Integrand size = 29, antiderivative size = 1432 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=-\frac {\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+e x+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}} \left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}{\left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right ),\frac {1}{4} \left (2+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {2 c^2 d+b \left (b+\sqrt {b^2-4 a c}\right ) f-c \left (b e+\sqrt {b^2-4 a c} e+2 a f\right )}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2} \sqrt {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}} \]
-(cos(2*arctan((2*c^2*d-b*c*e+b^2*f-2*a*c*f-(-b*f+c*e)*(-4*a*c+b^2)^(1/2)) ^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(b^2*d+b*(-a*e+d*(-4*a*c+b^2)^ (1/2))-a*(2*c*d-2*a*f+e*(-4*a*c+b^2)^(1/2)))^(1/4)/(b+2*c*x+(-4*a*c+b^2)^( 1/2))^(1/2)))^2)^(1/2)/cos(2*arctan((2*c^2*d-b*c*e+b^2*f-2*a*c*f-(-b*f+c*e )*(-4*a*c+b^2)^(1/2))^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(b^2*d+b* (-a*e+d*(-4*a*c+b^2)^(1/2))-a*(2*c*d-2*a*f+e*(-4*a*c+b^2)^(1/2)))^(1/4)/(b +2*c*x+(-4*a*c+b^2)^(1/2))^(1/2)))*EllipticF(sin(2*arctan((2*c^2*d-b*c*e+b ^2*f-2*a*c*f-(-b*f+c*e)*(-4*a*c+b^2)^(1/2))^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^( 1/2)))^(1/2)/(b^2*d+b*(-a*e+d*(-4*a*c+b^2)^(1/2))-a*(2*c*d-2*a*f+e*(-4*a*c +b^2)^(1/2)))^(1/4)/(b+2*c*x+(-4*a*c+b^2)^(1/2))^(1/2))),1/2*(2+(2*a*f-b*e +2*c*d)*(b+(-4*a*c+b^2)^(1/2))/(b^2*d+b*(-a*e+d*(-4*a*c+b^2)^(1/2))-a*(2*c *d-2*a*f+e*(-4*a*c+b^2)^(1/2)))^(1/2)/(2*c^2*d+b*f*(b+(-4*a*c+b^2)^(1/2))- c*(b*e+2*a*f+e*(-4*a*c+b^2)^(1/2)))^(1/2))^(1/2))*(b+2*c*x+(-4*a*c+b^2)^(1 /2))^(3/2)*(b^2*d+b*(-a*e+d*(-4*a*c+b^2)^(1/2))-a*(2*c*d-2*a*f+e*(-4*a*c+b ^2)^(1/2)))^(1/4)*(2*a+x*(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((f*x^2+e*x+d)*(4*a *c-(b+(-4*a*c+b^2)^(1/2))^2)^2/(b+2*c*x+(-4*a*c+b^2)^(1/2))^2/(4*a^2*f-2*a *e*(b+(-4*a*c+b^2)^(1/2))+d*(b+(-4*a*c+b^2)^(1/2))^2))^(1/2)*(1+(2*a+x*(b+ (-4*a*c+b^2)^(1/2)))*(2*c^2*d-b*c*e+b^2*f-2*a*c*f-(-b*f+c*e)*(-4*a*c+b^2)^ (1/2))^(1/2)/(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b^2*d+b*(-a*e+d*(-4*a*c+b^2)^(1 /2))-a*(2*c*d-2*a*f+e*(-4*a*c+b^2)^(1/2)))^(1/2))*((1+(2*a+x*(b+(-4*a*c...
