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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.00 \[ -\frac {\sin ^{-1}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{\sqrt {c}} \]

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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fricas [B]  time = 1.00, size = 141, normalized size = 6.13 \[ \left [-\frac {\sqrt {-c} \log \left (4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - {\left (2 \, c x - b\right )} \sqrt {-c} \sqrt {-\frac {4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}} - 2 \, c\right )}{2 \, c}, -\frac {\arctan \left (\frac {{\left (2 \, c x - b\right )} \sqrt {c} \sqrt {-\frac {4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}{c}}}{4 \, c^{2} x^{2} - 4 \, b c x + b^{2} - 4 \, c}\right )}{\sqrt {c}}\right ] \]

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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giac [B]  time = 1.08, size = 53, normalized size = 2.30 \[ -\frac {\log \left ({\left | {\left (2 \, \sqrt {-c} x - \sqrt {-4 \, c x^{2} + 4 \, b x - \frac {b^{2} - 4 \, c}{c}}\right )} \sqrt {-c} + b \right |}\right )}{\sqrt {-c}} \]

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)

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maple [B]  time = 0.09, size = 44, normalized size = 1.91 \[ \frac {\arctan \left (\frac {2 \left (x -\frac {b}{2 c}\right ) \sqrt {c}}{\sqrt {-4 c \,x^{2}+4 b x -\frac {b^{2}-4 c}{c}}}\right )}{\sqrt {c}} \]

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 [A]  time = 3.08, size = 19, normalized size = 0.83 \[ -\frac {\arcsin \left (-\frac {2 \, c x - b}{2 \, \sqrt {c}}\right )}{\sqrt {c}} \]

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]

-arcsin(-1/2*(2*c*x - b)/sqrt(c))/sqrt(c)

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mupad [B]  time = 0.41, size = 46, normalized size = 2.00 \[ \frac {\ln \left (\frac {b-2\,c\,x}{\sqrt {-c}}+\sqrt {4\,b\,x+\frac {4\,c-b^2}{c}-4\,c\,x^2}\right )}{\sqrt {-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \int \frac {1}{\sqrt {- \frac {b^{2}}{c} + 4 b x - 4 c x^{2} + 4}}\, dx \]

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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