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

Optimal. Leaf size=72 \[ \frac{b \tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2}}-\frac{a+b x}{2 \left (a^2-b^2\right ) \left (2 a x+b x^2+b\right )} \]

[Out]

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

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Rubi [A]  time = 0.0962778, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b \tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2}}-\frac{a+b x}{2 \left (a^2-b^2\right ) \left (2 a x+b x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*a*x + b*x^2)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5646, size = 58, normalized size = 0.81 \[ \frac{b \operatorname{atanh}{\left (\frac{a + b x}{\sqrt{a^{2} - b^{2}}} \right )}}{2 \left (a^{2} - b^{2}\right )^{\frac{3}{2}}} - \frac{2 a + 2 b x}{4 \left (a^{2} - b^{2}\right ) \left (2 a x + b x^{2} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0715335, size = 72, normalized size = 1. \[ \frac{a+b x}{2 \left (b^2-a^2\right ) \left (2 a x+b x^2+b\right )}+\frac{b \tan ^{-1}\left (\frac{a+b x}{\sqrt{b^2-a^2}}\right )}{2 \left (b^2-a^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 86, normalized size = 1.2 \[{\frac{2\,bx+2\,a}{ \left ( -4\,{a}^{2}+4\,{b}^{2} \right ) \left ( b{x}^{2}+2\,ax+b \right ) }}+2\,{\frac{b}{ \left ( -4\,{a}^{2}+4\,{b}^{2} \right ) \sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,bx+2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+2*a*x+b)^2,x)

[Out]

(2*b*x+2*a)/(-4*a^2+4*b^2)/(b*x^2+2*a*x+b)+2*b/(-4*a^2+4*b^2)/(-a^2+b^2)^(1/2)*a
rctan(1/2*(2*b*x+2*a)/(-a^2+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((b*x^2 + 2*a*x + b)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224718, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (-\frac{2 \, a^{3} - 2 \, a b^{2} + 2 \,{\left (a^{2} b - b^{3}\right )} x -{\left (b^{2} x^{2} + 2 \, a b x + 2 \, a^{2} - b^{2}\right )} \sqrt{a^{2} - b^{2}}}{b x^{2} + 2 \, a x + b}\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b x + a\right )}}{4 \,{\left (a^{2} b - b^{3} +{\left (a^{2} b - b^{3}\right )} x^{2} + 2 \,{\left (a^{3} - a b^{2}\right )} x\right )} \sqrt{a^{2} - b^{2}}}, -\frac{{\left (b^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b x + a\right )}}{a^{2} - b^{2}}\right ) + \sqrt{-a^{2} + b^{2}}{\left (b x + a\right )}}{2 \,{\left (a^{2} b - b^{3} +{\left (a^{2} b - b^{3}\right )} x^{2} + 2 \,{\left (a^{3} - a b^{2}\right )} x\right )} \sqrt{-a^{2} + b^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((b^2*x^2 + 2*a*b*x + b^2)*log(-(2*a^3 - 2*a*b^2 + 2*(a^2*b - b^3)*x - (b^
2*x^2 + 2*a*b*x + 2*a^2 - b^2)*sqrt(a^2 - b^2))/(b*x^2 + 2*a*x + b)) + 2*sqrt(a^
2 - b^2)*(b*x + a))/((a^2*b - b^3 + (a^2*b - b^3)*x^2 + 2*(a^3 - a*b^2)*x)*sqrt(
a^2 - b^2)), -1/2*((b^2*x^2 + 2*a*b*x + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*x + a)/
(a^2 - b^2)) + sqrt(-a^2 + b^2)*(b*x + a))/((a^2*b - b^3 + (a^2*b - b^3)*x^2 + 2
*(a^3 - a*b^2)*x)*sqrt(-a^2 + b^2))]

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Sympy [A]  time = 2.58932, size = 228, normalized size = 3.17 \[ - \frac{b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} \log{\left (x + \frac{- a^{4} b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + a b - b^{5} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}}}{b^{2}} \right )}}{4} + \frac{b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} \log{\left (x + \frac{a^{4} b \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} - 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}} + a b + b^{5} \sqrt{\frac{1}{\left (a - b\right )^{3} \left (a + b\right )^{3}}}}{b^{2}} \right )}}{4} - \frac{a + b x}{2 a^{2} b - 2 b^{3} + x^{2} \left (2 a^{2} b - 2 b^{3}\right ) + x \left (4 a^{3} - 4 a b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(1/((a - b)**3*(a + b)**3))*log(x + (-a**4*b*sqrt(1/((a - b)**3*(a + b)**
3)) + 2*a**2*b**3*sqrt(1/((a - b)**3*(a + b)**3)) + a*b - b**5*sqrt(1/((a - b)**
3*(a + b)**3)))/b**2)/4 + b*sqrt(1/((a - b)**3*(a + b)**3))*log(x + (a**4*b*sqrt
(1/((a - b)**3*(a + b)**3)) - 2*a**2*b**3*sqrt(1/((a - b)**3*(a + b)**3)) + a*b
+ b**5*sqrt(1/((a - b)**3*(a + b)**3)))/b**2)/4 - (a + b*x)/(2*a**2*b - 2*b**3 +
x**2*(2*a**2*b - 2*b**3) + x*(4*a**3 - 4*a*b**2))

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GIAC/XCAS [A]  time = 0.209048, size = 101, normalized size = 1.4 \[ -\frac{b \arctan \left (\frac{b x + a}{\sqrt{-a^{2} + b^{2}}}\right )}{2 \,{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}}} - \frac{b x + a}{2 \,{\left (b x^{2} + 2 \, a x + b\right )}{\left (a^{2} - b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b*arctan((b*x + a)/sqrt(-a^2 + b^2))/((a^2 - b^2)*sqrt(-a^2 + b^2)) - 1/2*(
b*x + a)/((b*x^2 + 2*a*x + b)*(a^2 - b^2))