\(\int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx\) [7]

Optimal result

Integrand size = 26, antiderivative size = 1077 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx =\text {Too large to display} \] Output:

-(b^2*d+b*(-4*a*c+b^2)^(1/2)*d-2*a*(-a*f+c*d))^(1/4)*(b+(-4*a*c+b^2)^(1/2) 
+2*c*x)^(3/2)*(2*a+(b+(-4*a*c+b^2)^(1/2))*x)^(1/2)*((4*a*c-(b+(-4*a*c+b^2) 
^(1/2))^2)^2*(f*x^2+d)/((b+(-4*a*c+b^2)^(1/2))^2*d+4*a^2*f)/(b+(-4*a*c+b^2 
)^(1/2)+2*c*x)^2)^(1/2)*(1+(2*c^2*d-2*a*c*f+b*(b+(-4*a*c+b^2)^(1/2))*f)^(1 
/2)*(2*a+(b+(-4*a*c+b^2)^(1/2))*x)/(b^2*d+b*(-4*a*c+b^2)^(1/2)*d-2*a*(-a*f 
+c*d))^(1/2)/(b+(-4*a*c+b^2)^(1/2)+2*c*x))*((1-4*(b+(-4*a*c+b^2)^(1/2))*(a 
*f+c*d)*(2*a+(b+(-4*a*c+b^2)^(1/2))*x)/((b+(-4*a*c+b^2)^(1/2))^2*d+4*a^2*f 
)/(b+(-4*a*c+b^2)^(1/2)+2*c*x)+(4*c^2*d+(b+(-4*a*c+b^2)^(1/2))^2*f)*(2*a+( 
b+(-4*a*c+b^2)^(1/2))*x)^2/((b+(-4*a*c+b^2)^(1/2))^2*d+4*a^2*f)/(b+(-4*a*c 
+b^2)^(1/2)+2*c*x)^2)/(1+(2*c^2*d-2*a*c*f+b*(b+(-4*a*c+b^2)^(1/2))*f)^(1/2 
)*(2*a+(b+(-4*a*c+b^2)^(1/2))*x)/(b^2*d+b*(-4*a*c+b^2)^(1/2)*d-2*a*(-a*f+c 
*d))^(1/2)/(b+(-4*a*c+b^2)^(1/2)+2*c*x))^2)^(1/2)*InverseJacobiAM(2*arctan 
((2*c^2*d-2*a*c*f+b*(b+(-4*a*c+b^2)^(1/2))*f)^(1/4)*(2*a+(b+(-4*a*c+b^2)^( 
1/2))*x)^(1/2)/(b^2*d+b*(-4*a*c+b^2)^(1/2)*d-2*a*(-a*f+c*d))^(1/4)/(b+(-4* 
a*c+b^2)^(1/2)+2*c*x)^(1/2)),1/2*(2+2*(b+(-4*a*c+b^2)^(1/2))*(a*f+c*d)/(2* 
c^2*d-2*a*c*f+b*(b+(-4*a*c+b^2)^(1/2))*f)^(1/2)/(b^2*d+b*(-4*a*c+b^2)^(1/2 
)*d-2*a*(-a*f+c*d))^(1/2))^(1/2))/(4*a*c-(b+(-4*a*c+b^2)^(1/2))^2)/(2*c^2* 
d-2*a*c*f+b*(b+(-4*a*c+b^2)^(1/2))*f)^(1/4)/(c*x^2+b*x+a)^(1/2)/(f*x^2+d)^ 
(1/2)/(1-4*(b+(-4*a*c+b^2)^(1/2))*(a*f+c*d)*(2*a+(b+(-4*a*c+b^2)^(1/2))*x) 
/((b+(-4*a*c+b^2)^(1/2))^2*d+4*a^2*f)/(b+(-4*a*c+b^2)^(1/2)+2*c*x)+(4*c...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.98 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=-\frac {2 \sqrt {2} \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (-i \sqrt {d}+\sqrt {f} x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (i \sqrt {d}+\sqrt {f} x\right )}{\left (-2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {\frac {c \left (-i \sqrt {d} \left (\sqrt {b^2-4 a c}+2 c x\right )+\sqrt {f} \left (-2 a+\sqrt {b^2-4 a c} x\right )+b \left (-i \sqrt {d}-\sqrt {f} x\right )\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right ),\frac {c d-i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}{c d+i \sqrt {b^2-4 a c} \sqrt {d} \sqrt {f}+a f}\right )}{\left (-2 i c \sqrt {d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \sqrt {\frac {i c \sqrt {b^2-4 a c} \left (\sqrt {d}+i \sqrt {f} x\right )}{\left (2 i c \sqrt {d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {d+f x^2} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(-2*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*((-I)*Sqrt[d] + Sqrt[f]*x)*Sq 
rt[-((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[ArcSin[Sq 
rt[(((-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*Sq 
rt[d] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[f])*Sqrt[(I*c*Sqrt[b^2 - 4*a*c]*(Sqr 
t[d] + I*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[d + f*x^2]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (warning: unable to verify)

Time = 1.26 (sec) , antiderivative size = 1077, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1324, 732, 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.

