-((c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2))) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2))","A",6,3,26,0.1154,1,"{1424, 245, 1422}"
73,1,368,0,0.7068934,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)} \, dx","Int[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))),x]","-\frac{c x \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c x \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)}","-\frac{c x \left(-2 c e \left(d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(\sqrt{b^2-4 a c}+b\right)+2 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c x \left(-2 c e \left(-d \sqrt{b^2-4 a c}+a e+b d\right)+b e^2 \left(b-\sqrt{b^2-4 a c}\right)+2 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)}",1,"-((c*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2)) - (c*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2) + (e^2*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^2) + (e^2*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a*e^2))","A",7,3,26,0.1154,1,"{1424, 245, 1422}"
74,1,552,0,1.0233463,"\int \frac{1}{\left(d+e x^n\right)^3 \left(a+b x^n+c x^{2 n}\right)} \, dx","Int[1/((d + e*x^n)^3*(a + b*x^n + c*x^(2*n))),x]","\frac{e^2 x \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(-d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)-b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+2 c^3 d^3\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^3 \left(a e^2-b d e+c d^2\right)}","\frac{e^2 x \left(-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(3 b \left(d \sqrt{b^2-4 a c}+a e\right)+a e \sqrt{b^2-4 a c}+3 b^2 d\right)-b^2 e^3 \left(\sqrt{b^2-4 a c}+b\right)+2 c^3 d^3\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}-\frac{c x \left(-3 c^2 d e \left(-d \sqrt{b^2-4 a c}+2 a e+b d\right)+c e^2 \left(-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right)-b^2 e^3 \left(b-\sqrt{b^2-4 a c}\right)+2 c^3 d^3\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^3}+\frac{e^2 x (2 c d-b e) \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^2 \left(a e^2-b d e+c d^2\right)^2}+\frac{e^2 x \, _2F_1\left(3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d^3 \left(a e^2-b d e+c d^2\right)}",1,"-((c*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 - 4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(Sqrt[b^2 - 4*a*c]*d + a*e)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3)) - (c*(2*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d - Sqrt[b^2 - 4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3) + (e^2*(2*c*d - b*e)*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a*e^2)^2) + (e^2*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^3*(c*d^2 - b*d*e + a*e^2))","A",8,3,26,0.1154,1,"{1424, 245, 1422}"
75,1,750,0,2.9382688,"\int \frac{\left(d+e x^n\right)^3}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^2,x]","\frac{x \left(\frac{b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)-a b^3 e^3 (1-3 n)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}+(1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a c n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left((1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)-\frac{b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)-a b^3 e^3 (1-3 n)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a c n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(-\left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right)-a b e \left(a e^2+3 c d^2\right)-2 a c d \left(c d^2-3 a e^2\right)+b^2 c d^3\right)}{a c n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}+\frac{e^2 x \left(\frac{6 c d-3 b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{e^2 x \left(e-\frac{3 (2 c d-b e)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c \left(\sqrt{b^2-4 a c}+b\right)}","\frac{x \left(\frac{b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)-a b^3 e^3 (1-3 n)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}+(1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a c n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}+\frac{x \left((1-n) \left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)-\frac{b^2 c d \left(3 a e^2 (1-3 n)-c d^2 (1-n)\right)-a b^3 e^3 (1-3 n)+2 a b c e \left(a e^2 (2-5 n)+3 c d^2 n\right)+4 a c^2 d (1-2 n) \left(c d^2-3 a e^2\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a c n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(-\left(a b^2 e^3-b c d \left(3 a e^2+c d^2\right)+2 a c e \left(3 c d^2-a e^2\right)\right)\right)-a b e \left(a e^2+3 c d^2\right)-2 a c d \left(c d^2-3 a e^2\right)+b^2 c d^3\right)}{a c n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}+\frac{e^2 x \left(\frac{6 c d-3 b e}{\sqrt{b^2-4 a c}}+e\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{c \left(b-\sqrt{b^2-4 a c}\right)}+\frac{e^2 x \left(e-\frac{3 (2 c d-b e)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{c \left(\sqrt{b^2-4 a c}+b\right)}",1,"(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (e^2*(e + (6*c*d - 3*b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(c*(b - Sqrt[b^2 - 4*a*c])) + (((a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*(1 - n) + (b^2*c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c*d^2 - 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) + (e^2*(e - (3*(2*c*d - b*e))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(c*(b + Sqrt[b^2 - 4*a*c])) + (((a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*(1 - n) - (b^2*c*d*(3*a*e^2*(1 - 3*n) - c*d^2*(1 - n)) - a*b^3*e^3*(1 - 3*n) + 4*a*c^2*d*(c*d^2 - 3*a*e^2)*(1 - 2*n) + 2*a*b*c*e*(a*e^2*(2 - 5*n) + 3*c*d^2*n))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)","A",9,4,26,0.1538,1,"{1436, 1430, 1422, 245}"