Time = 4.80 (sec) , antiderivative size = 670, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=-\frac {\left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {-\frac {c \left (4 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}-2 \sqrt {b^2-4 a c} f x+2 c \sqrt {e^2-4 d f} x-e \left (\sqrt {b^2-4 a c}+2 c x\right )+b \left (-e+\sqrt {e^2-4 d f}+2 f x\right )\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right ),\frac {2 c d-b e+2 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}{2 c d-b e+2 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}\right )}{\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \sqrt {\frac {c \sqrt {b^2-4 a c} \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {a+x (b+c x)} \sqrt {d+x (e+f x)}} \]
-(((-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Sqrt[- ((c*Sqrt[b^2 - 4*a*c]*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(((b + Sqrt[b^2 - 4 *a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))]* Sqrt[-((c*(4*a*f + Sqrt[b^2 - 4*a*c]*Sqrt[e^2 - 4*d*f] - 2*Sqrt[b^2 - 4*a* c]*f*x + 2*c*Sqrt[e^2 - 4*d*f]*x - e*(Sqrt[b^2 - 4*a*c] + 2*c*x) + b*(-e + Sqrt[e^2 - 4*d*f] + 2*f*x)))/(((b + Sqrt[b^2 - 4*a*c])*f + c*(-e + Sqrt[e ^2 - 4*d*f]))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))]*EllipticF[ArcSin[Sqrt[(( (-b + Sqrt[b^2 - 4*a*c])*f + c*(e - Sqrt[e^2 - 4*d*f]))*(b + Sqrt[b^2 - 4* a*c] + 2*c*x))/(((b + Sqrt[b^2 - 4*a*c])*f + c*(-e + Sqrt[e^2 - 4*d*f]))*( -b + Sqrt[b^2 - 4*a*c] - 2*c*x))]], (2*c*d - b*e + 2*a*f - Sqrt[b^2 - 4*a* c]*Sqrt[e^2 - 4*d*f])/(2*c*d - b*e + 2*a*f + Sqrt[b^2 - 4*a*c]*Sqrt[e^2 - 4*d*f])])/(((-b + Sqrt[b^2 - 4*a*c])*f + c*(e - Sqrt[e^2 - 4*d*f]))*Sqrt[( c*Sqrt[b^2 - 4*a*c]*(-e + Sqrt[e^2 - 4*d*f] - 2*f*x))/(((b + Sqrt[b^2 - 4* a*c])*f + c*(-e + Sqrt[e^2 - 4*d*f]))*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]*S qrt[a + x*(b + c*x)]*Sqrt[d + x*(e + f*x)]))
Time = 1.72 (sec) , antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1323, 1280, 1416}
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx\) |
| \(\Big \downarrow \) 1323 |
| \(\displaystyle \frac {\sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \int \frac {1}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {f x^2+e x+d}}dx}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1280 |
| \(\displaystyle -\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )^{3/2} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \sqrt {\frac {\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right )^2 \left (d+e x+f x^2\right )}{\left (\sqrt {b^2-4 a c}+b+2 c x\right )^2 \left (4 a^2 f+d \left (\sqrt {b^2-4 a c}+b\right )^2-2 a e \left (\sqrt {b^2-4 a c}+b\right )\right )}} \int \frac {1}{\sqrt {\frac {\left (4 d c^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e c+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}}d\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}}}}{\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right ) \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\sqrt [4]{d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (f x^2+e x+d\right )}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}} \left (\frac {\sqrt {f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e c+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right ),\frac {1}{4} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {2 d c^2-\left (b e+\sqrt {b^2-4 a c} e+2 a f\right ) c+b \left (b+\sqrt {b^2-4 a c}\right ) f}}+2\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {c x^2+b x+a} \sqrt {f x^2+e x+d} \sqrt {\frac {\left (4 d c^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e c+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}}\) |
-(((b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f))^(1/4)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^(3/2)*Sqrt[2*a + (b + Sqr t[b^2 - 4*a*c])*x]*Sqrt[((4*a*c - (b + Sqrt[b^2 - 4*a*c])^2)^2*(d + e*x + f*x^2))/(((b + Sqrt[b^2 - 4*a*c])^2*d - 2*a*(b + Sqrt[b^2 - 4*a*c])*e + 4* a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^2)]*(1 + (Sqrt[2*c^2*d - b*c*e + b^ 2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*(2*a + (b + Sqrt[b^2 - 4*a* c])*x))/(Sqrt[b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))*Sqrt[(1 - ((b + Sqr t[b^2 - 4*a*c])*(2*c*d - b*e + 2*a*f)*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/( (b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f))*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + ((4*c^2*d - 2*c*(b + Sqrt[b^2 - 4*a*c])*e + (b + Sqrt[b^2 - 4*a*c])^2*f)*(2*a + (b + Sqrt[b^2 - 4*a*c])*x )^2)/(((b + Sqrt[b^2 - 4*a*c])^2*d - 2*a*(b + Sqrt[b^2 - 4*a*c])*e + 4*a^2 *f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^2))/(1 + (Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*(2*a + (b + Sqrt[b^2 - 4*a*c]) *x))/(Sqrt[b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*(2*c*d + Sqrt[b^2 - 4 *a*c]*e - 2*a*f)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))^2]*EllipticF[2*ArcTan[ ((2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f))^(1/4) *Sqrt[2*a + (b + Sqrt[b^2 - 4*a*c])*x])/((b^2*d + b*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f))^(1/4)*Sqrt[b + Sqrt[b^...
