Integrand size = 26, antiderivative size = 955 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\frac {2}{3} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 a e \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 a \sqrt {d+e x} \left (c+b x+a x^2\right )}-\frac {\sqrt {2} c \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d},\arcsin \left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}\right )}{\sqrt {a} \left (c+b x+a x^2\right )} \]
2/3*x*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)+1/3*(a*d+b*e)*x*EllipticE(1/2*((b+ 2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+ b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^( 1/2)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/ 2)/a/e/(a*x^2+b*x+c)/(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/ 3*d*(a*d+b*e)*x*EllipticF(1/2*((b+2*a*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^( 1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2 ))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+c/x^2+b/x)^(1/2)*(-a*(a*x^2+b*x+ c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/ a/e/(a*x^2+b*x+c)/(e*x+d)^(1/2)+4/3*(b*d+c*e)*x*EllipticF(1/2*((b+2*a*x+(- 4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/ 2)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a+ c/x^2+b/x)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*(a*(e*x+d)/(2*a*d-e *(b+(-4*a*c+b^2)^(1/2))))^(1/2)/a/(a*x^2+b*x+c)/(e*x+d)^(1/2)-c*x*Elliptic Pi(2^(1/2)*a^(1/2)*(e*x+d)^(1/2)/(2*a*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2),1/ 2*(2*a*d-b*e+e*(-4*a*c+b^2)^(1/2))/a/d,((b-2*a*d/e-(-4*a*c+b^2)^(1/2))/(b- 2*a*d/e+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*(a+c/x^2+b/x)^(1/2)*(1-2*a*(e* x+d)/(2*a*d-e*(b-(-4*a*c+b^2)^(1/2))))^(1/2)*(2*a*d-e*(b-(-4*a*c+b^2)^(1/2 )))^(1/2)*(1-2*a*(e*x+d)/(2*a*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/(a*x^2+b* x+c)/a^(1/2)
Result contains complex when optimal does not.
Time = 28.32 (sec) , antiderivative size = 1258, normalized size of antiderivative = 1.32 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\frac {2}{3} x \sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}}+\frac {x (d+e x)^{3/2} \sqrt {a+\frac {c+b x}{x^2}} \left (\frac {4 e^2 (a d+b e) \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}{(d+e x)^2}-\frac {i \sqrt {2} (a d+b e) \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}+\frac {i \sqrt {2} \left (b e \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a \left (3 b d e-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}+\frac {6 i \sqrt {2} a c e^2 \sqrt {\frac {-2 c e^2+2 a d e x+b e (d-e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2-2 a d e x+b e (-d+e x)+\sqrt {\left (b^2-4 a c\right ) e^2} (d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticPi}\left (\frac {d \left (2 a d-b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (-b d+c e)\right )},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{6 a e^2 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))} \]
(2*x*Sqrt[d + e*x]*Sqrt[a + (c + b*x)/x^2])/3 + (x*(d + e*x)^(3/2)*Sqrt[a + (c + b*x)/x^2]*((4*e^2*(a*d + b*e)*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a *d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c + x*(b + a*x)))/(d + e*x)^2 - (I*S qrt[2]*(a*d + b*e)*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b* e*(-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b ^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b *d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], - ((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c )*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(b*e*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^ 2]) + a*(3*b*d*e - 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b *e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*c*e^2 - 2*a*d*e*x + b*e* (-d + e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d *e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -(( -2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)* e^2]))])/Sqrt[d + e*x] + ((6*I)*Sqrt[2]*a*c*e^2*Sqrt[(-2*c*e^2 + 2*a*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/((2*a*d - b*e + S...
