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

Optimal. Leaf size=31 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b^2-4 a b^3}+2 b^2 x}{b}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0291565, antiderivative size = 58, normalized size of antiderivative = 1.87, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {616, 31} \[ \frac{\log \left (\sqrt{b^2-4 a b^3}+2 b^2 x+b\right )}{b}-\frac{\log \left (-\sqrt{b^2-4 a b^3}-2 b^2 x+b\right )}{b} \]

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

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a b-\sqrt{b^2-4 a b^3} x-b^2 x^2} \, dx &=-\left (b \int \frac{1}{\frac{1}{2} \left (-b-\sqrt{b^2-4 a b^3}\right )-b^2 x} \, dx\right )+b \int \frac{1}{\frac{1}{2} \left (b-\sqrt{b^2-4 a b^3}\right )-b^2 x} \, dx\\ &=-\frac{\log \left (b-\sqrt{b^2-4 a b^3}-2 b^2 x\right )}{b}+\frac{\log \left (b+\sqrt{b^2-4 a b^3}+2 b^2 x\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0311477, size = 32, normalized size = 1.03 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{-b^2 (4 a b-1)}+2 b^2 x}{b}\right )}{b} \]

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 = 0.147, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{1}{b}{\it Artanh} \left ({\frac{2\,{b}^{2}x+\sqrt{-{b}^{2} \left ( 4\,ab-1 \right ) }}{b}} \right ) } \end{align*}

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 [B]  time = 1.22274, size = 82, normalized size = 2.65 \begin{align*} -\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}}} \end{align*}

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, 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 [B]  time = 2.3588, size = 128, normalized size = 4.13 \begin{align*} \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} \end{align*}

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, 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 [B]  time = 0.450968, size = 56, normalized size = 1.81 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.21375, size = 73, normalized size = 2.35 \begin{align*} -\frac{\log \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 |}} \end{align*}

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, algorithm="giac")

[Out]

-log(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)