Integrand size = 29, antiderivative size = 757 \[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=-\frac {(b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{4 c d}-\frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{2 x}+\frac {\sqrt {b^2-4 a c} (b d+c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{4 \sqrt {2} c d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac {\sqrt {b^2-4 a c} (b d-5 c e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {2} c \sqrt {d+e x} \left (c+b x+a x^2\right )}+\frac {\sqrt {b^2-4 a c} \left (b^2 d^2-2 b c d e-c \left (4 a d^2-c e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} c \left (b+\sqrt {b^2-4 a c}\right ) d \sqrt {d+e x} \left (c+b x+a x^2\right )} \] Output:
-1/4*(b*d+c*e)*(a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/c/d-1/2*(a+c/x^2+b/x)^(1/ 2)*(e*x+d)^(1/2)/x+1/8*(-4*a*c+b^2)^(1/2)*(b*d+c*e)*(a+c/x^2+b/x)^(1/2)*x* (e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*a* x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+ (-4*a*c+b^2)^(1/2))*e))^(1/2))*2^(1/2)/c/d/(a*(e*x+d)/(2*a*d-(b+(-4*a*c+b^ 2)^(1/2))*e))^(1/2)/(a*x^2+b*x+c)-1/4*(-4*a*c+b^2)^(1/2)*(b*d-5*c*e)*(a+c/ x^2+b/x)^(1/2)*x*(a*(e*x+d)/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-a*(a *x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*a*x+b)/(-4*a*c+b^2)^(1 /2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+(-4*a*c+b^2)^(1/2))* e))^(1/2))*2^(1/2)/c/(e*x+d)^(1/2)/(a*x^2+b*x+c)+1/2*(-4*a*c+b^2)^(1/2)*(b ^2*d^2-2*b*c*d*e-c*(4*a*d^2-c*e^2))*(a+c/x^2+b/x)^(1/2)*x*(a*(e*x+d)/(2*a* d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*E llipticPi(1/2*(1+(2*a*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2 )^(1/2)/(b+(-4*a*c+b^2)^(1/2)),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+(-4*a*c+ b^2)^(1/2))*e))^(1/2))*2^(1/2)/c/(b+(-4*a*c+b^2)^(1/2))/d/(e*x+d)^(1/2)/(a *x^2+b*x+c)
Result contains complex when optimal does not.
Time = 34.42 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\frac {x \sqrt {a+\frac {c+b x}{x^2}} \left (-\frac {8 c d^3}{x^2}-\frac {8 c d^2 e}{x}-\frac {4 d^2 (b d+c e)}{x}-\frac {i (d+e x)^{3/2} \sqrt {1-\frac {2 \left (a d^2+e (-b d+c e)\right )}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (a d^2+e (-b d+c e)\right )}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (d (b d+c e) \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (b^2 d^2 e+b d \left (-5 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (4 a d^2+2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+2 e \left (b^2 d^2-2 b c d e+c \left (-4 a d^2+c e^2\right )\right ) \operatorname {EllipticPi}\left (\frac {d \left (2 a d-b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (-b d+c e)\right )},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}\right )}{16 c d^2 \sqrt {d+e x}} \] Input:
Integrate[(Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/x^2,x]
Output:
(x*Sqrt[a + (c + b*x)/x^2]*((-8*c*d^3)/x^2 - (8*c*d^2*e)/x - (4*d^2*(b*d + c*e))/x - (I*(d + e*x)^(3/2)*Sqrt[1 - (2*(a*d^2 + e*(-(b*d) + c*e)))/((2* a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(a*d^2 + e*(- (b*d) + c*e)))/((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(d*(b *d + c*e)*(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqr t[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]))] - (b^2*d^2*e + b*d*(-5*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(4*a*d^2 + 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*Ellip ticF[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]))] + 2*e*(b^2*d^2 - 2*b*c*d* e + c*(-4*a*d^2 + c*e^2))*EllipticPi[(d*(2*a*d - b*e - Sqrt[(b^2 - 4*a*c)* e^2]))/(2*(a*d^2 + e*(-(b*d) + c*e))), 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]))]))/(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)))))/(16*c*d^2*Sqrt[d + e*x])
Time = 3.07 (sec) , antiderivative size = 1486, normalized size of antiderivative = 1.96, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {1897, 1271, 2154, 1282, 2154, 25, 27, 1172, 321, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{x^2} \, dx\) |
| \(\Big \downarrow \) 1897 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \int \frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{x^3}dx}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1271 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \int \frac {3 a e x^2+2 (a d+b e) x+b d+c e}{x^2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left ((b d+c e) \int \frac {1}{x^2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\int \frac {2 a d+2 b e+3 a e x}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1282 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (\int \frac {2 a d+2 b e+3 a e x}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {\int \frac {-a e x^2+b d+c e}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (\int \frac {3 a e}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\int -\frac {a e x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (\int \frac {3 a e}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-\int \frac {a e x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (3 a e \int \frac {1}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \int \frac {x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (\frac {6 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \int \frac {x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \int \frac {x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )+\frac {6 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left (2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \left (\frac {\int \frac {\sqrt {d+e x}}{\sqrt {a x^2+b x+c}}dx}{e}-\frac {d \int \frac {1}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{e}\right )}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )+\frac {6 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+(b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{4} \left ((b d+c e) \left (-\frac {(b d+c e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx-a e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {a x^2+b x+c} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} d \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )+2 (a d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\frac {6 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}+\frac {2 (a d+b e) \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \int \frac {1}{x \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {a x^2+b x+c}}+(b d+c e) \left (-\frac {\frac {(b d+c e) \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \int \frac {1}{x \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {a x^2+b x+c}}-a e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \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 )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}-\frac {4 (a d+b e) \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \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}}+(b d+c e) \left (-\frac {-a e \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}} 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} d \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}}\right )-\frac {2 (b d+c e) \sqrt {b+2 a x-\sqrt {b^2-4 a c}} \sqrt {b+2 a x+\sqrt {b^2-4 a c}} \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}}}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \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 )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}-\frac {4 (a d+b 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}} \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 {b+\frac {2 a (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}}+(b d+c e) \left (-\frac {-a e \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}} 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} d \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}}\right )-\frac {2 (b d+c 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}} \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 {b+\frac {2 a (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 a d}{e}}}}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {1}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \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 )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}-\frac {4 (a d+b 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}} \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 {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}}}+(b d+c e) \left (-\frac {-a e \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}} 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} d \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}}\right )-\frac {2 (b d+c 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}} \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 {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}}}}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\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}{4} \left (\frac {6 \sqrt {2} \sqrt {b^2-4 a c} e \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 )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} (a d+b e) \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} d \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}}}+(b d+c e) \left (-\frac {-a e \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}} 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} d \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}}\right )-\frac {\sqrt {2} (b d+c e) \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} d \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}}}}{2 c d}-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{c d x}\right )\right )-\frac {\sqrt {d+e x} \sqrt {a x^2+b x+c}}{2 x^2}\right )}{\sqrt {a x^2+b x+c}}\) |
Input:
Int[(Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x])/x^2,x]
Output:
(Sqrt[a + c/x^2 + b/x]*x*(-1/2*(Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/x^2 + ((6*Sqrt[2]*Sqrt[b^2 - 4*a*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[ArcS in[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*S qrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[d + e*x]*S qrt[c + b*x + a*x^2]) - (2*Sqrt[2]*(a*d + b*e)*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)]*S qrt[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 - Sqrt[b^2 - 4*a*c])*e]], (2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[a]*d*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]) + (b*d + c*e)*(-(( Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/(c*d*x)) - (-(a*e*((Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*Ellipt icE[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*Sqr t[(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]*d*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqr...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sq rt[a + b*x + c*x^2]/(e*(m + 1))), x] - Simp[1/(2*e*(m + 1)) Int[((d + e*x )^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Intege rQ[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[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x ]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x)^ (m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b* e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b* g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
Int[(x_)^(m_.)*((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[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, 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. \(1596\) vs. \(2(670)=1340\).
Time = 1.64 (sec) , antiderivative size = 1597, normalized size of antiderivative = 2.11
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1597\) |
| default | \(\text {Expression too large to display}\) | \(4957\) |
Input:
int((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-1/4*(e*x+d)^(1/2)*(b*d*x+c*e*x+2*c*d)/x/c/d*((a*x^2+b*x+c)/x^2)^(1/2)+1/8 /c/d*(-2*(4*a*c*d^2-b^2*d^2+2*b*c*d*e-c^2*e^2)*(d/e-1/2*(b+(-4*a*c+b^2)^(1 /2))/a)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4 *a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b +(-4*a*c+b^2)^(1/2))/a)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^ 3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^(1/2)/d*e*EllipticPi(((x+d/e)/(d/e-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)/d*e, ((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)) ^(1/2))+2*a*c*e^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+d/e)/(d/e-1/2*(b+ (-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2 *(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e +1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x +c*d)^(1/2)*((-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)*EllipticE(((x+d/e)/(d/e- 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)/ (-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+1/2*(-b+(-4*a*c+b^2)^(1/2))/a *EllipticF(((x+d/e)/(d/e-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)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)))+2*a *b*d*e*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2 )^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a* c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e+1/2*(b+...
Timed out. \[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\int \frac {\sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x^{2}}\, dx \] Input:
integrate((a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2)/x**2,x)
Output:
Integral(sqrt(d + e*x)*sqrt(a + b/x + c/x**2)/x**2, x)
\[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x^{2}} \,d x } \] Input:
integrate((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)/x^2, x)
\[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}}{x^{2}} \,d x } \] Input:
integrate((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)/x^2, x)
Timed out. \[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\int \frac {\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{x^2} \,d x \] Input:
int(((d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2))/x^2,x)
Output:
int(((d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2))/x^2, x)
\[ \int \frac {\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \sqrt {d+e x}}{x^2} \, dx=\int \frac {\sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, \sqrt {e x +d}}{x^{2}}d x \] Input:
int((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x)
Output:
int((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2)/x^2,x)