Time = 1.68 (sec) , antiderivative size = 857, normalized size of antiderivative = 0.90, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1779, 1272, 25, 2154, 1269, 1172, 321, 327, 1279, 187, 413, 413, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \, dx\) |
| \(\Big \downarrow \) 1779 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \int \frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{x}dx}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1272 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}-\frac {1}{3} \int -\frac {(a d+b e) x^2+2 (b d+c e) x+3 c d}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \int \frac {(a d+b e) x^2+2 (b d+c e) x+3 c d}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (3 c d \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\int \frac {2 b d+2 c e+(a d+b e) x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (-\frac {\left (a d^2-e (b d+2 c e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{e}+3 c d \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\frac {(a d+b e) \int \frac {\sqrt {d+e x}}{\sqrt {a x^2+b x+c}}dx}{e}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+3 c d \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+3 c d \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (3 c d \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (\frac {3 c d \sqrt {-\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \int \frac {1}{x \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (-\frac {6 c d \sqrt {-\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \int -\frac {1}{e x \sqrt {b+\frac {2 a (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 a d}{e}} \sqrt {b+\frac {2 a (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}}d\sqrt {d+e x}}{\sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (-\frac {6 c d \sqrt {-\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {1-\frac {2 a (d+e x)}{2 a d-e \left (b-\sqrt {b^2-4 a c}\right )}} \int -\frac {1}{e x \sqrt {b+\frac {2 a (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 a d}{e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {a x^2+b x+c} \sqrt {-\sqrt {b^2-4 a c}+\frac {2 a (d+e x)}{e}-\frac {2 a d}{e}+b}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{3} \left (-\frac {6 c d \sqrt {-\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {1-\frac {2 a (d+e x)}{2 a d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int -\frac {1}{e x \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {a x^2+b x+c} \sqrt {-\sqrt {b^2-4 a c}+\frac {2 a (d+e x)}{e}-\frac {2 a d}{e}+b} \sqrt {\sqrt {b^2-4 a c}+\frac {2 a (d+e x)}{e}-\frac {2 a d}{e}+b}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (a d+b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{3} \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} (a d+b e) \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-e (b d+2 c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}-\frac {3 \sqrt {2} c \sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (\frac {2 a d-b e+\sqrt {b^2-4 a c} e}{2 a d},\arcsin \left (\frac {\sqrt {2} \sqrt {a} \sqrt {d+e x}}{\sqrt {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {2 a d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {a} \sqrt {a x^2+b x+c} \sqrt {b+\frac {2 a (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 a d}{e}} \sqrt {b+\frac {2 a (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 a d}{e}}}\right )+\frac {2}{3} \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
(Sqrt[a + c/x^2 + b/x]*x*((2*Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/3 + ((Sq rt[2]*Sqrt[b^2 - 4*a*c]*(a*d + b*e)*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x ^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x) /Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[ b^2 - 4*a*c])*e)])/(a*e*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c] )*e)]*Sqrt[c + b*x + a*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b* d + 2*c*e))*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[- ((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/( 2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(a*e*Sqrt[d + e*x]*Sqrt[c + b*x + a*x ^2]) - (3*Sqrt[2]*c*Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[b - Sqrt[ b^2 - 4*a*c] + 2*a*x]*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*a*x]*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e*x))/(2* a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a *c]*e)/(2*a*d), ArcSin[(Sqrt[2]*Sqrt[a]*Sqrt[d + e*x])/Sqrt[2*a*d - (b - S qrt[b^2 - 4*a*c])*e]], (2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*a*d - (b + S qrt[b^2 - 4*a*c])*e)])/(Sqrt[a]*Sqrt[c + b*x + a*x^2]*Sqrt[b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e + (2*a*(d + e*x))/e]*Sqrt[b + Sqrt[b^2 - 4*a*c] - (2*a* d)/e + (2*a*(d + e*x))/e]))/3))/Sqrt[c + b*x + a*x^2]
3.1.83.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*( Sqrt[a + b*x + c*x^2]/(e*(2*m + 5))), x] - Simp[1/(e*(2*m + 5)) Int[((d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IntegerQ[2*m] && !LtQ[m, -1]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]) Int[1/((d + e*x )*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_ )^(n_.))^(q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + b/x^n + c/x^(2* n))^FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p]) Int[((d + e*x^n)^q*( c + b*x^n + a*x^(2*n))^p)/x^(2*n*p), x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && !IntegerQ[p] && !IntegerQ[q] & & PosQ[n]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(3022\) vs. \(2(836)=1672\).
Time = 0.45 (sec) , antiderivative size = 3023, normalized size of antiderivative = 3.17
1/3*((a*x^2+b*x+c)/x^2)^(1/2)*x*(e*x+d)^(1/2)*(2^(1/2)*(-(e*x+d)*a/(e*(-4* a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*d*a-b *e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c +b^2)^(1/2)-2*d*a+b*e))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2 )^(1/2)-2*d*a+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e)/(2*d*a-b*e+e* (-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*d^2*e-2^(1/2)*(-(e*x+d)*a /(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*((-2*a*x+(-4*a*c+b^2)^(1/2)-b)*e/ (2*d*a-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^(1/2))*e/(e *(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*a/(e*(-4 *a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e)/(2*d* a-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*b*d*e^2-2*2^(1/2)*( -(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*((-2*a*x+(-4*a*c+b^2)^( 1/2)-b)*e/(2*d*a-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a*c+b^2)^( 1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d )*a/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*d*a+ b*e)/(2*d*a-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*c*e^3+3*2 ^(1/2)*(-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*((-2*a*x+(-4*a* c+b^2)^(1/2)-b)*e/(2*d*a-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*a*x+(-4*a* c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2)*EllipticF(2^(1/2)* (-(e*x+d)*a/(e*(-4*a*c+b^2)^(1/2)-2*d*a+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(...
Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\text {Timed out} \]
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\int \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \]
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} \,d x } \]
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} \,d x } \]
Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x} \, dx=\int \sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \]