3.99 \(\int \frac{1}{a b+\sqrt{b^2-4 a b^3} x-b^2 x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 \tanh ^{-1}\left (\frac{2 b^2 x-\sqrt{b^2-4 a b^3}}{b}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0736025, antiderivative size = 58, normalized size of antiderivative = 1.76, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\log \left (-\sqrt{b^2-4 a b^3}+2 b^2 x+b\right )}{b}-\frac{\log \left (\sqrt{b^2-4 a b^3}-2 b^2 x+b\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[(a*b + Sqrt[b^2 - 4*a*b^3]*x - b^2*x^2)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.47441, size = 49, normalized size = 1.48 \[ - \frac{\log{\left (- 2 b^{2} x + b + \sqrt{b^{2} \left (- 4 a b + 1\right )} \right )}}{b} + \frac{\log{\left (2 b^{2} x + b - \sqrt{b^{2} \left (- 4 a b + 1\right )} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*b-b**2*x**2+x*(-4*a*b**3+b**2)**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.0563528, size = 34, normalized size = 1.03 \[ \frac{2 \tanh ^{-1}\left (\frac{2 b^2 x-\sqrt{-b^2 (4 a b-1)}}{b}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*b + Sqrt[b^2 - 4*a*b^3]*x - b^2*x^2)^(-1),x]

[Out]

(2*ArcTanh[(-Sqrt[-(b^2*(-1 + 4*a*b))] + 2*b^2*x)/b])/b

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Maple [A]  time = 0.013, size = 31, normalized size = 0.9 \[ -2\,{\frac{1}{b}{\it Artanh} \left ({\frac{-2\,{b}^{2}x+\sqrt{-{b}^{2} \left ( 4\,ab-1 \right ) }}{b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*b-b^2*x^2+x*(-4*a*b^3+b^2)^(1/2)),x)

[Out]

-2/b*arctanh((-2*b^2*x+(-b^2*(4*a*b-1))^(1/2))/b)

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Maxima [A]  time = 0.703031, size = 88, normalized size = 2.67 \[ -\frac{\log \left (\frac{2 \, b^{2} x - \sqrt{-4 \, a b^{3} + b^{2}} - \sqrt{b^{2}}}{2 \, b^{2} x - \sqrt{-4 \, a b^{3} + b^{2}} + \sqrt{b^{2}}}\right )}{\sqrt{b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b^2*x^2 - a*b - sqrt(-4*a*b^3 + b^2)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235365, size = 85, normalized size = 2.58 \[ \frac{\log \left (\frac{2 \, b^{2} x + b - \sqrt{-4 \, a b^{3} + b^{2}}}{b}\right ) - \log \left (\frac{2 \, b^{2} x - b - \sqrt{-4 \, a b^{3} + b^{2}}}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b^2*x^2 - a*b - sqrt(-4*a*b^3 + b^2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.806361, size = 56, normalized size = 1.7 \[ - \frac{\log{\left (x - \frac{1}{2 b} - \frac{\sqrt{- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )} - \log{\left (x + \frac{1}{2 b} - \frac{\sqrt{- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*b-b**2*x**2+x*(-4*a*b**3+b**2)**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217982, size = 76, normalized size = 2.3 \[ -\frac{{\rm ln}\left (\frac{{\left | 2 \, b^{2} x - \sqrt{-4 \, a b + 1}{\left | b \right |} -{\left | b \right |} \right |}}{{\left | 2 \, b^{2} x - \sqrt{-4 \, a b + 1}{\left | b \right |} +{\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(b^2*x^2 - a*b - sqrt(-4*a*b^3 + b^2)*x),x, algorithm="giac")

[Out]

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