3.2.22.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* ((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr t[a + b*x + c*x^2])) Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e, f, g}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]*Sqrt[(d_) + (e_.)*(x_) + (f_. )*(x_)^2]), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b + r + 2 *c*x]*(Sqrt[2*a + (b + r)*x]/Sqrt[a + b*x + c*x^2]) Int[1/(Sqrt[b + r + 2 *c*x]*Sqrt[2*a + (b + r)*x]*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Time = 5.22 (sec) , antiderivative size = 905, normalized size of antiderivative = 0.63
| method | result | size |
| elliptic | \(\frac {2 \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+e x +d \right )}\, \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}^{2} \sqrt {\frac {\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, \sqrt {\frac {\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}, \sqrt {\frac {\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\right )}{\sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+e x +d}\, \left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {c f \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\) | \(905\) |
| default | \(\frac {8 \left (2 b c f \,x^{2}-2 e \,x^{2} c^{2}-2 c^{2} x^{2} \sqrt {-4 d f +e^{2}}-2 c f \,x^{2} \sqrt {-4 a c +b^{2}}+8 a c f x -2 b c e x -2 b c x \sqrt {-4 d f +e^{2}}-2 c e x \sqrt {-4 a c +b^{2}}-2 c x \sqrt {-4 d f +e^{2}}\, \sqrt {-4 a c +b^{2}}+2 b a f +2 a c e +2 a c \sqrt {-4 d f +e^{2}}+2 a f \sqrt {-4 a c +b^{2}}-b^{2} e -b^{2} \sqrt {-4 d f +e^{2}}-b e \sqrt {-4 a c +b^{2}}-b \sqrt {-4 d f +e^{2}}\, \sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {-\frac {\left (c \sqrt {-4 d f +e^{2}}-f \sqrt {-4 a c +b^{2}}-b f +c e \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (c \sqrt {-4 d f +e^{2}}+f \sqrt {-4 a c +b^{2}}-b f +c e \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}, \sqrt {\frac {\left (c \sqrt {-4 d f +e^{2}}+f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (c \sqrt {-4 d f +e^{2}}+f \sqrt {-4 a c +b^{2}}-b f +c e \right )}{\left (c \sqrt {-4 d f +e^{2}}-f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (c \sqrt {-4 d f +e^{2}}-f \sqrt {-4 a c +b^{2}}-b f +c e \right )}}\right ) \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right ) c}{\left (c \sqrt {-4 d f +e^{2}}+f \sqrt {-4 a c +b^{2}}-b f +c e \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right ) c}{\left (c \sqrt {-4 d f +e^{2}}-f \sqrt {-4 a c +b^{2}}+b f -c e \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {-\frac {\left (c \sqrt {-4 d f +e^{2}}-f \sqrt {-4 a c +b^{2}}-b f +c e \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (c \sqrt {-4 d f +e^{2}}+f \sqrt {-4 a c +b^{2}}-b f +c e \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+e x +d}}{\sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}{c f}}\, \sqrt {-4 a c +b^{2}}\, \left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+e x +d \right )}}\) | \(906\) |
2*((c*x^2+b*x+a)*(f*x^2+e*x+d))^(1/2)/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1 /2)*(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(e+(-4*d*f+e^2)^(1/2))/f)*((-1/2*(e +(-4*d*f+e^2)^(1/2))/f+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(x-1/2/c*(-b+(-4*a*c+ b^2)^(1/2)))/(-1/2*(e+(-4*d*f+e^2)^(1/2))/f-1/2/c*(-b+(-4*a*c+b^2)^(1/2))) /(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^ 2*((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x-1/2/f* (-e+(-4*d*f+e^2)^(1/2)))/(1/2/f*(-e+(-4*d*f+e^2)^(1/2))-1/2/c*(-b+(-4*a*c+ b^2)^(1/2)))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((-1/2*(b+(-4*a*c+b^2 )^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f) /(-1/2*(e+(-4*d*f+e^2)^(1/2))/f-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(x+1/2*(b+( -4*a*c+b^2)^(1/2))/c))^(1/2)/(-1/2*(e+(-4*d*f+e^2)^(1/2))/f+1/2*(b+(-4*a*c +b^2)^(1/2))/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2 )))/(c*f*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c )*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))^(1/2 )*EllipticF(((-1/2*(e+(-4*d*f+e^2)^(1/2))/f+1/2*(b+(-4*a*c+b^2)^(1/2))/c)* (x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(e+(-4*d*f+e^2)^(1/2))/f-1/2/c*(-b +(-4*a*c+b^2)^(1/2)))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(- 4*a*c+b^2)^(1/2))/c-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))*(1/2/c*(-b+(-4*a*c+b^2) ^(1/2))+1/2*(e+(-4*d*f+e^2)^(1/2))/f)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))-1/2/f *(-e+(-4*d*f+e^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*(e+(-4*d*f...
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}} \,d x } \]
integral(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)/(c*f*x^4 + (c*e + b*f )*x^3 + (c*d + b*e + a*f)*x^2 + a*d + (b*d + a*e)*x), x)
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=\int \frac {1}{\sqrt {a + b x + c x^{2}} \sqrt {d + e x + f x^{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}} \,d x } \]
\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,\sqrt {f\,x^2+e\,x+d}} \,d x \]