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

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

[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 = 0.0220785, 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} \[ -\frac{1}{448} \left (-4 x^2-9\right )^{7/2}-\frac{9}{160} \left (-4 x^2-9\right )^{5/2}-\frac{27}{64} \left (-4 x^2-9\right )^{3/2} \]

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 = 0.0123819, size = 27, normalized size = 0.59 \[ -\frac{1}{280} \left (-4 x^2-9\right )^{3/2} \left (10 x^4-18 x^2+27\right ) \]

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 = 0.003, size = 24, normalized size = 0.5 \begin{align*} -{\frac{10\,{x}^{4}-18\,{x}^{2}+27}{280} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}} \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*(10*x^4-18*x^2+27)*(-4*x^2-9)^(3/2)

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Maxima [A]  time = 3.08869, 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 = 1.27527, 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 = 1.91907, size = 68, normalized size = 1.48 \begin{align*} \frac{x^{6} \sqrt{- 4 x^{2} - 9}}{7} + \frac{9 x^{4} \sqrt{- 4 x^{2} - 9}}{140} - \frac{27 x^{2} \sqrt{- 4 x^{2} - 9}}{140} + \frac{243 \sqrt{- 4 x^{2} - 9}}{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(-4*x**2 - 9)/7 + 9*x**4*sqrt(-4*x**2 - 9)/140 - 27*x**2*sqrt(-4*x**2 - 9)/140 + 243*sqrt(-4*x**2 - 9
)/280

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Giac [C]  time = 2.18085, size = 46, normalized size = 1. \begin{align*} \frac{1}{448} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{7}{2}} - \frac{9}{160} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} + \frac{27}{64} i \,{\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*I*(4*x^2 + 9)^(7/2) - 9/160*I*(4*x^2 + 9)^(5/2) + 27/64*I*(4*x^2 + 9)^(3/2)