3.2184 \(\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-((b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*ArcTanh[(b + 2*c*x)/Sqrt
[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi [A]  time = 0.0667497, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(-2),x]

[Out]

-((b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*ArcTanh[(b + 2*c*x)/Sqrt
[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi in Sympy [A]  time = 7.36642, size = 60, normalized size = 0.91 \[ \frac{4 c \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{b + 2 c x}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(3/2) - (b + 2*c*x)/
((-4*a*c + b**2)*(a + b*x + c*x**2))

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Mathematica [A]  time = 0.146819, size = 70, normalized size = 1.06 \[ -\frac{\frac{4 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(-2),x]

[Out]

-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/
Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c))

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Maple [A]  time = 0.004, size = 68, normalized size = 1. \[{\frac{2\,cx+b}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}+4\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x+a)^2,x)

[Out]

(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)+4*c/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*
c-b^2)^(1/2))

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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((c*x^2 + b*x + a)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212103, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{4 \,{\left (c^{2} x^{2} + b c x + a c\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-2),x, algorithm="fricas")

[Out]

[-(2*(c^2*x^2 + b*c*x + a*c)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x - (2*c^
2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) + sqrt(b^2
- 4*a*c)*(2*c*x + b))/((a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c
)*x)*sqrt(b^2 - 4*a*c)), -(4*(c^2*x^2 + b*c*x + a*c)*arctan(-sqrt(-b^2 + 4*a*c)*
(2*c*x + b)/(b^2 - 4*a*c)) + sqrt(-b^2 + 4*a*c)*(2*c*x + b))/((a*b^2 - 4*a^2*c +
(b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 2.68581, size = 265, normalized size = 4.02 \[ - 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + \frac{b + 2 c x}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-32*a**2*c**3*sqrt(-1/(4*a*c - b**2)**3
) + 16*a*b**2*c**2*sqrt(-1/(4*a*c - b**2)**3) - 2*b**4*c*sqrt(-1/(4*a*c - b**2)*
*3) + 2*b*c)/(4*c**2)) + 2*c*sqrt(-1/(4*a*c - b**2)**3)*log(x + (32*a**2*c**3*sq
rt(-1/(4*a*c - b**2)**3) - 16*a*b**2*c**2*sqrt(-1/(4*a*c - b**2)**3) + 2*b**4*c*
sqrt(-1/(4*a*c - b**2)**3) + 2*b*c)/(4*c**2)) + (b + 2*c*x)/(4*a**2*c - a*b**2 +
x**2*(4*a*c**2 - b**2*c) + x*(4*a*b*c - b**3))

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GIAC/XCAS [A]  time = 0.204332, size = 103, normalized size = 1.56 \[ -\frac{4 \, c \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-2),x, algorithm="giac")

[Out]

-4*c*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(-b^2 + 4*a*c)) -
(2*c*x + b)/((c*x^2 + b*x + a)*(b^2 - 4*a*c))