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

Optimal. Leaf size=228 \[ \frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e (e f-d g)}-\frac{\sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{g (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e g} \]

[Out]

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

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Rubi [A]  time = 0.683496, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e (e f-d g)}-\frac{\sqrt{a g^2-b f g+c f^2} \tanh ^{-1}\left (\frac{-2 a g+x (2 c f-b g)+b f}{2 \sqrt{a+b x+c x^2} \sqrt{a g^2-b f g+c f^2}}\right )}{g (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e g} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 96.7341, size = 199, normalized size = 0.87 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{e g} - \frac{\sqrt{a g^{2} - b f g + c f^{2}} \operatorname{atanh}{\left (\frac{2 a g - b f + x \left (b g - 2 c f\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a g^{2} - b f g + c f^{2}}} \right )}}{g \left (d g - e f\right )} + \frac{\sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e \left (d g - e f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(e*g) - sqrt(a*g**
2 - b*f*g + c*f**2)*atanh((2*a*g - b*f + x*(b*g - 2*c*f))/(2*sqrt(a + b*x + c*x*
*2)*sqrt(a*g**2 - b*f*g + c*f**2)))/(g*(d*g - e*f)) + sqrt(a*e**2 - b*d*e + c*d*
*2)*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2
- b*d*e + c*d**2)))/(e*(d*g - e*f))

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Mathematica [A]  time = 0.653289, size = 299, normalized size = 1.31 \[ \frac{-g \sqrt{a e^2-b d e+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )+g \log (d+e x) \sqrt{e (a e-b d)+c d^2}-\sqrt{c} d g \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-e \log (f+g x) \sqrt{a g^2-b f g+c f^2}+e \sqrt{a g^2-b f g+c f^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a g^2-b f g+c f^2}+2 a g-b f+b g x-2 c f x\right )+\sqrt{c} e f \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{e g (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c*d^2 + e*(-(b*d) + a*e)]*g*Log[d + e*x] - e*Sqrt[c*f^2 - b*f*g + a*g^2]*L
og[f + g*x] + Sqrt[c]*e*f*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - Sqr
t[c]*d*g*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - Sqrt[c*d^2 - b*d*e +
a*e^2]*g*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*S
qrt[a + x*(b + c*x)]] + e*Sqrt[c*f^2 - b*f*g + a*g^2]*Log[-(b*f) + 2*a*g - 2*c*f
*x + b*g*x + 2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + x*(b + c*x)]])/(e*g*(e*f - d
*g))

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Maple [B]  time = 0.037, size = 1529, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(d*g-e*f)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)+1/
2/(d*g-e*f)*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)/g*
(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/c^(1/2)*b-1/(d*g-e*f)/g*ln((1/2*(b*g-2*c
*f)/g+c*(x+f/g))/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/
g^2)^(1/2))*c^(1/2)*f-1/(d*g-e*f)/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b
*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)
^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*a+1/(d*g-e*f
)/g/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*
(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*
g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*b*f-1/(d*g-e*f)/g^2/((a*g^2-b*f*g+c*f^2)/g
^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*
f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2
))/(x+f/g))*c*f^2-1/(d*g-e*f)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2)-1/2/(d*g-e*f)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)*b+1/(d*g-e*f)/
e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d+1/(d*g-e*f)/((a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+
d/e))*a-1/(d*g-e*f)/e/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/
e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*
c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*d+1/(d*g-e*f)/e^2/((a*
e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+
2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*
e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError