1,1,169,389,0.017000," ","int(1/(e*x+d)/(c*x^4+a)^(1/2),x)","\frac{\frac{\sqrt{-\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, \sqrt{\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, e \EllipticPi \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, x , -\frac{i \sqrt{a}\, e^{2}}{\sqrt{c}\, d^{2}}, \frac{\sqrt{-\frac{i \sqrt{c}}{\sqrt{a}}}}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, \sqrt{c \,x^{4}+a}\, d}-\frac{\arctanh \left(\frac{\frac{2 c \,d^{2} x^{2}}{e^{2}}+2 a}{2 \sqrt{a +\frac{c \,d^{4}}{e^{4}}}\, \sqrt{c \,x^{4}+a}}\right)}{2 \sqrt{a +\frac{c \,d^{4}}{e^{4}}}}}{e}"," ",0,"1/e*(-1/2/(c*d^4/e^4+a)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(c*d^4/e^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(I/a^(1/2)*c^(1/2))^(1/2)/d*e*(-I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi((I/a^(1/2)*c^(1/2))^(1/2)*x,-I*a^(1/2)/c^(1/2)/d^2*e^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))","C"
2,1,421,601,0.017000," ","int(1/(e*x+d)^2/(c*x^4+a)^(1/2),x)","-\frac{\sqrt{-\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, \sqrt{\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, c \,d^{2} \EllipticF \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, x , i\right)}{\left(a \,e^{4}+c \,d^{4}\right) \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, \sqrt{c \,x^{4}+a}}+\frac{i \sqrt{-\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, \sqrt{\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, \left(-\EllipticE \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, x , i\right)+\EllipticF \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, x , i\right)\right) \sqrt{a}\, \sqrt{c}\, e^{2}}{\left(a \,e^{4}+c \,d^{4}\right) \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, \sqrt{c \,x^{4}+a}}+\frac{2 \left(\frac{\sqrt{-\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, \sqrt{\frac{i \sqrt{c}\, x^{2}}{\sqrt{a}}+1}\, e \EllipticPi \left(\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, x , -\frac{i \sqrt{a}\, e^{2}}{\sqrt{c}\, d^{2}}, \frac{\sqrt{-\frac{i \sqrt{c}}{\sqrt{a}}}}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}\right)}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\, \sqrt{c \,x^{4}+a}\, d}-\frac{\arctanh \left(\frac{\frac{2 c \,d^{2} x^{2}}{e^{2}}+2 a}{2 \sqrt{a +\frac{c \,d^{4}}{e^{4}}}\, \sqrt{c \,x^{4}+a}}\right)}{2 \sqrt{a +\frac{c \,d^{4}}{e^{4}}}}\right) c \,d^{3}}{\left(a \,e^{4}+c \,d^{4}\right) e}-\frac{\sqrt{c \,x^{4}+a}\, e^{3}}{\left(a \,e^{4}+c \,d^{4}\right) \left(e x +d \right)}"," ",0,"-e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)-c*d^2/(a*e^4+c*d^4)/(I/a^(1/2)*c^(1/2))^(1/2)*(-I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4+a)^(1/2)*EllipticF((I/a^(1/2)*c^(1/2))^(1/2)*x,I)+I*c^(1/2)*e^2/(a*e^4+c*d^4)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(-I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4+a)^(1/2)*(EllipticF((I/a^(1/2)*c^(1/2))^(1/2)*x,I)-EllipticE((I/a^(1/2)*c^(1/2))^(1/2)*x,I))+2*c*d^3/(a*e^4+c*d^4)/e*(-1/2/(a+c*d^4/e^4)^(1/2)*arctanh(1/2*(2*c*d^2/e^2*x^2+2*a)/(a+c*d^4/e^4)^(1/2)/(c*x^4+a)^(1/2))+1/(I/a^(1/2)*c^(1/2))^(1/2)/d*e*(-I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*c^(1/2)*x^2+1)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi((I/a^(1/2)*c^(1/2))^(1/2)*x,-I*a^(1/2)/c^(1/2)/d^2*e^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))","C"
3,1,281,488,0.020000," ","int(1/(e*x+d)/(c*x^4+b*x^2+a)^(1/2),x)","\frac{\frac{\sqrt{2}\, \sqrt{-\frac{\left(-b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{2 a}+1}\, \sqrt{\frac{\left(b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{2 a}+1}\, e \EllipticPi \left(\frac{\sqrt{2}\, \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, x}{2}, \frac{2 a \,e^{2}}{\left(-b +\sqrt{-4 a c +b^{2}}\right) d^{2}}, \frac{\sqrt{-\frac{b +\sqrt{-4 a c +b^{2}}}{2 a}}\, \sqrt{2}}{\sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}}\right)}{\sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}\, d}-\frac{\arctanh \left(\frac{b \,x^{2}+\frac{2 c \,d^{2} x^{2}}{e^{2}}+2 a +\frac{b \,d^{2}}{e^{2}}}{2 \sqrt{a +\frac{b \,d^{2}}{e^{2}}+\frac{c \,d^{4}}{e^{4}}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}}\right)}{2 \sqrt{a +\frac{b \,d^{2}}{e^{2}}+\frac{c \,d^{4}}{e^{4}}}}}{e}"," ",0,"1/e*(-1/2/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)*arctanh(1/2*(2*c*d^2/e^2*x^2+b*x^2+b*d^2/e^2+2*a)/(c*d^4/e^4+b*d^2/e^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)/d*e*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x,2/(-b+(-4*a*c+b^2)^(1/2))*a/d^2*e^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)))","A"
