3.122 \(\int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx\)

Optimal. Leaf size=1432 \[ -\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}} F\left (2 \tan ^{-1}\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}} \]

[Out]

-(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+b^2)^(1/2)))^2*(4*c^2*d-2*c*e*(b+(-4*a*c+b^2)^(1/2))+f*(b+(-4*
a*c+b^2)^(1/2))^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)-(2*a*f-b*e+2*c*d)*(b+(-4*a*c+b^2)^(1/2))*(2*a+x*(b+(-4*a*c+b^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*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))^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))^(1/4)/(4*a*c-(b+(-4*a*c+b^2)^(1/2))^2)/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1/2)/(1+(2*a+
x*(b+(-4*a*c+b^2)^(1/2)))^2*(4*c^2*d-2*c*e*(b+(-4*a*c+b^2)^(1/2))+f*(b+(-4*a*c+b^2)^(1/2))^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)-(2*a*f-b*e+2*c*d)*(b+(-4*a*c+
b^2)^(1/2))*(2*a+x*(b+(-4*a*c+b^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)

________________________________________________________________________________________

Rubi [A]  time = 6.22, antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {992, 935, 1103} \[ -\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}} F\left (2 \tan ^{-1}\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}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x + c*x^2]*Sqrt[d + e*x + f*x^2]),x]

[Out]

-(((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 + Sqrt[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 + Sqrt[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 + Sqr
t[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^2 - 4*a*c] + 2*c*x])], (2 + ((b + Sqrt[b^2 - 4*a*c])*(2*c*d - b
*e + 2*a*f))/(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)]*Sqrt[2*c^2
*d + b*(b + Sqrt[b^2 - 4*a*c])*f - c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)]))/4])/((4*a*c - (b + Sqrt[b^2 - 4*a*
c])^2)*(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[a + b*x + c*x^2]*Sqrt[d
+ e*x + f*x^2]*Sqrt[1 - ((b + Sqrt[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
)]))

Rule 935

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :
> Dist[(-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)*Sqrt[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] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 992

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]}, Dist[(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]

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx &=\frac {\left (\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}\right ) \int \frac {1}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {d+e x+f x^2}} \, dx}{\sqrt {a+b x+c x^2}}\\ &=-\frac {\left (2 \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}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {\left (4 c \left (b+\sqrt {b^2-4 a c}\right ) d-4 a c e-\left (b+\sqrt {b^2-4 a c}\right )^2 e+4 a \left (b+\sqrt {b^2-4 a c}\right ) f\right ) x^2}{\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}+\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 ) x^4}{\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}}} \, dx,x,\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}}\\ &=-\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}} F\left (2 \tan ^{-1}\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}}}\\ \end {align*}

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Mathematica [A]  time = 2.34, size = 670, normalized size = 0.47 \[ -\frac {\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )\right )}} \sqrt {-\frac {c \left (\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}-e \left (\sqrt {b^2-4 a c}+2 c x\right )-2 f x \sqrt {b^2-4 a c}+4 a f+b \left (\sqrt {e^2-4 d f}-e+2 f x\right )+2 c x \sqrt {e^2-4 d f}\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt {b^2-4 a c}+b\right )+c \left (\sqrt {e^2-4 d f}-e\right )\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\left (\sqrt {b^2-4 a c}-b\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (\sqrt {e^2-4 d f}-e\right )\right ) \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}}\right )|\frac {2 c d-b e+2 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}{2 c d-b e+2 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}\right )}{\sqrt {a+x (b+c x)} \sqrt {d+x (e+f x)} \left (f \left (\sqrt {b^2-4 a c}-b\right )+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \sqrt {\frac {c \sqrt {b^2-4 a c} \left (\sqrt {e^2-4 d f}-e-2 f x\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt {b^2-4 a c}+b\right )+c \left (\sqrt {e^2-4 d f}-e\right )\right )}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[a + b*x + c*x^2]*Sqrt[d + e*x + f*x^2]),x]

[Out]

-(((-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 + Sq
rt[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))]*Sqrt[a + x*(b + c*x)]*Sqrt[d + x*(e + f*x)]))

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fricas [F]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}}{c f x^{4} + {\left (c e + b f\right )} x^{3} + {\left (c d + b e + a f\right )} x^{2} + a d + {\left (b d + a e\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)), x)