76,1,543,0,1.8183154,"\int \frac{\left(d+e x^n\right)^2}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^2,x]","-\frac{x \left((1-n) \left(a b e^2-4 a c d e+b c d^2\right)-\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}-\frac{x \left(\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}+(1-n) \left(a b e^2-4 a c d e+b c d^2\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(a b e^2-4 a c d e+b c d^2\right)-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{2 e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}","-\frac{x \left((1-n) \left(a b e^2-4 a c d e+b c d^2\right)-\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right)}-\frac{x \left(\frac{b^2 \left(a e^2 (1-3 n)-c d^2 (1-n)\right)+4 a b c d e n+4 a c (1-2 n) \left(c d^2-a e^2\right)}{\sqrt{b^2-4 a c}}+(1-n) \left(a b e^2-4 a c d e+b c d^2\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(\sqrt{b^2-4 a c}+b\right)}+\frac{x \left(x^n \left(a b e^2-4 a c d e+b c d^2\right)-2 a b d e-2 a \left(c d^2-a e^2\right)+b^2 d^2\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{2 e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 e^2 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{b \sqrt{b^2-4 a c}-4 a c+b^2}",1,"(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (2*e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) - (b^2*(a*e^2*(1 - 3*n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*n) - (2*e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (((b*c*d^2 - 4*a*c*d*e + a*b*e^2)*(1 - n) + (b^2*(a*e^2*(1 - 3*n) - c*d^2*(1 - n)) + 4*a*c*(c*d^2 - a*e^2)*(1 - 2*n) + 4*a*b*c*d*e*n)/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b + Sqrt[b^2 - 4*a*c])*n)","A",9,5,26,0.1923,1,"{1436, 1430, 1422, 245, 1347}"
77,1,328,0,0.6301195,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^2,x]","-\frac{c x \left(-(1-n) \sqrt{b^2-4 a c} (b d-2 a e)+2 a b e n+2 a c d (2-4 n)+b^2 (-d) (1-n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left((1-n) \sqrt{b^2-4 a c} (b d-2 a e)+2 a b e n+4 a c d (1-2 n)+b^2 (-d) (1-n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}","-\frac{c x \left(-b \left(d (1-n) \sqrt{b^2-4 a c}-2 a e n\right)+2 a \left(e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right)+b^2 (-(d-d n))\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(b \left(d (1-n) \sqrt{b^2-4 a c}+2 a e n\right)+2 a \left(c d (2-4 n)-e (1-n) \sqrt{b^2-4 a c}\right)+b^2 (-d) (1-n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)}",1,"(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (c*(2*a*c*d*(2 - 4*n) - b^2*d*(1 - n) - Sqrt[b^2 - 4*a*c]*(b*d - 2*a*e)*(1 - n) + 2*a*b*e*n)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (c*(4*a*c*d*(1 - 2*n) - b^2*d*(1 - n) + Sqrt[b^2 - 4*a*c]*(b*d - 2*a*e)*(1 - n) + 2*a*b*e*n)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n)","A",4,3,24,0.1250,1,"{1430, 1422, 245}"
78,1,726,0,1.9267942,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))^2),x]","-\frac{c x \left(\frac{2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)+b^3 (-e) (1-n)}{\sqrt{b^2-4 a c}}+(1-n) \left(2 a c e+b^2 (-e)+b c d\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right) \left(a e^2-b d e+c d^2\right)}+\frac{x \left(c x^n \left(2 a c e+b^2 (-e)+b c d\right)+3 a b c e-2 a c^2 d+b^2 c d-b^3 e\right)}{a n \left(b^2-4 a c\right) \left(a e^2-b d e+c d^2\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{c e^2 x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c e^2 x \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{c x \left(b^2 (1-n) \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(2 a e (2-3 n)-d (1-n) \sqrt{b^2-4 a c}\right)-2 a c \left(e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right)+b^3 (-e) (1-n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}","-\frac{c x \left(\frac{2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)+b^3 (-e) (1-n)}{\sqrt{b^2-4 a c}}+(1-n) \left(2 a c e+b^2 (-e)+b c d\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b-\sqrt{b^2-4 a c}\right) \left(a e^2-b d e+c d^2\right)}+\frac{x \left(c x^n \left(2 a c e+b^2 (-e)+b c d\right)+3 a b c e-2 a c^2 d+b^2 c d-b^3 e\right)}{a n \left(b^2-4 a c\right) \left(a e^2-b d e+c d^2\right) \left(a+b x^n+c x^{2 n}\right)}-\frac{c e^2 x \left(2 c d-e \left(\sqrt{b^2-4 a c}+b\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{\left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}-\frac{c e^2 x \left(2 c d-e \left(b-\sqrt{b^2-4 a c}\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)^2}+\frac{c x \left(b^2 (1-n) \left(e \sqrt{b^2-4 a c}+c d\right)+b c \left(2 a e (2-3 n)-d (1-n) \sqrt{b^2-4 a c}\right)-2 a c \left(e (1-n) \sqrt{b^2-4 a c}+2 c d (1-2 n)\right)+b^3 (-e) (1-n)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a n \left(b^2-4 a c\right) \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right) \left(a e^2-b d e+c d^2\right)}+\frac{e^4 x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right)}{d \left(a e^2-b d e+c d^2\right)^2}",1,"(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*n*(a + b*x^n + c*x^(2*n))) - (c*e^2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2) - (c*((2*a*b*c*e*(2 - 3*n) - 4*a*c^2*d*(1 - 2*n) + b^2*c*d*(1 - n) - b^3*e*(1 - n))/Sqrt[b^2 - 4*a*c] + (b*c*d - b^2*e + 2*a*c*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)*n) - (c*e^2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2) + (c*(b*c*(2*a*e*(2 - 3*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(2*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)*n) + (e^4*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^2)","A",10,4,26,0.1538,1,"{1436, 245, 1430, 1422}"
79,1,1129,0,3.3414203,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^2} \, dx","Int[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^2),x]","\frac{2 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d \left(c d^2-b e d+a e^2\right)^3}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d^2 \left(c d^2-b e d+a e^2\right)^2}-\frac{2 c \left(3 c^2 d^2+b \left(b+\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d+2 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{2 c \left(3 c^2 d^2+b \left(b-\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d-2 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right)-3 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(3 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}","\frac{2 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d \left(c d^2-b e d+a e^2\right)^3}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^4}{d^2 \left(c d^2-b e d+a e^2\right)^2}-\frac{2 c \left(3 c^2 d^2+b \left(b+\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d+2 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{2 c \left(3 c^2 d^2+b \left(b-\sqrt{b^2-4 a c}\right) e^2-c e \left(3 b d-2 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right)-3 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(e^2 (1-n) b^4-e \left(2 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (5-7 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-n)\right) b^2+c \left(3 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-\sqrt{b^2-4 a c} d (1-n)\right)-c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}",1,"-((x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b*x^n + c*x^(2*n)))) - (2*c*e^2*(3*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d + 2*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*(4*a*c^2*(e*(a*e*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) + 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2 - 3*n) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^4*e^2*(1 - n) - b^3*e*(2*c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) - (2*c*e^2*(3*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - c*e*(3*b*d - 2*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*(4*a*c^2*(e*(a*e*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(5 - 7*n) - 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(2 - 3*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) + 3*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + b^4*e^2*(1 - n) - b^3*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) + (2*e^4*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a*e^2)^2)","A",11,4,26,0.1538,1,"{1436, 245, 1430, 1422}"
80,1,1707,0,5.2527307,"\int \frac{\left(d+e x^n\right)^3}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)^3/(a + b*x^n + c*x^(2*n))^3,x]","\frac{\left(-e (1-n) b^3+\left(3 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)+3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a c \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{\left(-e (1-n) b^3+\left(3 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a c \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{x \left(c \left(-e b^2+3 c d b-2 a c e\right) x^n-6 a c^2 d+3 b^2 c d-b^3 e+a b c e\right) e^2}{a c^2 \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)-\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)+\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}-\frac{x \left(c \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right) x^n+a b^2 c^2 d \left(3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)+4 a^2 c^3 d \left(c d^2-3 a e^2\right) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left(3 c d^2 (2-3 n)-5 a e^2 n\right)-3 a b^3 c e \left(c d^2-3 a e^2 n\right)+b^4 c d \left(c (1-2 n) d^2+6 a e^2 n\right)\right)}{2 a^2 c^2 \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(-\left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right) x^n+b^2 c d^3-2 a c d \left(c d^2-3 a e^2\right)-a b e \left(3 c d^2+a e^2\right)\right)}{2 a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}","\frac{\left(-e (1-n) b^3+\left(3 c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)+3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a c \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{\left(-e (1-n) b^3+\left(3 c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a c \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{x \left(c \left(-e b^2+3 c d b-2 a c e\right) x^n-6 a c^2 d+3 b^2 c d-b^3 e+a b c e\right) e^2}{a c^2 \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)-\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}+\frac{\left((1-n) \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right)+\frac{2 a e^3 (1-n) n b^5-c d (1-n) \left(c (1-2 n) d^2+6 a e^2 n\right) b^4+a c e \left(3 c (1-n) d^2+a e^2 \left(30 n^2-19 n+1\right)\right) b^3+6 a c^2 d \left(c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-4 a^2 c^2 e \left(3 c \left(-3 n^2-n+1\right) d^2+a e^2 \left(19 n^2-11 n+1\right)\right) b-8 a^2 c^3 d \left(c d^2-3 a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 c \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}-\frac{x \left(c \left(-2 a e^3 n b^4+c d \left(c (1-2 n) d^2+6 a e^2 n\right) b^3-a c e \left(3 c d^2-a e^2 (2 n+1)\right) b^2-2 a c^2 d \left(c (2-7 n) d^2+3 a e^2 n\right) b+4 a^2 c^2 e \left(3 c d^2-a e^2\right) (1-3 n)\right) x^n+a b^2 c^2 d \left(3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)+4 a^2 c^3 d \left(c d^2-3 a e^2\right) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left(3 c d^2 (2-3 n)-5 a e^2 n\right)-3 a b^3 c e \left(c d^2-3 a e^2 n\right)+b^4 c d \left(c (1-2 n) d^2+6 a e^2 n\right)\right)}{2 a^2 c^2 \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(-\left(a b^2 e^3+2 a c \left(3 c d^2-a e^2\right) e-b c d \left(c d^2+3 a e^2\right)\right) x^n+b^2 c d^3-2 a c d \left(c d^2-3 a e^2\right)-a b e \left(3 c d^2+a e^2\right)\right)}{2 a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(2*a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(3*b^2*c*d - 6*a*c^2*d - b^3*e + a*b*c*e + c*(3*b*c*d - b^2*e - 