3.851 \(\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=285 \[ \frac{2 \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \]

[Out]

(-2*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2
- 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*
x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (2*Sqrt[2*c*
d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*
Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b
^2 - 4*a*c]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Rubi [A]  time = 1.19717, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{2 \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2
- 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*
x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (2*Sqrt[2*c*
d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*
Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b
^2 - 4*a*c]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Rubi in Sympy [A]  time = 178.437, size = 267, normalized size = 0.94 \[ - \frac{2 \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{b g - 2 c f - g \sqrt{- 4 a c + b^{2}}}}{\sqrt{f + g x} \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{- 4 a c + b^{2}} \sqrt{b g - 2 c f - g \sqrt{- 4 a c + b^{2}}}} + \frac{2 \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}}{\sqrt{f + g x} \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{- 4 a c + b^{2}} \sqrt{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))*atanh(sqrt(d + e*x)*sqrt(b*g - 2*c*
f - g*sqrt(-4*a*c + b**2))/(sqrt(f + g*x)*sqrt(b*e - 2*c*d - e*sqrt(-4*a*c + b**
2))))/(sqrt(-4*a*c + b**2)*sqrt(b*g - 2*c*f - g*sqrt(-4*a*c + b**2))) + 2*sqrt(b
*e - 2*c*d + e*sqrt(-4*a*c + b**2))*atanh(sqrt(d + e*x)*sqrt(b*g - 2*c*f + g*sqr
t(-4*a*c + b**2))/(sqrt(f + g*x)*sqrt(b*e - 2*c*d + e*sqrt(-4*a*c + b**2))))/(sq
rt(-4*a*c + b**2)*sqrt(b*g - 2*c*f + g*sqrt(-4*a*c + b**2)))

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Mathematica [B]  time = 4.75224, size = 925, normalized size = 3.25 \[ \frac{\frac{\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \log \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g}}-\frac{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \log \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{2 d f c^2-\left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) e g}}-\frac{\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \log \left ((d g+e (f+2 g x)) b^2-\sqrt{b^2-4 a c} (d g+e (f+2 g x)) b+2 c \left (\sqrt{b^2-4 a c} e f x-2 a e (f+2 g x)+d \left (2 \sqrt{b^2-4 a c} f-2 a g+\sqrt{b^2-4 a c} g x\right )\right )+2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g}}+\frac{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \log \left (-(d g+e (f+2 g x)) b^2-\sqrt{b^2-4 a c} (d g+e (f+2 g x)) b+2 c \left (\sqrt{b^2-4 a c} e f x+2 a e (f+2 g x)+d \left (2 \sqrt{b^2-4 a c} f+2 a g+\sqrt{b^2-4 a c} g x\right )\right )+2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 d f c^2-\left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{2 d f c^2-\left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) e g}}}{\sqrt{2} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/Sqrt
[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + Sqrt[b^2
- 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))] - ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)
*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x])/Sqrt[2*c^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*
e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g
)] - ((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Log[2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[
2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + Sqrt[b^2
- 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))]*Sqrt[d + e*x]*Sqrt[f + g*x] + b^2*(d*g
+ e*(f + 2*g*x)) - b*Sqrt[b^2 - 4*a*c]*(d*g + e*(f + 2*g*x)) + 2*c*(Sqrt[b^2 - 4
*a*c]*e*f*x - 2*a*e*(f + 2*g*x) + d*(2*Sqrt[b^2 - 4*a*c]*f - 2*a*g + Sqrt[b^2 -
4*a*c]*g*x))])/Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*
a*c]*e*f + Sqrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))] + ((2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e)*Log[2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f + b*(b + Sqrt
[b^2 - 4*a*c])*e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c
]*d*g + 2*a*e*g)]*Sqrt[d + e*x]*Sqrt[f + g*x] - b^2*(d*g + e*(f + 2*g*x)) - b*Sq
rt[b^2 - 4*a*c]*(d*g + e*(f + 2*g*x)) + 2*c*(Sqrt[b^2 - 4*a*c]*e*f*x + 2*a*e*(f
+ 2*g*x) + d*(2*Sqrt[b^2 - 4*a*c]*f + 2*a*g + Sqrt[b^2 - 4*a*c]*g*x))])/Sqrt[2*c
^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*
g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c])

