3.1.13 \(\int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx\) [13]

Optimal. Leaf size=1077 \[ -\frac {\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \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+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\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+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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )|\frac {1}{2} \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {a+b x+c x^2} \sqrt {d+f x^2} \sqrt {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\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+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.04, antiderivative size = 1077, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1007, 950, 1117} \begin {gather*} -\frac {\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \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+d\right )}{\left (4 f a^2+\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 {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2+\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+\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 {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right )|\frac {1}{2} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)}}+1\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {c x^2+b x+a} \sqrt {f x^2+d} \sqrt {\frac {\left (4 d c^2+\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+\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 {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - 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 + f*x^2))/(((b + Sqrt[b^2 - 4*a*c])^2*
d + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^2)]*(1 + (Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*(
2*a + (b + Sqrt[b^2 - 4*a*c])*x))/(Sqrt[b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]*(b + Sqrt[b^2 - 4*a*c
] + 2*c*x)))*Sqrt[(1 - (4*(b + Sqrt[b^2 - 4*a*c])*(c*d + a*f)*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/(((b + Sqrt[b
^2 - 4*a*c])^2*d + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + ((4*c^2*d + (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 + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)^2))/(1
+ (Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/(Sqrt[b^2*d + b*S
qrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)))^2]*EllipticF[2*ArcTan[((2*c^2*d - 2*a*
c*f + b*(b + Sqrt[b^2 - 4*a*c])*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 - 2*a*(c*d - a*f))^(1/4)*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*c*x])], (1 + ((b + Sqrt[b^2 - 4*a*c])*(c*d + a*f))/(
Sqrt[2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f]*Sqrt[b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*(c*d - a*f)]))/
2])/((4*a*c - (b + Sqrt[b^2 - 4*a*c])^2)*(2*c^2*d - 2*a*c*f + b*(b + Sqrt[b^2 - 4*a*c])*f)^(1/4)*Sqrt[a + b*x
+ c*x^2]*Sqrt[d + f*x^2]*Sqrt[1 - (4*(b + Sqrt[b^2 - 4*a*c])*(c*d + a*f)*(2*a + (b + Sqrt[b^2 - 4*a*c])*x))/((
(b + Sqrt[b^2 - 4*a*c])^2*d + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)) + ((4*c^2*d + (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 + 4*a^2*f)*(b + Sqrt[b^2 - 4*a*c] + 2*
c*x)^2)]))

Rule 950

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2*(d +
e*x)*(Sqrt[(e*f - d*g)^2*((a + c*x^2)/((c*f^2 + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqrt[a + c*x^2])), Subst[I
nt[1/Sqrt[1 - (2*c*d*f + 2*a*e*g)*(x^2/(c*f^2 + a*g^2)) + (c*d^2 + a*e^2)*(x^4/(c*f^2 + a*g^2))], x], x, Sqrt[
f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 1007

Int[1/(Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]*Sqrt[(d_) + (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]*Sqr
t[2*a + (b + r)*x]*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1117