2*a*c*e)*x^n))/(a*c^2*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*d*(3*a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) + 4*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 4*n) - 2*a*b^5*e^3*n + 2*a^2*b*c^2*e*(3*c*d^2*(2 - 3*n) - 5*a*e^2*n) - 3*a*b^3*c*e*(c*d^2 - 3*a*e^2*n) + b^4*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) + c*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n)))*x^n))/(2*a^2*c^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) + (e^2*(b*c*(2*a*e*(2 - 5*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(6*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) - (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) + (e^2*(b*c*(2*a*e*(2 - 5*n) - 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(6*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) + (2*a*b^5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)) - 4*a^2*b*c^2*e*(3*c*d^2*(1 - n - 3*n^2) + a*e^2*(1 - 11*n + 19*n^2)) + a*b^3*c*e*(3*c*d^2*(1 - n) + a*e^2*(1 - 19*n + 30*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*c*(b^2 - 4*a*c)^2*(b + Sqrt[b^2 - 4*a*c])*n^2)","A",11,4,26,0.1538,1,"{1436, 1430, 1422, 245}"
81,1,1191,0,4.0099444,"\int \frac{\left(d+e x^n\right)^2}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^3,x]","-\frac{\left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{\left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{x \left(b c x^n+b^2-2 a c\right) e^2}{a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{\left((1-n) \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)+\frac{-(1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}-\frac{\left((1-n) \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)-\frac{-(1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}+\frac{x \left(c \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right) x^n+2 a b^3 c d e-a b^2 c \left(a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)-4 a^2 c^2 \left(c d^2-a e^2\right) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left(c (1-2 n) d^2+2 a e^2 n\right)\right)}{2 a^2 c \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(\left(b c d^2-4 a c e d+a b e^2\right) x^n+b^2 d^2-2 a b d e-2 a \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}","-\frac{\left(-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) n}-\frac{\left(-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 n)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) n}+\frac{x \left(b c x^n+b^2-2 a c\right) e^2}{a c \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)}-\frac{\left((1-n) \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)+\frac{-(1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) n^2}-\frac{\left((1-n) \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right)-\frac{-(1-n) \left(c (1-2 n) d^2+2 a e^2 n\right) b^4+2 a c d e (1-n) b^3+2 a c \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(15 n^2-10 n+1\right)\right) b^2-8 a^2 c^2 d e \left(-3 n^2-n+1\right) b-8 a^2 c^2 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) n^2}+\frac{x \left(c \left(-\left(c (1-2 n) d^2+2 a e^2 n\right) b^3+2 a c d e b^2+2 a c \left(c (2-7 n) d^2+a e^2 n\right) b-8 a^2 c^2 d e (1-3 n)\right) x^n+2 a b^3 c d e-a b^2 c \left(a e^2 (1-9 n)-5 c d^2 (1-3 n)\right)-4 a^2 c^2 \left(c d^2-a e^2\right) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left(c (1-2 n) d^2+2 a e^2 n\right)\right)}{2 a^2 c \left(b^2-4 a c\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(\left(b c d^2-4 a c e d+a b e^2\right) x^n+b^2 d^2-2 a b d e-2 a \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b*e^2)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2 - 2*a*c + b*c*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (x*(2*a*b^3*c*d*e - a*b^2*c*(a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) - 4*a^2*c^2*(c*d^2 - a*e^2)*(1 - 4*n) - 4*a^2*b*c^2*d*e*(2 - 3*n) - b^4*(c*d^2*(1 - 2*n) + 2*a*e^2*n) + c*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n))*x^n))/(2*a^2*c*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n)) + (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2*n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*n^2) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n)) - (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2*n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b + Sqrt[b^2 - 4*a*c])*n^2)","A",11,5,26,0.1923,1,"{1436, 1430, 1422, 245, 1345}"
82,1,713,0,1.6634951,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^3} \, dx","Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^3,x]","\frac{x \left(c x^n \left(-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right)-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)+a b^3 e-b^4 d (1-2 n)\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(a+b x^n+c x^{2 n}\right)}+\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}+2 c d \left(8 n^2-6 n+1\right)\right)-2 a b c \left(2 a e \left(-3 n^2-n+1\right)-d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}\right)+b^3 (1-n) \left(a e-d (1-2 n) \sqrt{b^2-4 a c}\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right)+b^4 (-d) \left(2 n^2-3 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}-2 c d \left(8 n^2-6 n+1\right)\right)+2 a b c \left(d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}+2 a e \left(-3 n^2-n+1\right)\right)-b^3 (1-n) \left(d (1-2 n) \sqrt{b^2-4 a c}+a e\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right)+b^4 d \left(2 n^2-3 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{2 a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)^2}","\frac{x \left(c x^n \left(-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right)-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)+a b^3 e-b^4 d (1-2 n)\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(a+b x^n+c x^{2 n}\right)}+\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}+2 c d \left(8 n^2-6 n+1\right)\right)-2 