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Maple [B]  time = 0.06, size = 5484, normalized size = 19.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 40.9607, size = 6036, normalized size = 21.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g + ((b^2*c - 4*a*c^2)*f^2 - (
b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)
/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a
^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2
*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2))*log(-(2*b*d^2*
f*g - 2*a*d^2*g^2 - 2*(b*d*e - a*e^2)*f^2 + sqrt(2)*((b^2 - 4*a*c)*e*f^2 - (b^2
- 4*a*c)*d*f*g + ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g
+ 3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2 - 4*a^3*c)*g^3)*sqrt((e^2*f^2 - 2*d*e
*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 -
2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*
c)*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g +
((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*
f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*
g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b
^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a
^2*c)*g^2)) - (b*e^2*f^2 - 4*a*e^2*f*g - (b*d^2 - 4*a*d*e)*g^2)*x - (2*(b^2*c -
4*a*c^2)*d*f^3 - 2*(b^3 - 4*a*b*c)*d*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^2
*c - 4*a*c^2)*e*f^3 + (a*b^2 - 4*a^2*c)*d*g^3 + ((b^2*c - 4*a*c^2)*d - (b^3 - 4*
a*b*c)*e)*f^2*g - ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*f*g^2)*x)*sqrt((e^2*
f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*
g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b
^2 - 4*a^3*c)*g^4)))/x) - 1/4*sqrt(2)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g +
((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*
f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*
g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b
^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a
^2*c)*g^2))*log(-(2*b*d^2*f*g - 2*a*d^2*g^2 - 2*(b*d*e - a*e^2)*f^2 - sqrt(2)*((
b^2 - 4*a*c)*e*f^2 - (b^2 - 4*a*c)*d*f*g + ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a
*b^2*c - 8*a^2*c^2)*f^2*g + 3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2 - 4*a^3*c)*
g^3)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c -
4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)
*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(((2*c*d - b
*e)*f - (b*d - 2*a*e)*g + ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2
- 4*a^2*c)*g^2)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 -
2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 -
4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4
*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)) - (b*e^2*f^2 - 4*a*e^2*f*g - (b*d^2 - 4*a*
d*e)*g^2)*x - (2*(b^2*c - 4*a*c^2)*d*f^3 - 2*(b^3 - 4*a*b*c)*d*f^2*g + 2*(a*b^2
- 4*a^2*c)*d*f*g^2 + ((b^2*c - 4*a*c^2)*e*f^3 + (a*b^2 - 4*a^2*c)*d*g^3 + ((b^2*
c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*f^2*g - ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*
c)*e)*f*g^2)*x)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 -
2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 -
4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/x) + 1/4*sqrt(2)*sqrt(((2*c*d - b
*e)*f - (b*d - 2*a*e)*g - ((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2
- 4*a^2*c)*g^2)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 -
2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 -
4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4
*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2))*log(-(2*b*d^2*f*g - 2*a*d^2*g^2 - 2*(b*d*e
- a*e^2)*f^2 + sqrt(2)*((b^2 - 4*a*c)*e*f^2 - (b^2 - 4*a*c)*d*f*g - ((b^3*c - 4
*a*b*c^2)*f^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g + 3*(a*b^3 - 4*a^2*b*c)*f*g^
2 - 2*(a^2*b^2 - 4*a^3*c)*g^3)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 -
4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g
^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))*sqrt(e*x + d)*sqrt
(g*x + f)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g - ((b^2*c - 4*a*c^2)*f^2 - (b^
3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(
(b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2
*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c
- 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)) - (b*e^2*f^2 - 4
*a*e^2*f*g - (b*d^2 - 4*a*d*e)*g^2)*x + (2*(b^2*c - 4*a*c^2)*d*f^3 - 2*(b^3 - 4*
a*b*c)*d*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^2*c - 4*a*c^2)*e*f^3 + (a*b^2
- 4*a^2*c)*d*g^3 + ((b^2*c - 4*a*c^2)*d - (b^3 - 4*a*b*c)*e)*f^2*g - ((b^3 - 4*
a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*f*g^2)*x)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(
(b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2
*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/x) - 1/
4*sqrt(2)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g - ((b^2*c - 4*a*c^2)*f^2 - (b^
3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(
(b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*a*b^2*c - 8*a^2
*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)*g^4)))/((b^2*c
- 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2))*log(-(2*b*d^2*f*
g - 2*a*d^2*g^2 - 2*(b*d*e - a*e^2)*f^2 - sqrt(2)*((b^2 - 4*a*c)*e*f^2 - (b^2 -
4*a*c)*d*f*g - ((b^3*c - 4*a*b*c^2)*f^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g +
3*(a*b^3 - 4*a^2*b*c)*f*g^2 - 2*(a^2*b^2 - 4*a^3*c)*g^3)*sqrt((e^2*f^2 - 2*d*e*f
*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g + (b^4 - 2*
a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2 - 4*a^3*c)
*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(((2*c*d - b*e)*f - (b*d - 2*a*e)*g - ((
b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2*c)*g^2)*sqrt((e^2*f^
2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g
+ (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2
- 4*a^3*c)*g^4)))/((b^2*c - 4*a*c^2)*f^2 - (b^3 - 4*a*b*c)*f*g + (a*b^2 - 4*a^2
*c)*g^2)) - (b*e^2*f^2 - 4*a*e^2*f*g - (b*d^2 - 4*a*d*e)*g^2)*x + (2*(b^2*c - 4*
a*c^2)*d*f^3 - 2*(b^3 - 4*a*b*c)*d*f^2*g + 2*(a*b^2 - 4*a^2*c)*d*f*g^2 + ((b^2*c
- 4*a*c^2)*e*f^3 + (a*b^2 - 4*a^2*c)*d*g^3 + ((b^2*c - 4*a*c^2)*d - (b^3 - 4*a*
b*c)*e)*f^2*g - ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*f*g^2)*x)*sqrt((e^2*f^
2 - 2*d*e*f*g + d^2*g^2)/((b^2*c^2 - 4*a*c^3)*f^4 - 2*(b^3*c - 4*a*b*c^2)*f^3*g
+ (b^4 - 2*a*b^2*c - 8*a^2*c^2)*f^2*g^2 - 2*(a*b^3 - 4*a^2*b*c)*f*g^3 + (a^2*b^2
- 4*a^3*c)*g^4)))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError