3.235 \(\int \frac {1}{\sqrt {4+3 x-2 x^2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {619, 216} \[ -\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

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 {4+3 x-2 x^2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{41}}} \, dx,x,3-4 x\right )}{\sqrt {82}}\\ &=-\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ -\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.10, size = 32, normalized size = 1.68 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-2 x^2+3 x+4}-2}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

Sqrt[2]*ArcTan[(Sqrt[2]*x)/(-2 + Sqrt[4 + 3*x - 2*x^2])]

________________________________________________________________________________________

fricas [B]  time = 0.86, size = 33, normalized size = 1.74 \[ -\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x^{2} + 3 \, x + 4} - 2 \, \sqrt {2}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(2)*arctan(1/2*(sqrt(2)*sqrt(-2*x^2 + 3*x + 4) - 2*sqrt(2))/x)

________________________________________________________________________________________

giac [A]  time = 0.66, size = 16, normalized size = 0.84 \[ \frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {1}{41} \, \sqrt {41} {\left (4 \, x - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*arcsin(1/41*sqrt(41)*(4*x - 3))

________________________________________________________________________________________

maple [A]  time = 0.36, size = 15, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {2}\, \arcsin \left (\frac {4 \sqrt {41}\, \left (x -\frac {3}{4}\right )}{41}\right )}{2}\)	\(15\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +3 \RootOf \left (\textit {\_Z}^{2}+2\right )+4 \sqrt {-2 x^{2}+3 x +4}\right )}{2}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^2+3*x+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*arcsin(4/41*41^(1/2)*(x-3/4))

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 16, normalized size = 0.84 \[ -\frac {1}{2} \, \sqrt {2} \arcsin \left (-\frac {1}{41} \, \sqrt {41} {\left (4 \, x - 3\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*arcsin(-1/41*sqrt(41)*(4*x - 3))

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 16, normalized size = 0.84 \[ \frac {\sqrt {2}\,\mathrm {asin}\left (\frac {\sqrt {41}\,\left (4\,x-3\right )}{41}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x - 2*x^2 + 4)^(1/2),x)

[Out]

(2^(1/2)*asin((41^(1/2)*(4*x - 3))/41))/2

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 2 x^{2} + 3 x + 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**2+3*x+4)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**2 + 3*x + 4), x)

________________________________________________________________________________________