a b c \left(2 a e \left(-3 n^2-n+1\right)-d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}\right)+b^3 (1-n) \left(a e-d (1-2 n) \sqrt{b^2-4 a c}\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right)+b^4 (-d) \left(2 n^2-3 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(-b \sqrt{b^2-4 a c}-4 a c+b^2\right)}-\frac{c x \left(-4 a^2 c \left(e \left(3 n^2-4 n+1\right) \sqrt{b^2-4 a c}-2 c d \left(8 n^2-6 n+1\right)\right)+2 a b c \left(d \left(7 n^2-9 n+2\right) \sqrt{b^2-4 a c}+2 a e \left(-3 n^2-n+1\right)\right)-b^3 (1-n) \left(d (1-2 n) \sqrt{b^2-4 a c}+a e\right)+a b^2 (1-n) \left(e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right)+b^4 d \left(2 n^2-3 n+1\right)\right) \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 n^2 \left(b^2-4 a c\right)^2 \left(b \sqrt{b^2-4 a c}-4 a c+b^2\right)}+\frac{x \left(c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right)}{2 a n \left(b^2-4 a c\right) \left(a+b x^n+c x^{2 n}\right)^2}",1,"(x*(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (x*(a*b^3*e - 4*a^2*c^2*d*(1 - 4*n) + 5*a*b^2*c*d*(1 - 3*n) - 2*a^2*b*c*e*(2 - 3*n) - b^4*d*(1 - 2*n) + c*(a*b^2*e + 2*a*b*c*d*(2 - 7*n) - 4*a^2*c*e*(1 - 3*n) - b^3*d*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) + (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e + 6*c*d*(1 - 3*n))*(1 - n) + b^3*(a*e - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) - b^4*d*(1 - 3*n + 2*n^2) - 2*a*b*c*(2*a*e*(1 - n - 3*n^2) - Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*n^2) - (c*(a*b^2*(Sqrt[b^2 - 4*a*c]*e - 6*c*d*(1 - 3*n))*(1 - n) - b^3*(a*e + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) + b^4*d*(1 - 3*n + 2*n^2) + 2*a*b*c*(2*a*e*(1 - n - 3*n^2) + Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2)) - 4*a^2*c*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) - 2*c*d*(1 - 6*n + 8*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n^2)","A",5,3,24,0.1250,1,"{1430, 1422, 245}"
83,1,1708,0,5.0744189,"\int \frac{1}{\left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^3} \, dx","Int[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))^3),x]","\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(-e (1-n) b^3+\left(c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(-e (1-n) b^3+\left(c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}-\frac{c \left(-e \left(2 n^2-3 n+1\right) b^5+\left(c d-\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4+c \left(a e (7-18 n)+\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)+2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)-2 c d \left(8 n^2-6 n+1\right)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{c \left(e \left(2 n^2-3 n+1\right) b^5-\left(c d+\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4-c \left(a e (7-18 n)-\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)-2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)+2 c d \left(8 n^2-6 n+1\right)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{x \left(-c \left(-e (1-2 n) b^4+c d (1-2 n) b^3+a c e (5-14 n) b^2-2 a c^2 d (2-7 n) b-4 a^2 c^2 e (1-3 n)\right) x^n+2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right) n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) n \left(b x^n+c x^{2 n}+a\right)^2}","\frac{x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b+\sqrt{b^2-4 a c}\right) e\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(2 c d-\left(b-\sqrt{b^2-4 a c}\right) e\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3}+\frac{c \left(-e (1-n) b^3+\left(c d-\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)+\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{c \left(-e (1-n) b^3+\left(c d+\sqrt{b^2-4 a c} e\right) (1-n) b^2+c \left(2 a e (2-3 n)-\sqrt{b^2-4 a c} d (1-n)\right) b-2 a c \left(2 c d (1-2 n)+\sqrt{b^2-4 a c} e (1-n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)}-\frac{c \left(-e \left(2 n^2-3 n+1\right) b^5+\left(c d-\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4+c \left(a e (7-18 n)+\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)+2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)-2 c d \left(8 n^2-6 n+1\right)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{c \left(e \left(2 n^2-3 n+1\right) b^5-\left(c d+\sqrt{b^2-4 a c} e\right) \left(2 n^2-3 n+1\right) b^4-c \left(a e (7-18 n)-\sqrt{b^2-4 a c} d (1-2 n)\right) (1-n) b^3+a c \left(\sqrt{b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right) (1-n) b^2-2 a c^2 \left(\sqrt{b^2-4 a c} d \left(7 n^2-9 n+2\right)-2 a e \left(13 n^2-13 n+3\right)\right) b-4 a^2 c^2 \left(\sqrt{b^2-4 a c} e \left(3 n^2-4 n+1\right)+2 c d \left(8 n^2-6 n+1\right)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right) n^2}+\frac{x \left(-c \left(-e (1-2 n) b^4+c d (1-2 n) b^3+a c e (5-14 n) b^2-2 a c^2 d (2-7 n) b-4 a^2 c^2 e (1-3 n)\right) x^n+2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right) n^2 \left(b x^n+c x^{2 n}+a\right)}+\frac{x \left(c \left(-e b^2+c d b+2 a c e\right) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right) n \left(b x^n+c x^{2 n}+a\right)^2}",1,"(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^n))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b*x^n + c*x^(2*n))) + (x*(2*a^2*b*c^2*e*(4 - 11*n) - 3*a*b^3*c*e*(2 - 5*n) - 4*a^2*c^3*d*(1 - 4*n) + 5*a*b^2*c^2*d*(1 - 3*n) - b^4*c*d*(1 - 2*n) + b^5*(e - 2*e*n) - c*(a*b^2*c*e*(5 - 14*n) - 2*a*b*c^2*d*(2 - 7*n) - 4*a^2*c^2*e*(1 - 3*n) + b^3*c*d*(1 - 2*n) - b^4*e*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*n^2*(a + b*x^n + c*x^(2*n))) - (c*e^4*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*e^2*(b*c*(2*a*e*(2 - 3*n) + Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(2*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) - (c*(a*b^2*c*(Sqrt[b^2 - 4*a*c]*e*(5 - 14*n) - 6*c*d*(1 - 3*n))*(1 - n) + b^3*c*(a*e*(7 - 18*n) + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) - b^5*e*(1 - 3*n + 2*n^2) + b^4*(c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - 3*n + 2*n^2) - 4*a^2*c^2*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) - 2*c*d*(1 - 6*n + 8*n^2)) - 2*a*b*c^2*(Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2) + 2*a*e*(3 - 13*n + 13*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)*n^2) - (c*e^4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3) + (c*e^2*(b*c*(2*a*e*(2 - 3*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(2*c*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) + (c*(a*b^2*c*(Sqrt[b^2 - 4*a*c]*e*(5 - 14*n) + 6*c*d*(1 - 3*n))*(1 - n) - b^3*c*(a*e*(7 - 18*n) - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) + b^5*e*(1 - 3*n + 2*n^2) - b^4*(c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - 3*n + 2*n^2) - 4*a^2*c^2*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n + 8*n^2)) - 2*a*b*c^2*(Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2) - 2*a*e*(3 - 13*n + 13*n^2)))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)*n^2) + (e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3)","A",15,4,26,0.1538,1,"{1436, 245, 1430, 1422}"
84,1,2446,0,8.9352792,"\int \frac{1}{\left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^3} \, dx","Int[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3),x]","\frac{3 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^4}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d^2 \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(10 c^2 d^2+3 b \left(b+\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d+3 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}-\frac{c \left(10 c^2 d^2+3 b \left(b-\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d-3 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d-2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right)-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d+2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(5 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}-\frac{x \left(c \left(-2 e^2 b^3+5 c d e b^2-c \left(3 c d^2-5 a e^2\right) b-8 a c^2 d e\right) x^n-2 b^4 e^2-14 a b c^2 d e+5 b^3 c d e-b^2 c \left(3 c d^2-7 a e^2\right)+2 a c^2 \left(3 c d^2-a e^2\right)\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n \left(b x^n+c x^{2 n}+a\right)}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)-\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)+\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}-\frac{x \left(c \left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) x^n+a b^2 c^2 \left(a e^2 (13-37 n)-5 c d^2 (1-3 n)\right)-b^4 c \left(a e^2 (7-17 n)-c d^2 (1-2 n)\right)-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left(c d^2-a e^2\right) (1-4 n)+b^6 e^2 (1-2 n)-2 b^5 c d e (1-2 n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)^2}","\frac{3 (2 c d-b e) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d \left(c d^2-b e d+a e^2\right)^4}+\frac{x \, _2F_1\left(2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right) e^6}{d^2 \left(c d^2-b e d+a e^2\right)^3}-\frac{c \left(10 c^2 d^2+3 b \left(b+\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d+3 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}-\frac{c \left(10 c^2 d^2+3 b \left(b-\sqrt{b^2-4 a c}\right) e^2-2 c e \left(5 b d-3 \sqrt{b^2-4 a c} d+a e\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^4}{\left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^4}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d-2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(c d \left(4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right)-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right) b+4 a c^2 \left(e \left(a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2-\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}+\frac{c \left(2 e^2 (1-n) b^4-e \left(5 c d+2 \sqrt{b^2-4 a c} e\right) (1-n) b^3-c \left(e \left(a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-n)\right) b^2+c \left(5 a \sqrt{b^2-4 a c} (1-n) e^2+c d \left(4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right)\right) b+4 a c^2 \left(e \left(a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right)-3 c d^2 (1-2 n)\right)\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right) e^2}{a \left(b^2-4 a c\right) \left(b^2+\sqrt{b^2-4 a c} b-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n}-\frac{x \left(c \left(-2 e^2 b^3+5 c d e b^2-c \left(3 c d^2-5 a e^2\right) b-8 a c^2 d e\right) x^n-2 b^4 e^2-14 a b c^2 d e+5 b^3 c d e-b^2 c \left(3 c d^2-7 a e^2\right)+2 a c^2 \left(3 c d^2-a e^2\right)\right) e^2}{a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^3 n \left(b x^n+c x^{2 n}+a\right)}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)-\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b-\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}+\frac{c \left(\left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) (1-n)+\frac{-e^2 \left(2 n^2-3 n+1\right) b^6+2 c d e \left(2 n^2-3 n+1\right) b^5+c \left(4 a e^2 (2-5 n)-c d^2 (1-2 n)\right) (1-n) b^4-2 a c^2 d e \left(18 n^2-25 n+7\right) b^3+2 a c^2 \left(3 c d^2 \left(3 n^2-4 n+1\right)-a e^2 \left(35 n^2-38 n+9\right)\right) b^2+8 a^2 c^3 d e \left(13 n^2-13 n+3\right) b-8 a^2 c^3 \left(c d^2-a e^2\right) \left(8 n^2-6 n+1\right)}{\sqrt{b^2-4 a c}}\right) x \, _2F_1\left(1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(b+\sqrt{b^2-4 a c}\right) \left(c d^2-b e d+a e^2\right)^2 n^2}-\frac{x \left(c \left(e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left(2 a e^2 (3-8 n)-c d^2 (1-2 n)\right) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left(a e^2 (4-13 n)-c d^2 (2-7 n)\right) b-8 a^2 c^3 d e (1-3 n)\right) x^n+a b^2 c^2 \left(a e^2 (13-37 n)-5 c d^2 (1-3 n)\right)-b^4 c \left(a e^2 (7-17 n)-c d^2 (1-2 n)\right)-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left(c d^2-a e^2\right) (1-4 n)+b^6 e^2 (1-2 n)-2 b^5 c d e (1-2 n)\right)}{2 a^2 \left(b^2-4 a c\right)^2 \left(c d^2-b e d+a e^2\right)^2 n^2 \left(b x^n+c x^{2 n}+a\right)}-\frac{x \left(c \left(-e^2 b^3+2 c d e b^2-c \left(c d^2-3 a e^2\right) b-4 a c^2 d e\right) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left(c d^2-4 a e^2\right)+2 a c^2 \left(c d^2-a e^2\right)\right)}{2 a \left(b^2-4 a c\right) \left(c d^2-b e d+a e^2\right)^2 n \left(b x^n+c x^{2 n}+a\right)^2}",1,"-(x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^n))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b*x^n + c*x^(2*n))^2) - (e^2*x*(5*b^3*c*d*e - 14*a*b*c^2*d*e - 2*b^4*e^2 - b^2*c*(3*c*d^2 - 7*a*e^2) + 2*a*c^2*(3*c*d^2 - a*e^2) + c*(5*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(3*c*d^2 - 5*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*(a*e^2*(13 - 37*n) - 5*c*d^2*(1 - 3*n)) - b^4*c*(a*e^2*(7 - 17*n) - c*d^2*(1 - 2*n)) - 4*a^2*b*c^3*d*e*(4 - 11*n) + 6*a*b^3*c^2*d*e*(2 - 5*n) + 4*a^2*c^3*(c*d^2 - a*e^2)*(1 - 4*n) - 2*b^5*c*d*e*(1 - 2*n) + b^6*e^2*(1 - 2*n) + c*(2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*n^2*(a + b*x^n + c*x^(2*n))) - (c*e^4*(10*c^2*d^2 + 3*b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d + 3*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) + 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) + 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(5 - 8*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*b^4*e^2*(1 - n) - b^3*e*(5*c*d - 2*Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3*n) + (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) - (b^4*c*(4*a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2) - 8*a^2*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*n + 18*n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n^2) - (c*e^4*(10*c^2*d^2 + 3*b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d - 3*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) - 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) - 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(5 - 8*n) - 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) + 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*b^4*e^2*(1 - n) - b^3*e*(5*c*d + 2*Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^3*n) + (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) + (b^4*c*(4*a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2) - 8*a^2*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*n + 18*n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b + Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n^2) + (3*e^6*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^4) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a*e^2)^3)","A",16,4,26,0.1538,1,"{1436, 245, 1430, 1422}"
85,1,292,0,0.3551065,"\int \left(d+e x^n\right) \sqrt{a+b x^n+c x^{2 n}} \, dx","Int[(d + e*x^n)*Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{1}{2},-\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}","\frac{d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{1}{2},-\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{1}{2},-\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(e*x^(1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[1 + n^(-1), -1/2, -1/2, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]) + (d*x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -1/2, -1/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])","A",6,5,26,0.1923,1,"{1432, 1348, 429, 1385, 510}"
86,1,294,0,0.3479934,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^{3/2} \, dx","Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{a d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{a e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{3}{2},-\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}","\frac{a d x \sqrt{a+b x^n+c x^{2 n}} F_1\left(\frac{1}{n};-\frac{3}{2},-\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}+\frac{a e x^{n+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left(1+\frac{1}{n};-\frac{3}{2},-\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}}",1,"(a*e*x^(1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[1 + n^(-1), -3/2, -3/2, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]) + (a*d*x*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[n^(-1), -3/2, -3/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])","A",6,5,26,0.1923,1,"{1432, 1348, 429, 1385, 510}"
87,1,292,0,0.3381179,"\int \frac{d+e x^n}{\sqrt{a+b x^n+c x^{2 n}}} \, dx","Int[(d + e*x^n)/Sqrt[a + b*x^n + c*x^(2*n)],x]","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{a+b x^n+c x^{2 n}}}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{\sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{(n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(e*x^(1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 1/2, 1/2, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]) + (d*x*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 1/2, 1/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[a + b*x^n + c*x^(2*n)]","A",6,5,26,0.1923,1,"{1432, 1348, 429, 1385, 510}"
88,1,298,0,0.3458196,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^{3/2}} \, dx","Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^(3/2),x]","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{3}{2},\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{3}{2},\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a (n+1) \sqrt{a+b x^n+c x^{2 n}}}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{3}{2},\frac{3}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{3}{2},\frac{3}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a (n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(e*x^(1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 3/2, 3/2, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*(1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]) + (d*x*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 3/2, 3/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*Sqrt[a + b*x^n + c*x^(2*n)])","A",6,5,26,0.1923,1,"{1432, 1348, 429, 1385, 510}"