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maple [A]  time = 0.27, size = 928, normalized size = 0.65 \[ \frac {8 \left (2 b \,f^{2} x^{2}-2 c e f \,x^{2}+2 b e f x -8 c d f x +2 \sqrt {-4 d f +e^{2}}\, c f \,x^{2}+2 \sqrt {-4 a c +b^{2}}\, f^{2} x^{2}-2 b d f +b \,e^{2}+2 \sqrt {-4 d f +e^{2}}\, b f x -2 c d e +2 \sqrt {-4 a c +b^{2}}\, e f x +\sqrt {-4 d f +e^{2}}\, b e -2 \sqrt {-4 d f +e^{2}}\, c d -2 \sqrt {-4 a c +b^{2}}\, d f +\sqrt {-4 a c +b^{2}}\, e^{2}+2 \sqrt {-4 d f +e^{2}}\, \sqrt {-4 a c +b^{2}}\, f x +\sqrt {-4 d f +e^{2}}\, \sqrt {-4 a c +b^{2}}\, e \right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right ) f}{\left (b f -c e +\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (2 f x +e +\sqrt {-4 d f +e^{2}}\right )}}\, \sqrt {-\frac {\sqrt {-4 d f +e^{2}}\, \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) f}{\left (b f -c e +\sqrt {-4 d f +e^{2}}\, c -\sqrt {-4 a c +b^{2}}\, f \right ) \left (2 f x +e +\sqrt {-4 d f +e^{2}}\right )}}\, \sqrt {\frac {\left (-b f +c e +\sqrt {-4 d f +e^{2}}\, c -\sqrt {-4 a c +b^{2}}\, f \right ) \left (-2 f x -e +\sqrt {-4 d f +e^{2}}\right )}{\left (b f -c e +\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (2 f x +e +\sqrt {-4 d f +e^{2}}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+e x +d}\, \EllipticF \left (\sqrt {\frac {\left (-b f +c e +\sqrt {-4 d f +e^{2}}\, c -\sqrt {-4 a c +b^{2}}\, f \right ) \left (-2 f x -e +\sqrt {-4 d f +e^{2}}\right )}{\left (b f -c e +\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (2 f x +e +\sqrt {-4 d f +e^{2}}\right )}}, \sqrt {\frac {\left (-b f +c e +\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (b f -c e +\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right )}{\left (b f -c e +\sqrt {-4 d f +e^{2}}\, c -\sqrt {-4 a c +b^{2}}\, f \right ) \left (-b f +c e +\sqrt {-4 d f +e^{2}}\, c -\sqrt {-4 a c +b^{2}}\, f \right )}}\right )}{\sqrt {\frac {\left (-2 f x -e +\sqrt {-4 d f +e^{2}}\right ) \left (2 f x +e +\sqrt {-4 d f +e^{2}}\right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{c f}}\, \sqrt {-4 d f +e^{2}}\, \left (b f -c e -\sqrt {-4 d f +e^{2}}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \sqrt {c f \,x^{4}+b f \,x^{3}+c e \,x^{3}+a f \,x^{2}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1/2),x)

[Out]

8*(2*b*f^2*x^2-2*c*e*f*x^2+2*x^2*c*f*(-4*d*f+e^2)^(1/2)+2*(-4*a*c+b^2)^(1/2)*f^2*x^2+2*b*e*f*x+2*x*b*f*(-4*d*f
+e^2)^(1/2)-8*c*d*f*x+2*x*e*f*(-4*a*c+b^2)^(1/2)+2*x*f*(-4*d*f+e^2)^(1/2)*(-4*a*c+b^2)^(1/2)-2*b*d*f+b*e^2+b*e
*(-4*d*f+e^2)^(1/2)-2*c*d*e-2*c*d*(-4*d*f+e^2)^(1/2)-2*(-4*a*c+b^2)^(1/2)*d*f+e^2*(-4*a*c+b^2)^(1/2)+e*(-4*d*f
+e^2)^(1/2)*(-4*a*c+b^2)^(1/2))*EllipticF((((-4*d*f+e^2)^(1/2)*c-(-4*a*c+b^2)^(1/2)*f-b*f+c*e)*(-2*f*x+(-4*d*f
+e^2)^(1/2)-e)/((-4*d*f+e^2)^(1/2)*c+(-4*a*c+b^2)^(1/2)*f+b*f-c*e)/(2*f*x+(-4*d*f+e^2)^(1/2)+e))^(1/2),(((-4*d
*f+e^2)^(1/2)*c+(-4*a*c+b^2)^(1/2)*f-b*f+c*e)*((-4*d*f+e^2)^(1/2)*c+(-4*a*c+b^2)^(1/2)*f+b*f-c*e)/((-4*d*f+e^2
)^(1/2)*c-(-4*a*c+b^2)^(1/2)*f+b*f-c*e)/((-4*d*f+e^2)^(1/2)*c-(-4*a*c+b^2)^(1/2)*f-b*f+c*e))^(1/2))*((-4*d*f+e
^2)^(1/2)*(2*c*x+b+(-4*a*c+b^2)^(1/2))*f/((-4*d*f+e^2)^(1/2)*c+(-4*a*c+b^2)^(1/2)*f+b*f-c*e)/(2*f*x+(-4*d*f+e^
2)^(1/2)+e))^(1/2)*(-(-4*d*f+e^2)^(1/2)*(-2*c*x-b+(-4*a*c+b^2)^(1/2))*f/((-4*d*f+e^2)^(1/2)*c-(-4*a*c+b^2)^(1/
2)*f+b*f-c*e)/(2*f*x+(-4*d*f+e^2)^(1/2)+e))^(1/2)*(((-4*d*f+e^2)^(1/2)*c-(-4*a*c+b^2)^(1/2)*f-b*f+c*e)*(-2*f*x
+(-4*d*f+e^2)^(1/2)-e)/((-4*d*f+e^2)^(1/2)*c+(-4*a*c+b^2)^(1/2)*f+b*f-c*e)/(2*f*x+(-4*d*f+e^2)^(1/2)+e))^(1/2)
*(c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)^(1/2)/(1/c/f*(-2*f*x+(-4*d*f+e^2)^(1/2)-e)*(2*f*x+(-4*d*f+e^2)^(1/2)+e)*(-2
*c*x-b+(-4*a*c+b^2)^(1/2))*(2*c*x+b+(-4*a*c+b^2)^(1/2)))^(1/2)/(-4*d*f+e^2)^(1/2)/((-4*a*c+b^2)^(1/2)*f-(-4*d*
f+e^2)^(1/2)*c+b*f-c*e)/(c*f*x^4+b*f*x^3+c*e*x^3+a*f*x^2+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x+a)^(1/2)/(f*x^2+e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,\sqrt {f\,x^2+e\,x+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2)^(1/2)),x)

[Out]

int(1/((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x + c x^{2}} \sqrt {d + e x + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x+a)**(1/2)/(f*x**2+e*x+d)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x + c*x**2)*sqrt(d + e*x + f*x**2)), x)

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