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

\(\Big \downarrow \) 1324

\(\displaystyle \frac {\sqrt {\sqrt {b^2-4 a c}+b+2 c x} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \int \frac {1}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {f x^2+d}}dx}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 732

\(\displaystyle -\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )^{3/2} \sqrt {x \left (\sqrt {b^2-4 a c}+b\right )+2 a} \sqrt {\frac {\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right )^2 \left (d+f x^2\right )}{\left (\sqrt {b^2-4 a c}+b+2 c x\right )^2 \left (4 a^2 f+d \left (\sqrt {b^2-4 a c}+b\right )^2\right )}} \int \frac {1}{\sqrt {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}}d\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+2 c x+\sqrt {b^2-4 a c}}}}{\left (4 a c-\left (\sqrt {b^2-4 a c}+b\right )^2\right ) \sqrt {d+f x^2} \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\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}} \operatorname {EllipticF}\left (2 \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}}\)

Input:

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

Output:

-(((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 + Sqr 
t[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)))^2]*Ellipti 
cF[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[...
 

Defintions of rubi rules used

rule 732

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)* 
(x_)^2]), x_Symbol] :> Simp[-2*(c + d*x)*(Sqrt[(d*e - c*f)^2*((a + b*x^2)/( 
(b*e^2 + a*f^2)*(c + d*x)^2))]/((d*e - c*f)*Sqrt[a + b*x^2]))   Subst[Int[1 
/Sqrt[Simp[1 - (2*b*c*e + 2*a*d*f)*(x^2/(b*e^2 + a*f^2)) + (b*c^2 + a*d^2)* 
(x^4/(b*e^2 + a*f^2)), x]], x], x, Sqrt[e + f*x]/Sqrt[c + d*x]], x] /; Free 
Q[{a, b, c, d, e, f}, x]
 

rule 1324

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]}, 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 + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f}, x] && Ne 
Q[b^2 - 4*a*c, 0]
 

rule 1416

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]
 

Maple [A] (warning: unable to verify)

Time = 7.34 (sec) , antiderivative size = 721, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {16 \left (b c f \,x^{2}-2 c^{2} x^{2} \sqrt {-d f}-c f \,x^{2} \sqrt {-4 a c +b^{2}}+4 a c f x -2 b c x \sqrt {-d f}-2 c x \sqrt {-d f}\, \sqrt {-4 a c +b^{2}}+a b f +2 a c \sqrt {-d f}+a f \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-d f}-b \sqrt {-d f}\, \sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {\left (2 \sqrt {-d f}\, c -\sqrt {-4 a c +b^{2}}\, f -f b \right ) \left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right )}{\left (2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f -f b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )}}, \sqrt {\frac {\left (2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f +f b \right ) \left (2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f -f b \right )}{\left (2 \sqrt {-d f}\, c -\sqrt {-4 a c +b^{2}}\, f +f b \right ) \left (2 \sqrt {-d f}\, c -\sqrt {-4 a c +b^{2}}\, f -f b \right )}}\right ) \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (f x +\sqrt {-d f}\right ) c}{\left (2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f -f b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, \left (-f x +\sqrt {-d f}\right ) c}{\left (2 \sqrt {-d f}\, c -\sqrt {-4 a c +b^{2}}\, f +f b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )}}\, \sqrt {-\frac {\left (2 \sqrt {-d f}\, c -\sqrt {-4 a c +b^{2}}\, f -f b \right ) \left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right )}{\left (2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f -f b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}}{\sqrt {\frac {\left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) \left (-f x +\sqrt {-d f}\right ) \left (f x +\sqrt {-d f}\right )}{c f}}\, \sqrt {-4 a c +b^{2}}\, \left (-2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f +f b \right ) \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}}\)	\(721\)
elliptic	\(\frac {2 \sqrt {\left (c \,x^{2}+b x +a \right ) \left (f \,x^{2}+d \right )}\, \left (\frac {\sqrt {-d f}}{f}+\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\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 )}}\, \operatorname {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 {\sqrt {-d f}}{f}+\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 (\frac {\sqrt {-d f}}{f}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\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 )}}\)	\(787\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \] Input:

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

Output:

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)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int \frac {1}{\sqrt {d + f x^{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int \frac {1}{\sqrt {f\,x^2+d}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx=\int \frac {1}{\sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}}d x \] Input:

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

Output:

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