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

Optimal. Leaf size=1432 \[ \text{result too large to display} \]

[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 + Sqr
t[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 + Sq
rt[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^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)]))

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Rubi [A]  time = 13.0442, antiderivative size = 1432, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\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}} \]

Warning: Unable to verify antiderivative.

[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 + Sqr
t[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 + Sq
rt[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^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)]))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x+a)**(1/2)/(f*x**2+e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 5.02743, 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*Sq
rt[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))]*Sqrt[
a + x*(b + c*x)]*Sqrt[d + x*(e + f*x)]))

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Maple [A]  time = 0.331, size = 928, normalized size = 0.7 \[ \text{result too large to display} \]

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*x*f*d+2*(-4*a*c+b^2)^(1/2)*e*f*x+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+(-4*a*c+b^2)^(1/2)*e^2
+e*(-4*d*f+e^2)^(1/2)*(-4*a*c+b^2)^(1/2))*EllipticF(((c*(-4*d*f+e^2)^(1/2)-f*(-4
*a*c+b^2)^(1/2)-b*f+c*e)*(-2*f*x+(-4*d*f+e^2)^(1/2)-e)/(c*(-4*d*f+e^2)^(1/2)+f*(
-4*a*c+b^2)^(1/2)+b*f-c*e)/(2*f*x+(-4*d*f+e^2)^(1/2)+e))^(1/2),((c*(-4*d*f+e^2)^
(1/2)+f*(-4*a*c+b^2)^(1/2)-b*f+c*e)*(c*(-4*d*f+e^2)^(1/2)+f*(-4*a*c+b^2)^(1/2)+b
*f-c*e)/(c*(-4*d*f+e^2)^(1/2)-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/(c*(-4*d*f+e^2)^(1/2
)-f*(-4*a*c+b^2)^(1/2)-b*f+c*e))^(1/2))*((-4*d*f+e^2)^(1/2)*(b+2*c*x+(-4*a*c+b^2
)^(1/2))*f/(c*(-4*d*f+e^2)^(1/2)+f*(-4*a*c+b^2)^(1/2)+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+(-4*a*c+b^2)^(1/2)-b)*f/(c*(-4*d
*f+e^2)^(1/2)-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/(2*f*x+(-4*d*f+e^2)^(1/2)+e))^(1/2)*
((c*(-4*d*f+e^2)^(1/2)-f*(-4*a*c+b^2)^(1/2)-b*f+c*e)*(-2*f*x+(-4*d*f+e^2)^(1/2)-
e)/(c*(-4*d*f+e^2)^(1/2)+f*(-4*a*c+b^2)^(1/2)+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+(-4*a*c+b^2)^(1/2)-b)*(b+2*c*x+(-4*
a*c+b^2)^(1/2)))^(1/2)/(-4*d*f+e^2)^(1/2)/(-c*(-4*d*f+e^2)^(1/2)+f*(-4*a*c+b^2)^
(1/2)+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., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{f x^{2} + e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(f*x^2 + e*x + d)),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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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{f x^{2} + e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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