3.452 \(\int x^5 \sqrt{9-4 x^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{448} \left (9-4 x^2\right )^{7/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{27}{64} \left (9-4 x^2\right )^{3/2} \]

[Out]

(-27*(9 - 4*x^2)^(3/2))/64 + (9*(9 - 4*x^2)^(5/2))/160 - (9 - 4*x^2)^(7/2)/448

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Rubi [A]  time = 0.0205247, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{1}{448} \left (9-4 x^2\right )^{7/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{27}{64} \left (9-4 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[9 - 4*x^2],x]

[Out]

(-27*(9 - 4*x^2)^(3/2))/64 + (9*(9 - 4*x^2)^(5/2))/160 - (9 - 4*x^2)^(7/2)/448

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \sqrt{9-4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{9-4 x} x^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16} \sqrt{9-4 x}-\frac{9}{8} (9-4 x)^{3/2}+\frac{1}{16} (9-4 x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{27}{64} \left (9-4 x^2\right )^{3/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{1}{448} \left (9-4 x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0120286, size = 27, normalized size = 0.59 \[ -\frac{1}{280} \left (9-4 x^2\right )^{3/2} \left (10 x^4+18 x^2+27\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[9 - 4*x^2],x]

[Out]

-((9 - 4*x^2)^(3/2)*(27 + 18*x^2 + 10*x^4))/280

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Maple [A]  time = 0.002, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 10\,{x}^{4}+18\,{x}^{2}+27 \right ) }{280}\sqrt{-4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(-4*x^2+9)^(1/2),x)

[Out]

1/280*(-3+2*x)*(3+2*x)*(10*x^4+18*x^2+27)*(-4*x^2+9)^(1/2)

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Maxima [A]  time = 3.88775, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{28} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{4} - \frac{9}{140} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{2} - \frac{27}{280} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

-1/28*(-4*x^2 + 9)^(3/2)*x^4 - 9/140*(-4*x^2 + 9)^(3/2)*x^2 - 27/280*(-4*x^2 + 9)^(3/2)

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Fricas [A]  time = 1.49467, size = 77, normalized size = 1.67 \begin{align*} \frac{1}{280} \,{\left (40 \, x^{6} - 18 \, x^{4} - 54 \, x^{2} - 243\right )} \sqrt{-4 \, x^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/280*(40*x^6 - 18*x^4 - 54*x^2 - 243)*sqrt(-4*x^2 + 9)

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Sympy [A]  time = 1.92776, size = 61, normalized size = 1.33 \begin{align*} \frac{x^{6} \sqrt{9 - 4 x^{2}}}{7} - \frac{9 x^{4} \sqrt{9 - 4 x^{2}}}{140} - \frac{27 x^{2} \sqrt{9 - 4 x^{2}}}{140} - \frac{243 \sqrt{9 - 4 x^{2}}}{280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(-4*x**2+9)**(1/2),x)

[Out]

x**6*sqrt(9 - 4*x**2)/7 - 9*x**4*sqrt(9 - 4*x**2)/140 - 27*x**2*sqrt(9 - 4*x**2)/140 - 243*sqrt(9 - 4*x**2)/28
0

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Giac [A]  time = 2.33858, size = 70, normalized size = 1.52 \begin{align*} \frac{1}{448} \,{\left (4 \, x^{2} - 9\right )}^{3} \sqrt{-4 \, x^{2} + 9} + \frac{9}{160} \,{\left (4 \, x^{2} - 9\right )}^{2} \sqrt{-4 \, x^{2} + 9} - \frac{27}{64} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/448*(4*x^2 - 9)^3*sqrt(-4*x^2 + 9) + 9/160*(4*x^2 - 9)^2*sqrt(-4*x^2 + 9) - 27/64*(-4*x^2 + 9)^(3/2)