3.453 \(\int x^4 \sqrt {9-4 x^2} \, dx\)

Optimal. Leaf size=63 \[ -\frac {81}{256} \sqrt {9-4 x^2} x+\frac {1}{6} \sqrt {9-4 x^2} x^5-\frac {3}{32} \sqrt {9-4 x^2} x^3+\frac {729}{512} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

[Out]

729/512*arcsin(2/3*x)-81/256*x*(-4*x^2+9)^(1/2)-3/32*x^3*(-4*x^2+9)^(1/2)+1/6*x^5*(-4*x^2+9)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {279, 321, 216} \[ \frac {1}{6} \sqrt {9-4 x^2} x^5-\frac {3}{32} \sqrt {9-4 x^2} x^3-\frac {81}{256} \sqrt {9-4 x^2} x+\frac {729}{512} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[9 - 4*x^2],x]

[Out]

(-81*x*Sqrt[9 - 4*x^2])/256 - (3*x^3*Sqrt[9 - 4*x^2])/32 + (x^5*Sqrt[9 - 4*x^2])/6 + (729*ArcSin[(2*x)/3])/512

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 279

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 321

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^4 \sqrt {9-4 x^2} \, dx &=\frac {1}{6} x^5 \sqrt {9-4 x^2}+\frac {3}{2} \int \frac {x^4}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {3}{32} x^3 \sqrt {9-4 x^2}+\frac {1}{6} x^5 \sqrt {9-4 x^2}+\frac {81}{32} \int \frac {x^2}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {81}{256} x \sqrt {9-4 x^2}-\frac {3}{32} x^3 \sqrt {9-4 x^2}+\frac {1}{6} x^5 \sqrt {9-4 x^2}+\frac {729}{256} \int \frac {1}{\sqrt {9-4 x^2}} \, dx\\ &=-\frac {81}{256} x \sqrt {9-4 x^2}-\frac {3}{32} x^3 \sqrt {9-4 x^2}+\frac {1}{6} x^5 \sqrt {9-4 x^2}+\frac {729}{512} \sin ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.62 \[ \frac {1}{768} x \sqrt {9-4 x^2} \left (128 x^4-72 x^2-243\right )+\frac {729}{512} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[9 - 4*x^2],x]

[Out]

(x*Sqrt[9 - 4*x^2]*(-243 - 72*x^2 + 128*x^4))/768 + (729*ArcSin[(2*x)/3])/512

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fricas [A]  time = 0.90, size = 45, normalized size = 0.71 \[ \frac {1}{768} \, {\left (128 \, x^{5} - 72 \, x^{3} - 243 \, x\right )} \sqrt {-4 \, x^{2} + 9} - \frac {729}{256} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(128*x^5 - 72*x^3 - 243*x)*sqrt(-4*x^2 + 9) - 729/256*arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x)

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giac [A]  time = 1.08, size = 33, normalized size = 0.52 \[ \frac {1}{768} \, {\left (8 \, {\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt {-4 \, x^{2} + 9} x + \frac {729}{512} \, \arcsin \left (\frac {2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(8*(16*x^2 - 9)*x^2 - 243)*sqrt(-4*x^2 + 9)*x + 729/512*arcsin(2/3*x)

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maple [A]  time = 0.01, size = 46, normalized size = 0.73 \[ -\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}} x^{3}}{24}-\frac {9 \left (-4 x^{2}+9\right )^{\frac {3}{2}} x}{128}+\frac {81 \sqrt {-4 x^{2}+9}\, x}{256}+\frac {729 \arcsin \left (\frac {2 x}{3}\right )}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*x^3*(-4*x^2+9)^(3/2)-9/128*x*(-4*x^2+9)^(3/2)+81/256*x*(-4*x^2+9)^(1/2)+729/512*arcsin(2/3*x)

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maxima [A]  time = 3.00, size = 45, normalized size = 0.71 \[ -\frac {1}{24} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} x^{3} - \frac {9}{128} \, {\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} x + \frac {81}{256} \, \sqrt {-4 \, x^{2} + 9} x + \frac {729}{512} \, \arcsin \left (\frac {2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(-4*x^2 + 9)^(3/2)*x^3 - 9/128*(-4*x^2 + 9)^(3/2)*x + 81/256*sqrt(-4*x^2 + 9)*x + 729/512*arcsin(2/3*x)

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mupad [B]  time = 4.56, size = 32, normalized size = 0.51 \[ \frac {729\,\mathrm {asin}\left (\frac {2\,x}{3}\right )}{512}-\frac {\sqrt {\frac {9}{4}-x^2}\,\left (-\frac {2\,x^5}{3}+\frac {3\,x^3}{8}+\frac {81\,x}{64}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(9 - 4*x^2)^(1/2),x)

[Out]

(729*asin((2*x)/3))/512 - ((9/4 - x^2)^(1/2)*((81*x)/64 + (3*x^3)/8 - (2*x^5)/3))/2

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sympy [A]  time = 4.66, size = 167, normalized size = 2.65 \[ \begin {cases} \frac {2 i x^{7}}{3 \sqrt {4 x^{2} - 9}} - \frac {15 i x^{5}}{8 \sqrt {4 x^{2} - 9}} - \frac {27 i x^{3}}{64 \sqrt {4 x^{2} - 9}} + \frac {729 i x}{256 \sqrt {4 x^{2} - 9}} - \frac {729 i \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{512} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{9} > 1 \\- \frac {2 x^{7}}{3 \sqrt {9 - 4 x^{2}}} + \frac {15 x^{5}}{8 \sqrt {9 - 4 x^{2}}} + \frac {27 x^{3}}{64 \sqrt {9 - 4 x^{2}}} - \frac {729 x}{256 \sqrt {9 - 4 x^{2}}} + \frac {729 \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{512} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*x**7/(3*sqrt(4*x**2 - 9)) - 15*I*x**5/(8*sqrt(4*x**2 - 9)) - 27*I*x**3/(64*sqrt(4*x**2 - 9)) +
729*I*x/(256*sqrt(4*x**2 - 9)) - 729*I*acosh(2*x/3)/512, 4*Abs(x**2)/9 > 1), (-2*x**7/(3*sqrt(9 - 4*x**2)) + 1
5*x**5/(8*sqrt(9 - 4*x**2)) + 27*x**3/(64*sqrt(9 - 4*x**2)) - 729*x/(256*sqrt(9 - 4*x**2)) + 729*asin(2*x/3)/5
12, True))

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