\(\int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx\) [164]

Optimal result

Integrand size = 29, antiderivative size = 700 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=-\frac {\sqrt {2} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e-\sqrt {e^2-4 d f}+2 f x} \sqrt {-\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) \left (e+\sqrt {e^2-4 d f}\right )-2 \left (\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {e^2-4 d f} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}} \sqrt {a+b x+c x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {b^2-4 a c+b \sqrt {b^2-4 a c}} \sqrt {e-\sqrt {e^2-4 d f}+2 f x}}{\sqrt {\sqrt {b^2-4 a c} e-4 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}+b \left (e-\sqrt {e^2-4 d f}\right )} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right ),-\frac {4 c \left (b+\sqrt {b^2-4 a c}\right ) d-2 b \left (b+\sqrt {b^2-4 a c}\right ) e+4 a \left (b+\sqrt {b^2-4 a c}\right ) f-\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt {e^2-4 d f}}{2 \left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt {e^2-4 d f}}\right )}{\sqrt {b^2-4 a c+b \sqrt {b^2-4 a c}} \sqrt {\sqrt {b^2-4 a c} e-4 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}+b \left (e-\sqrt {e^2-4 d f}\right )} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 a f-\left (b+\sqrt {b^2-4 a c}\right ) \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}} \sqrt {d+e x+f x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 5.93 (sec) , antiderivative size = 670, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx=-\frac {\left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {-\frac {c \left (4 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}-2 \sqrt {b^2-4 a c} f x+2 c \sqrt {e^2-4 d f} x-e \left (\sqrt {b^2-4 a c}+2 c x\right )+b \left (-e+\sqrt {e^2-4 d f}+2 f x\right )\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right ),\frac {2 c d-b e+2 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}{2 c d-b e+2 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}\right )}{\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \sqrt {\frac {c \sqrt {b^2-4 a c} \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {a+x (b+c x)} \sqrt {d+x (e+f x)}} \] Input:

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

Output:

-(((-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)]))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1432\) vs. \(2(700)=1400\).

Time = 1.77 (sec) , antiderivative size = 1432, normalized size of antiderivative = 2.05, 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}}\)

Input:

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

Output:

-(((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^...
 

Defintions of rubi rules used

rule 1280

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]
 

rule 1323

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]
 

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 = 8.18 (sec) , antiderivative size = 905, normalized size of antiderivative = 1.29

method	result	size
elliptic	\(\frac {2 \sqrt {\left (f \,x^{2}+e x +d \right ) \left (c \,x^{2}+b x +a \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 {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\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 (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 )}^{2} \sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\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 {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\, \operatorname {EllipticF}\left (\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 (x -\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 (x +\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 (\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 {f \,x^{2}+e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \sqrt {c f \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 ) \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 )}}\)	\(905\)
default	\(-\frac {8 \left (-2 b \,f^{2} x^{2}+2 c e f \,x^{2}-2 c f \,x^{2} \sqrt {-4 d f +e^{2}}-2 f^{2} x^{2} \sqrt {-4 a c +b^{2}}-2 b e f x -2 b f x \sqrt {-4 d f +e^{2}}+8 c d f x -2 e f x \sqrt {-4 a c +b^{2}}-2 f x \sqrt {-4 a c +b^{2}}\, \sqrt {-4 d f +e^{2}}+2 b d f -b \,e^{2}-b e \sqrt {-4 d f +e^{2}}+2 d e c +2 c d \sqrt {-4 d f +e^{2}}+2 \sqrt {-4 a c +b^{2}}\, d f -\sqrt {-4 a c +b^{2}}\, e^{2}-e \sqrt {-4 a c +b^{2}}\, \sqrt {-4 d f +e^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {\left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right )}{\left (\sqrt {-4 a c +b^{2}}\, f +c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}, \sqrt {\frac {\left (\sqrt {-4 a c +b^{2}}\, f +c \sqrt {-4 d f +e^{2}}-f b +c e \right ) \left (\sqrt {-4 a c +b^{2}}\, f +c \sqrt {-4 d f +e^{2}}+f b -c e \right )}{\left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}-f b +c e \right ) \left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}+f b -c e \right )}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) f}{\left (\sqrt {-4 a c +b^{2}}\, f +c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, \left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) f}{\left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}-f b +c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {-\frac {\left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right )}{\left (\sqrt {-4 a c +b^{2}}\, f +c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+e x +d}}{\sqrt {\frac {\left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right ) \left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )}{c f}}\, \sqrt {-4 d f +e^{2}}\, \left (\sqrt {-4 a c +b^{2}}\, f -c \sqrt {-4 d f +e^{2}}+f b -c e \right ) \sqrt {\left (f \,x^{2}+e x +d \right ) \left (c \,x^{2}+b x +a \right )}}\)	\(906\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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