3.2.20 \(\int \frac {1}{\sqrt {2+5 x-3 x^2}} \, dx\) [120]

Optimal. Leaf size=17 \[ -\frac {\sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{\sqrt {3}} \]

[Out]

1/3*arcsin(-5/7+6/7*x)*3^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 17, 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 = {633, 222} \begin {gather*} -\frac {\text {ArcSin}\left (\frac {1}{7} (5-6 x)\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-(ArcSin[(5 - 6*x)/7]/Sqrt[3])

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 {2+5 x-3 x^2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{49}}} \, dx,x,5-6 x\right )}{7 \sqrt {3}}\\ &=-\frac {\sin ^{-1}\left (\frac {1}{7} (5-6 x)\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 30, normalized size = 1.76 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {6+15 x-9 x^2}}{1+3 x}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

(-2*ArcTan[Sqrt[6 + 15*x - 9*x^2]/(1 + 3*x)])/Sqrt[3]

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Maple [A]
time = 0.49, size = 12, normalized size = 0.71

method	result	size
default	\(\frac {\arcsin \left (-\frac {5}{7}+\frac {6 x}{7}\right ) \sqrt {3}}{3}\)	\(12\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (6 x \RootOf \left (\textit {\_Z}^{2}+3\right )+6 \sqrt {-3 x^{2}+5 x +2}-5 \RootOf \left (\textit {\_Z}^{2}+3\right )\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*arcsin(-5/7+6/7*x)*3^(1/2)

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Maxima [A]
time = 0.53, size = 11, normalized size = 0.65 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arcsin \left (-\frac {6}{7} \, x + \frac {5}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*arcsin(-6/7*x + 5/7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).
time = 1.27, size = 40, normalized size = 2.35 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 5\right )}}{6 \, {\left (3 \, x^{2} - 5 \, x - 2\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan(1/6*sqrt(3)*sqrt(-3*x^2 + 5*x + 2)*(6*x - 5)/(3*x^2 - 5*x - 2))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 3 x^{2} + 5 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2+5*x+2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**2 + 5*x + 2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (11) = 22\).
time = 0.90, size = 31, normalized size = 1.82 \begin {gather*} \frac {1}{12} \, \sqrt {-3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 5\right )} + \frac {49}{72} \, \sqrt {3} \arcsin \left (\frac {6}{7} \, x - \frac {5}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(-3*x^2 + 5*x + 2)*(6*x - 5) + 49/72*sqrt(3)*arcsin(6/7*x - 5/7)

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Mupad [B]
time = 0.17, size = 11, normalized size = 0.65 \begin {gather*} \frac {\sqrt {3}\,\mathrm {asin}\left (\frac {6\,x}{7}-\frac {5}{7}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x - 3*x^2 + 2)^(1/2),x)

[Out]

(3^(1/2)*asin((6*x)/7 - 5/7))/3

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