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

Optimal. Leaf size=69 \[ -\frac{a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}-\frac{b \tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]

[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 = 0.0699646, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{a-b x}{2 \left (a^2+b^2\right ) \left (2 a x-b x^2+b\right )}-\frac{b \tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}} \]

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 = 9.55234, size = 60, normalized size = 0.87 \[ - \frac{b \operatorname{atanh}{\left (\frac{a - b x}{\sqrt{a^{2} + b^{2}}} \right )}}{2 \left (a^{2} + b^{2}\right )^{\frac{3}{2}}} - \frac{2 a - 2 b x}{4 \left (a^{2} + b^{2}\right ) \left (2 a x - b x^{2} + b\right )} \]

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 = 0.110337, size = 78, normalized size = 1.13 \[ \frac{\frac{b x-a}{2 a x-b x^2+b}-\frac{b \tan ^{-1}\left (\frac{b x-a}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}}{2 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 84, normalized size = 1.2 \[{\frac{2\,bx-2\,a}{ \left ( -4\,{a}^{2}-4\,{b}^{2} \right ) \left ( b{x}^{2}-2\,ax-b \right ) }}-2\,{\frac{b}{ \left ( -4\,{a}^{2}-4\,{b}^{2} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,bx-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \]

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)*ar
ctanh(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 = 0.237248, size = 216, normalized size = 3.13 \[ -\frac{{\left (b^{2} x^{2} - 2 \, a b x - b^{2}\right )} \log \left (-\frac{2 \, a^{3} + 2 \, a b^{2} - 2 \,{\left (a^{2} b + b^{3}\right )} x -{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} + b^{2}\right )} \sqrt{a^{2} + b^{2}}}{b x^{2} - 2 \, a x - b}\right ) - 2 \, \sqrt{a^{2} + b^{2}}{\left (b x - a\right )}}{4 \,{\left (a^{2} b + b^{3} -{\left (a^{2} b + b^{3}\right )} x^{2} + 2 \,{\left (a^{3} + a b^{2}\right )} x\right )} \sqrt{a^{2} + b^{2}}} \]

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))

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Sympy [A]  time = 2.67383, size = 218, normalized size = 3.16 \[ - \frac{b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} \log{\left (x + \frac{- a^{4} b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - a b - b^{5} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}}}{b^{2}} \right )}}{4} + \frac{b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} \log{\left (x + \frac{a^{4} b \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} + 2 a^{2} b^{3} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}} - a b + b^{5} \sqrt{\frac{1}{\left (a^{2} + b^{2}\right )^{3}}}}{b^{2}} \right )}}{4} - \frac{- a + b x}{- 2 a^{2} b - 2 b^{3} + x^{2} \left (2 a^{2} b + 2 b^{3}\right ) + x \left (- 4 a^{3} - 4 a b^{2}\right )} \]

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((a**2 + b**2)**(-3))*log(x + (-a**4*b*sqrt((a**2 + b**2)**(-3)) - 2*a**2
*b**3*sqrt((a**2 + b**2)**(-3)) - a*b - b**5*sqrt((a**2 + b**2)**(-3)))/b**2)/4
+ b*sqrt((a**2 + b**2)**(-3))*log(x + (a**4*b*sqrt((a**2 + b**2)**(-3)) + 2*a**2
*b**3*sqrt((a**2 + b**2)**(-3)) - a*b + b**5*sqrt((a**2 + b**2)**(-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 = 0.21435, size = 122, normalized size = 1.77 \[ -\frac{b{\rm ln}\left (\frac{{\left | 2 \, b x - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{4 \,{\left (a^{2} + b^{2}\right )}^{\frac{3}{2}}} - \frac{b x - a}{2 \,{\left (b x^{2} - 2 \, a x - b\right )}{\left (a^{2} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*b*ln(abs(2*b*x - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*x - 2*a + 2*sqrt(a^2 + b^
2)))/(a^2 + b^2)^(3/2) - 1/2*(b*x - a)/((b*x^2 - 2*a*x - b)*(a^2 + b^2))