3.6.46 \(\int \frac {x^2}{\sqrt {9-4 x^2}} \, dx\) [546]

Optimal. Leaf size=27 \[ -\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{16} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

[Out]

9/16*arcsin(2/3*x)-1/8*x*(-4*x^2+9)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 222} \begin {gather*} \frac {9}{16} \text {ArcSin}\left (\frac {2 x}{3}\right )-\frac {1}{8} x \sqrt {9-4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[9 - 4*x^2],x]

[Out]

-1/8*(x*Sqrt[9 - 4*x^2]) + (9*ArcSin[(2*x)/3])/16

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {9-4 x^2}} \, dx &=-\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{8} \int \frac {1}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{16} \sin ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.48 \begin {gather*} -\frac {1}{8} x \sqrt {9-4 x^2}+\frac {9}{8} \tan ^{-1}\left (\frac {2 x}{-3+\sqrt {9-4 x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[9 - 4*x^2],x]

[Out]

-1/8*(x*Sqrt[9 - 4*x^2]) + (9*ArcTan[(2*x)/(-3 + Sqrt[9 - 4*x^2])])/8

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Maple [A]
time = 0.11, size = 20, normalized size = 0.74

method	result	size
default	\(\frac {9 \arcsin \left (\frac {2 x}{3}\right )}{16}-\frac {x \sqrt {-4 x^{2}+9}}{8}\)	\(20\)
risch	\(\frac {x \left (4 x^{2}-9\right )}{8 \sqrt {-4 x^{2}+9}}+\frac {9 \arcsin \left (\frac {2 x}{3}\right )}{16}\)	\(27\)
meijerg	\(\frac {9 i \left (\frac {2 i \sqrt {\pi }\, x \sqrt {1-\frac {4 x^{2}}{9}}}{3}-i \sqrt {\pi }\, \arcsin \left (\frac {2 x}{3}\right )\right )}{16 \sqrt {\pi }}\)	\(34\)
trager	\(-\frac {x \sqrt {-4 x^{2}+9}}{8}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+9}+2 x \right )}{16}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-4*x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

9/16*arcsin(2/3*x)-1/8*x*(-4*x^2+9)^(1/2)

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Maxima [A]
time = 0.52, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x + \frac {9}{16} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/8*sqrt(-4*x^2 + 9)*x + 9/16*arcsin(2/3*x)

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Fricas [A]
time = 1.11, size = 32, normalized size = 1.19 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x - \frac {9}{8} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

-1/8*sqrt(-4*x^2 + 9)*x - 9/8*arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x)

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Sympy [A]
time = 0.08, size = 22, normalized size = 0.81 \begin {gather*} - \frac {x \sqrt {9 - 4 x^{2}}}{8} + \frac {9 \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-4*x**2+9)**(1/2),x)

[Out]

-x*sqrt(9 - 4*x**2)/8 + 9*asin(2*x/3)/16

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Giac [A]
time = 0.85, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} + 9} x + \frac {9}{16} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

-1/8*sqrt(-4*x^2 + 9)*x + 9/16*arcsin(2/3*x)

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Mupad [B]
time = 0.02, size = 19, normalized size = 0.70 \begin {gather*} \frac {9\,\mathrm {asin}\left (\frac {2\,x}{3}\right )}{16}-\frac {x\,\sqrt {\frac {9}{4}-x^2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(9 - 4*x^2)^(1/2),x)

[Out]

(9*asin((2*x)/3))/16 - (x*(9/4 - x^2)^(1/2))/4

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