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+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+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+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {\left (4 c \left (b+\sqrt {b^2-4 a c}\right ) d+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+4 a^2 f}+\frac {\left (4 c^2 d+\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+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+f x^2}}\\ &=-\frac {\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \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+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\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+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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \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-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )|\frac {1}{2} \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {a+b x+c x^2} \sqrt {d+f x^2} \sqrt {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\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+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 [C] Result contains complex when optimal does not.
time = 3.13, size = 600, normalized size = 0.56 \begin {gather*} -\frac {2 \sqrt {2} \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (-i \sqrt {d}+\sqrt {f} x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (i \sqrt {d}+\sqrt {f} x\right )}{\left (-2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {\frac {c \left (-i \sqrt {d} \left (\sqrt {b^2-4 a c}+2 c x\right )+\sqrt {f} \left (-2 a+\sqrt {b^2-4 a c} x\right )+b \left (-i \sqrt {d}-\sqrt {f} x\right )\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right )|\frac {c d-i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}{c d+i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}\right )}{\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \sqrt {\frac {i c \sqrt {b^2-4 a c} \left (\sqrt {d}+i \sqrt {f} x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {d+f x^2} \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*((-I)*Sqrt[d] + Sqrt[f]*x)*Sqrt[-((c*Sqrt[b^2 - 4*a*c]*(I*Sqrt[d]
+ Sqrt[f]*x))/(((-2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)))]*Sqrt[
(c*((-I)*Sqrt[d]*(Sqrt[b^2 - 4*a*c] + 2*c*x) + Sqrt[f]*(-2*a + Sqrt[b^2 - 4*a*c]*x) + b*((-I)*Sqrt[d] - Sqrt[f
]*x)))/(((2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]*EllipticF[ArcSi
n[Sqrt[(((-2*I)*c*Sqrt[d] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(((2*I)*c*Sqrt[
d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]], (c*d - I*Sqrt[b^2 - 4*a*c]*Sqrt[d]*
Sqrt[f] + a*f)/(c*d + I*Sqrt[b^2 - 4*a*c]*Sqrt[d]*Sqrt[f] + a*f)])/(((-2*I)*c*Sqrt[d] + (-b + Sqrt[b^2 - 4*a*c
])*Sqrt[f])*Sqrt[(I*c*Sqrt[b^2 - 4*a*c]*(Sqrt[d] + I*Sqrt[f]*x))/(((2*I)*c*Sqrt[d] + (b + Sqrt[b^2 - 4*a*c])*S
qrt[f])*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))]*Sqrt[d + f*x^2]*Sqrt[a + x*(b + c*x)])

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Maple [A]
time = 0.27, size = 714, normalized size = 0.66

method	result	size
default	\(\frac {16 \left (b c f \,x^{2}-2 c^{2} x^{2} \sqrt {-d f}-c f \,x^{2} \sqrt {-4 a c +b^{2}}+4 a c f x -2 b c x \sqrt {-d f}-2 c x \sqrt {-4 a c +b^{2}}\, \sqrt {-d f}+a b f +2 a c \sqrt {-d f}+a f \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-d f}-b \sqrt {-4 a c +b^{2}}\, \sqrt {-d f}\right ) \EllipticF \left (\sqrt {\frac {\left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c +b f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}, \sqrt {\frac {\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c +b f \right ) \left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right )}{\left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c -b f \right ) \left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c +b f \right )}}\right ) \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (f x +\sqrt {-d f}\right ) c}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, \left (-f x +\sqrt {-d f}\right ) c}{\left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {\frac {\left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c +b f \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\left (f \sqrt {-4 a c +b^{2}}+2 \sqrt {-d f}\, c -b f \right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}}{\sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (-f x +\sqrt {-d f}\right ) \left (f x +\sqrt {-d f}\right )}{c f}}\, \sqrt {-4 a c +b^{2}}\, \left (f \sqrt {-4 a c +b^{2}}-2 \sqrt {-d f}\, c +b f \right ) \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}}\)	\(714\)
elliptic	\(\frac {2 \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}\, \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {\sqrt {-d f}}{f}\right ) \sqrt {\frac {\left (-\frac {\sqrt {-d f}}{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 {\sqrt {-d f}}{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 {\sqrt {-d f}}{f}\right )}{\left (\frac {\sqrt {-d f}}{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 {\sqrt {-d f}}{f}\right )}{\left (-\frac {\sqrt {-d f}}{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 )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {\sqrt {-d f}}{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 {\sqrt {-d f}}{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 {\sqrt {-d f}}{f}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {\sqrt {-d f}}{f}\right )}{\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {\sqrt {-d f}}{f}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {\sqrt {-d f}}{f}\right )}}\right )}{\sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}\, \left (-\frac {\sqrt {-d f}}{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 {\sqrt {-d f}}{f}\right ) \left (x +\frac {\sqrt {-d f}}{f}\right )}}\)	\(787\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + d)/(c*f*x^4 + b*f*x^3 + b*d*x + (c*d + a*f)*x^2 + a*d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + f x^{2}} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {f\,x^2+d}\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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