3.126 \(\int \frac{1}{\sqrt{\frac{-b^2+4 c}{4 c}+b x-c x^2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

[Out]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/Sqrt[c])

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Rubi [A]  time = 0.0133389, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {619, 216} \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(-b^2 + 4*c)/(4*c) + b*x - c*x^2],x]

[Out]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/Sqrt[c])

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\frac{-b^2+4 c}{4 c}+b x-c x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 c}}} \, dx,x,b-2 c x\right )}{2 c}\\ &=-\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.0272743, size = 23, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(-b^2 + 4*c)/(4*c) + b*x - c*x^2],x]

[Out]

-(ArcSin[(b - 2*c*x)/(2*Sqrt[c])]/Sqrt[c])

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Maple [B]  time = 0.247, size = 44, normalized size = 1.9 \begin{align*}{\arctan \left ( 2\,{\sqrt{c} \left ( x-1/2\,{\frac{b}{c}} \right ){\frac{1}{\sqrt{-4\,c{x}^{2}+4\,bx-{\frac{{b}^{2}-4\,c}{c}}}}}} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^(1/2)*arctan(2*c^(1/2)*(x-1/2*b/c)/(-4*c*x^2+4*b*x-(b^2-4*c)/c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.97964, size = 323, normalized size = 14.04 \begin{align*} \left [-\frac{\sqrt{-c} \log \left (4 \, c^{2} x^{2} - 4 \, b c x + b^{2} -{\left (2 \, c x - b\right )} \sqrt{-c} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}} - 2 \, c\right )}{2 \, c}, -\frac{\arctan \left (\frac{{\left (2 \, c x - b\right )} \sqrt{c} \sqrt{-\frac{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}}}{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}\right )}{\sqrt{c}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((-b^2+4*c)/c+4*b*x-4*c*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \int \frac{1}{\sqrt{- \frac{b^{2}}{c} + 4 b x - 4 c x^{2} + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((-b**2+4*c)/c+4*b*x-4*c*x**2)**(1/2),x)

[Out]

2*Integral(1/sqrt(-b**2/c + 4*b*x - 4*c*x**2 + 4), x)

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Giac [B]  time = 1.43435, size = 72, normalized size = 3.13 \begin{align*} -\frac{\log \left ({\left |{\left (2 \, \sqrt{-c} x - \sqrt{-4 \, c x^{2} + 4 \, b x - \frac{b^{2} - 4 \, c}{c}}\right )} \sqrt{-c} + b \right |}\right )}{\sqrt{-c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((-b^2+4*c)/c+4*b*x-4*c*x^2)^(1/2),x, algorithm="giac")

[Out]

-log(abs((2*sqrt(-c)*x - sqrt(-4*c*x^2 + 4*b*x - (b^2 - 4*c)/c))*sqrt(-c) + b))/sqrt(-c)