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

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Rubi [A]  time = 0.04, 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, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [B]  time = 1.29, size = 63, normalized size = 1.91 \[ \frac {\log \left (\frac {2 \, b^{2} x + b - \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right ) - \log \left (\frac {2 \, b^{2} x - b - \sqrt {-4 \, a b^{3} + b^{2}}}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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giac [A]  time = 0.53, size = 56, normalized size = 1.70 \[ -\frac {\log \left (\frac {{\left | 2 \, b^{2} x - \sqrt {-4 \, a b + 1} {\left | b \right |} - {\left | b \right |} \right |}}{{\left | 2 \, b^{2} x - \sqrt {-4 \, a b + 1} {\left | b \right |} + {\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]

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)

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maple [A]  time = 0.08, size = 31, normalized size = 0.94 \[ -\frac {2 \arctanh \left (\frac {-2 b^{2} x +\sqrt {-\left (4 a b -1\right ) b^{2}}}{b}\right )}{b} \]

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 = 1.39, size = 55, normalized size = 1.67 \[ -\frac {\log \left (\frac {2 \, b^{2} x - b - \sqrt {-4 \, a b^{3} + b^{2}}}{2 \, b^{2} x + b - \sqrt {-4 \, a b^{3} + b^{2}}}\right )}{b} \]

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, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.26, size = 38, normalized size = 1.15 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {b^2-4\,a\,b^3}}{\sqrt {b^2}}-\frac {2\,b^2\,x}{\sqrt {b^2}}\right )}{\sqrt {b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*atanh((b^2 - 4*a*b^3)^(1/2)/(b^2)^(1/2) - (2*b^2*x)/(b^2)^(1/2)))/(b^2)^(1/2)

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sympy [B]  time = 0.29, size = 56, normalized size = 1.70 \[ - \frac {\log {\left (x - \frac {1}{2 b} - \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )} - \log {\left (x + \frac {1}{2 b} - \frac {\sqrt {- 4 a b^{3} + b^{2}}}{2 b^{2}} \right )}}{b} \]

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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