3.5.55 \(\int x^2 \sqrt {9-4 x^2} \, dx\) [455]

Optimal. Leaf size=45 \[ -\frac {9}{32} x \sqrt {9-4 x^2}+\frac {1}{4} x^3 \sqrt {9-4 x^2}+\frac {81}{64} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

[Out]

81/64*arcsin(2/3*x)-9/32*x*(-4*x^2+9)^(1/2)+1/4*x^3*(-4*x^2+9)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {285, 327, 222} \begin {gather*} \frac {81}{64} \text {ArcSin}\left (\frac {2 x}{3}\right )-\frac {9}{32} \sqrt {9-4 x^2} x+\frac {1}{4} \sqrt {9-4 x^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[9 - 4*x^2],x]

[Out]

(-9*x*Sqrt[9 - 4*x^2])/32 + (x^3*Sqrt[9 - 4*x^2])/4 + (81*ArcSin[(2*x)/3])/64

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 285

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

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 x^2 \sqrt {9-4 x^2} \, dx &=\frac {1}{4} x^3 \sqrt {9-4 x^2}+\frac {9}{4} \int \frac {x^2}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {9}{32} x \sqrt {9-4 x^2}+\frac {1}{4} x^3 \sqrt {9-4 x^2}+\frac {81}{32} \int \frac {1}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {9}{32} x \sqrt {9-4 x^2}+\frac {1}{4} x^3 \sqrt {9-4 x^2}+\frac {81}{64} \sin ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 47, normalized size = 1.04 \begin {gather*} \frac {1}{32} x \sqrt {9-4 x^2} \left (-9+8 x^2\right )+\frac {81}{32} \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]

(x*Sqrt[9 - 4*x^2]*(-9 + 8*x^2))/32 + (81*ArcTan[(2*x)/(-3 + Sqrt[9 - 4*x^2])])/32

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Maple [A]
time = 0.13, size = 32, normalized size = 0.71

method	result	size
default	\(-\frac {x \left (-4 x^{2}+9\right )^{\frac {3}{2}}}{16}+\frac {81 \arcsin \left (\frac {2 x}{3}\right )}{64}+\frac {9 x \sqrt {-4 x^{2}+9}}{32}\)	\(32\)
risch	\(-\frac {x \left (8 x^{2}-9\right ) \left (4 x^{2}-9\right )}{32 \sqrt {-4 x^{2}+9}}+\frac {81 \arcsin \left (\frac {2 x}{3}\right )}{64}\)	\(34\)
meijerg	\(-\frac {81 i \left (-\frac {i \sqrt {\pi }\, x \left (-\frac {8 x^{2}}{3}+3\right ) \sqrt {1-\frac {4 x^{2}}{9}}}{9}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 x}{3}\right )}{2}\right )}{32 \sqrt {\pi }}\)	\(41\)
trager	\(\frac {x \left (8 x^{2}-9\right ) \sqrt {-4 x^{2}+9}}{32}+\frac {81 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+9}+2 x \right )}{64}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x*(-4*x^2+9)^(3/2)+81/64*arcsin(2/3*x)+9/32*x*(-4*x^2+9)^(1/2)

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Maxima [A]
time = 0.50, size = 31, normalized size = 0.69 \begin {gather*} -\frac {1}{16} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} x + \frac {9}{32} \, \sqrt {-4 \, x^{2} + 9} x + \frac {81}{64} \, \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]

-1/16*(-4*x^2 + 9)^(3/2)*x + 9/32*sqrt(-4*x^2 + 9)*x + 81/64*arcsin(2/3*x)

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Fricas [A]
time = 0.86, size = 40, normalized size = 0.89 \begin {gather*} \frac {1}{32} \, {\left (8 \, x^{3} - 9 \, x\right )} \sqrt {-4 \, x^{2} + 9} - \frac {81}{32} \, \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]

1/32*(8*x^3 - 9*x)*sqrt(-4*x^2 + 9) - 81/32*arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x)

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Sympy [C] Result contains complex when optimal does not.
time = 1.73, size = 122, normalized size = 2.71 \begin {gather*} \begin {cases} \frac {i x^{5}}{\sqrt {4 x^{2} - 9}} - \frac {27 i x^{3}}{8 \sqrt {4 x^{2} - 9}} + \frac {81 i x}{32 \sqrt {4 x^{2} - 9}} - \frac {81 i \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{64} & \text {for}\: \left |{x^{2}}\right | > \frac {9}{4} \\- \frac {x^{5}}{\sqrt {9 - 4 x^{2}}} + \frac {27 x^{3}}{8 \sqrt {9 - 4 x^{2}}} - \frac {81 x}{32 \sqrt {9 - 4 x^{2}}} + \frac {81 \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{64} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*x**5/sqrt(4*x**2 - 9) - 27*I*x**3/(8*sqrt(4*x**2 - 9)) + 81*I*x/(32*sqrt(4*x**2 - 9)) - 81*I*acos
h(2*x/3)/64, Abs(x**2) > 9/4), (-x**5/sqrt(9 - 4*x**2) + 27*x**3/(8*sqrt(9 - 4*x**2)) - 81*x/(32*sqrt(9 - 4*x*
*2)) + 81*asin(2*x/3)/64, True))

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Giac [A]
time = 0.64, size = 26, normalized size = 0.58 \begin {gather*} \frac {1}{32} \, {\left (8 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9} x + \frac {81}{64} \, \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]

1/32*(8*x^2 - 9)*sqrt(-4*x^2 + 9)*x + 81/64*arcsin(2/3*x)

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Mupad [B]
time = 0.03, size = 27, normalized size = 0.60 \begin {gather*} \frac {81\,\mathrm {asin}\left (\frac {2\,x}{3}\right )}{64}-\frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {9\,x}{8}-x^3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(9 - 4*x^2)^(1/2),x)

[Out]

(81*asin((2*x)/3))/64 - ((9/4 - x^2)^(1/2)*((9*x)/8 - x^3))/2

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