3.2.26 \(\int \frac {1}{\sqrt {\frac {-b^2+4 c}{4 c}+b x-c x^2}} \, dx\) [126]

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)

________________________________________________________________________________________

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 = {633, 222} \begin {gather*} -\frac {\text {ArcSin}\left (\frac {b-2 c x}{2 \sqrt {c}}\right )}{\sqrt {c}} \end {gather*}

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 222

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 633

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 {\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(23)=46\).
time = 0.22, size = 123, normalized size = 5.35 \begin {gather*} 2 \left (-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-c^2} x-\sqrt {c} \sqrt {4-\frac {b^2}{c}+4 b x-4 c x^2}}{b}\right )}{2 \sqrt {c}}-\frac {\log \left (2 c^2 x^2+c \left (-1-b x+\sqrt {-c} x \sqrt {4-\frac {b^2}{c}+4 b x-4 c x^2}\right )\right )}{4 \sqrt {-c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(17)=34\).
time = 0.60, size = 44, normalized size = 1.91

method	result	size
default	\(\frac {\arctan \left (\frac {2 \sqrt {c}\, \left (x -\frac {b}{2 c}\right )}{\sqrt {-4 c \,x^{2}+4 b x -\frac {b^{2}-4 c}{c}}}\right )}{\sqrt {c}}\)	\(44\)

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,method=_RETURNVERBOSE)

[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))

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 19, normalized size = 0.83 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, c x - b}{2 \, \sqrt {c}}\right )}{\sqrt {c}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (19) = 38\).
time = 1.46, size = 141, normalized size = 6.13 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
time = 1.55, size = 65, normalized size = 2.83 \begin {gather*} \frac {\sqrt {-c} \log \left (b \sqrt {-c} c - {\left (2 \, \sqrt {-c^{3}} x - \sqrt {-4 \, c^{3} x^{2} + 4 \, b c^{2} x - b^{2} c + 4 \, c^{2}}\right )} {\left | c \right |}\right )}{c} \end {gather*}

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]

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

________________________________________________________________________________________

Mupad [B]
time = 0.41, size = 46, normalized size = 2.00 \begin {gather*} \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}} \end {gather*}

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)

________________________________________________________________________________________