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

Optimal. Leaf size=71 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.082331, antiderivative size = 71, 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 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) \left (a+b x^2+c x\right )}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x + b*x^2)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.5301, size = 60, normalized size = 0.85 \[ \frac{4 b \operatorname{atanh}{\left (\frac{2 b x + c}{\sqrt{- 4 a b + c^{2}}} \right )}}{\left (- 4 a b + c^{2}\right )^{\frac{3}{2}}} - \frac{2 b x + c}{\left (- 4 a b + c^{2}\right ) \left (a + b x^{2} + c x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.104929, size = 70, normalized size = 0.99 \[ \frac{2 b x+c}{\left (4 a b-c^2\right ) (a+x (b x+c))}+\frac{4 b \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\left (4 a b-c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 68, normalized size = 1. \[{\frac{2\,bx+c}{ \left ( 4\,ab-{c}^{2} \right ) \left ( b{x}^{2}+cx+a \right ) }}+4\,{\frac{b}{ \left ( 4\,ab-{c}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,bx+c}{\sqrt{4\,ab-{c}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+c*x+a)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232097, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (b^{2} x^{2} + b c x + a b\right )} \log \left (-\frac{4 \, a b c - c^{3} + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x -{\left (2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2}\right )} \sqrt{-4 \, a b + c^{2}}}{b x^{2} + c x + a}\right ) - \sqrt{-4 \, a b + c^{2}}{\left (2 \, b x + c\right )}}{{\left (4 \, a^{2} b - a c^{2} +{\left (4 \, a b^{2} - b c^{2}\right )} x^{2} +{\left (4 \, a b c - c^{3}\right )} x\right )} \sqrt{-4 \, a b + c^{2}}}, -\frac{4 \,{\left (b^{2} x^{2} + b c x + a b\right )} \arctan \left (-\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right ) - \sqrt{4 \, a b - c^{2}}{\left (2 \, b x + c\right )}}{{\left (4 \, a^{2} b - a c^{2} +{\left (4 \, a b^{2} - b c^{2}\right )} x^{2} +{\left (4 \, a b c - c^{3}\right )} x\right )} \sqrt{4 \, a b - c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.78101, size = 265, normalized size = 3.73 \[ - 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + 2 b \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} b^{3} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c^{4} \sqrt{- \frac{1}{\left (4 a b - c^{2}\right )^{3}}} + 2 b c}{4 b^{2}} \right )} + \frac{2 b x + c}{4 a^{2} b - a c^{2} + x^{2} \left (4 a b^{2} - b c^{2}\right ) + x \left (4 a b c - c^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207574, size = 90, normalized size = 1.27 \[ \frac{4 \, b \arctan \left (\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{{\left (4 \, a b - c^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, b x + c}{{\left (b x^{2} + c x + a\right )}{\left (4 \, a b - c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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