4,1,771,789,0.016000," ","int(1/(e*x+d)^2/(c*x^4+b*x^2+a)^(1/2),x)","-\frac{\sqrt{2}\, \sqrt{-\frac{2 \left(-b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{a}+4}\, \sqrt{\frac{2 \left(b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{a}+4}\, \left(-\EllipticE \left(\frac{\sqrt{2}\, \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, x}{2}, \frac{\sqrt{\frac{2 \left(b +\sqrt{-4 a c +b^{2}}\right) b}{a c}-4}}{2}\right)+\EllipticF \left(\frac{\sqrt{2}\, \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, x}{2}, \frac{\sqrt{\frac{2 \left(b +\sqrt{-4 a c +b^{2}}\right) b}{a c}-4}}{2}\right)\right) a c \,e^{2}}{2 \left(a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right) \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}\, \left(b +\sqrt{-4 a c +b^{2}}\right)}-\frac{\sqrt{2}\, \sqrt{-\frac{2 \left(-b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{a}+4}\, \sqrt{\frac{2 \left(b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{a}+4}\, c \,d^{2} \EllipticF \left(\frac{\sqrt{2}\, \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, x}{2}, \frac{\sqrt{\frac{2 \left(b +\sqrt{-4 a c +b^{2}}\right) b}{a c}-4}}{2}\right)}{4 \left(a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right) \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}}-\frac{\sqrt{c \,x^{4}+b \,x^{2}+a}\, e^{3}}{\left(a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right) \left(e x +d \right)}+\frac{\left(b \,e^{2}+2 c \,d^{2}\right) \left(\frac{\sqrt{2}\, \sqrt{-\frac{\left(-b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{2 a}+1}\, \sqrt{\frac{\left(b +\sqrt{-4 a c +b^{2}}\right) x^{2}}{2 a}+1}\, e \EllipticPi \left(\frac{\sqrt{2}\, \sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, x}{2}, \frac{2 a \,e^{2}}{\left(-b +\sqrt{-4 a c +b^{2}}\right) d^{2}}, \frac{\sqrt{-\frac{b +\sqrt{-4 a c +b^{2}}}{2 a}}\, \sqrt{2}}{\sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}}\right)}{\sqrt{\frac{-b +\sqrt{-4 a c +b^{2}}}{a}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}\, d}-\frac{\arctanh \left(\frac{b \,x^{2}+\frac{2 c \,d^{2} x^{2}}{e^{2}}+2 a +\frac{b \,d^{2}}{e^{2}}}{2 \sqrt{a +\frac{b \,d^{2}}{e^{2}}+\frac{c \,d^{4}}{e^{4}}}\, \sqrt{c \,x^{4}+b \,x^{2}+a}}\right)}{2 \sqrt{a +\frac{b \,d^{2}}{e^{2}}+\frac{c \,d^{4}}{e^{4}}}}\right) d}{\left(a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right) e}"," ",0,"-e^3*(c*x^4+b*x^2+a)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)-1/4*c*d^2/(a*e^4+b*d^2*e^2+c*d^4)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2+4)^(1/2)*(2*(b+(-4*a*c+b^2)^(1/2))/a*x^2+4)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x,1/2*(2*(b+(-4*a*c+b^2)^(1/2))/a*b/c-4)^(1/2))-1/2*c*e^2/(a*e^4+b*d^2*e^2+c*d^4)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2+4)^(1/2)*(2*(b+(-4*a*c+b^2)^(1/2))/a*x^2+4)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x,1/2*(2*(b+(-4*a*c+b^2)^(1/2))/a*b/c-4)^(1/2))-EllipticE(1/2*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x,1/2*(2*(b+(-4*a*c+b^2)^(1/2))/a*b/c-4)^(1/2)))+d*(b*e^2+2*c*d^2)/(a*e^4+b*d^2*e^2+c*d^4)/e*(-1/2/(a+b*d^2/e^2+c*d^4/e^4)^(1/2)*arctanh(1/2*(b*x^2+2*c*d^2/e^2*x^2+2*a+b*d^2/e^2)/(a+b*d^2/e^2+c*d^4/e^4)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)/d*e*(-1/2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2+1)^(1/2)*(1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2+1)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x,2/(-b+(-4*a*c+b^2)^(1/2))*a/d^2*e^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)))","A"