89,1,298,0,0.3470539,"\int \frac{d+e x^n}{\left(a+b x^n+c x^{2 n}\right)^{5/2}} \, dx","Int[(d + e*x^n)/(a + b*x^n + c*x^(2*n))^(5/2),x]","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{5}{2},\frac{5}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{5}{2},\frac{5}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 (n+1) \sqrt{a+b x^n+c x^{2 n}}}","\frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(\frac{1}{n};\frac{5}{2},\frac{5}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left(1+\frac{1}{n};\frac{5}{2},\frac{5}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{a^2 (n+1) \sqrt{a+b x^n+c x^{2 n}}}",1,"(e*x^(1 + n)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1 + n^(-1), 5/2, 5/2, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a^2*(1 + n)*Sqrt[a + b*x^n + c*x^(2*n)]) + (d*x*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[n^(-1), 5/2, 5/2, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a^2*Sqrt[a + b*x^n + c*x^(2*n)])","A",6,5,26,0.1923,1,"{1432, 1348, 429, 1385, 510}"
90,0,0,0,0.0100255,"\int \left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p,x]","\int \left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p \, dx","\text{Int}\left(\left(d+e x^n\right)^q \left(a+b x^n+c x^{2 n}\right)^p,x\right)",0,"Defer[Int][(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]","A",0,0,0,0,-1,"{}"
91,1,606,0,0.6219731,"\int \left(d+e x^n\right)^3 \left(a+b x^n+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^3*(a + b*x^n + c*x^(2*n))^p,x]","\frac{3 d^2 e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+d^3 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{3 d e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(3+\frac{1}{n};-p,-p;4+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 n+1}","\frac{3 d^2 e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+d^3 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{3 d e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}+\frac{e^3 x^{3 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(3+\frac{1}{n};-p,-p;4+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{3 n+1}",1,"(3*d^2*e*x^(1 + n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (3*d*e^2*x^(1 + 2*n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[2 + n^(-1), -p, -p, 3 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + 2*n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (e^3*x^(1 + 3*n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[3 + n^(-1), -p, -p, 4 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + 3*n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (d^3*x*(a + b*x^n + c*x^(2*n))^p*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",10,5,26,0.1923,1,"{1436, 1348, 429, 1385, 510}"
92,1,447,0,0.4611826,"\int \left(d+e x^n\right)^2 \left(a+b x^n+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^p,x]","d^2 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{2 d e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}","d^2 x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{2 d e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}+\frac{e^2 x^{2 n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(2+\frac{1}{n};-p,-p;3+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{2 n+1}",1,"(2*d*e*x^(1 + n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (e^2*x^(1 + 2*n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[2 + n^(-1), -p, -p, 3 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + 2*n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (d^2*x*(a + b*x^n + c*x^(2*n))^p*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",8,5,26,0.1923,1,"{1436, 1348, 429, 1385, 510}"
93,1,288,0,0.2865606,"\int \left(d+e x^n\right) \left(a+b x^n+c x^{2 n}\right)^p \, dx","Int[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p,x]","d x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}","d x \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)+\frac{e x^{n+1} \left(\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right)^{-p} \left(\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right)^{-p} \left(a+b x^n+c x^{2 n}\right)^p F_1\left(1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right)}{n+1}",1,"(e*x^(1 + n)*(a + b*x^n + c*x^(2*n))^p*AppellF1[1 + n^(-1), -p, -p, 2 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p) + (d*x*(a + b*x^n + c*x^(2*n))^p*AppellF1[n^(-1), -p, -p, 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)","A",6,5,24,0.2083,1,"{1432, 1348, 429, 1385, 510}"
94,0,0,0,0.0112573,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","Int[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n),x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{d+e x^n},x\right)",0,"Defer[Int][(a + b*x^n + c*x^(2*n))^p/(d + e*x^n), x]","A",0,0,0,0,-1,"{}"
95,0,0,0,0.0113417,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","Int[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^2,x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^2},x\right)",0,"Defer[Int][(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^2, x]","A",0,0,0,0,-1,"{}"
96,0,0,0,0.0118203,"\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","Int[(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^3,x]","\int \frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3} \, dx","\text{Int}\left(\frac{\left(a+b x^n+c x^{2 n}\right)^p}{\left(d+e x^n\right)^3},x\right)",0,"Defer[Int][(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^3, x]","A",0,0,0,0,-1,"{}"