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

Optimal. Leaf size=46 \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]

[Out]

(-81*Sqrt[-9 - 4*x^2])/64 - (3*(-9 - 4*x^2)^(3/2))/32 - (-9 - 4*x^2)^(5/2)/320

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Rubi [A]  time = 0.0194125, 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} \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[-9 - 4*x^2],x]

[Out]

(-81*Sqrt[-9 - 4*x^2])/64 - (3*(-9 - 4*x^2)^(3/2))/32 - (-9 - 4*x^2)^(5/2)/320

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 \frac{x^5}{\sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-9-4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16 \sqrt{-9-4 x}}+\frac{9}{8} \sqrt{-9-4 x}+\frac{1}{16} (-9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{81}{64} \sqrt{-9-4 x^2}-\frac{3}{32} \left (-9-4 x^2\right )^{3/2}-\frac{1}{320} \left (-9-4 x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0084343, size = 27, normalized size = 0.59 \[ -\frac{1}{40} \sqrt{-4 x^2-9} \left (2 x^4-6 x^2+27\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[-9 - 4*x^2],x]

[Out]

-(Sqrt[-9 - 4*x^2]*(27 - 6*x^2 + 2*x^4))/40

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Maple [A]  time = 0.003, size = 24, normalized size = 0.5 \begin{align*} -{\frac{2\,{x}^{4}-6\,{x}^{2}+27}{40}\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]

-1/40*(2*x^4-6*x^2+27)*(-4*x^2-9)^(1/2)

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Maxima [A]  time = 3.95294, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{20} \, \sqrt{-4 \, x^{2} - 9} x^{4} + \frac{3}{20} \, \sqrt{-4 \, x^{2} - 9} x^{2} - \frac{27}{40} \, \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="maxima")

[Out]

-1/20*sqrt(-4*x^2 - 9)*x^4 + 3/20*sqrt(-4*x^2 - 9)*x^2 - 27/40*sqrt(-4*x^2 - 9)

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Fricas [A]  time = 1.29725, size = 61, normalized size = 1.33 \begin{align*} -\frac{1}{40} \,{\left (2 \, x^{4} - 6 \, x^{2} + 27\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]

-1/40*(2*x^4 - 6*x^2 + 27)*sqrt(-4*x^2 - 9)

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Sympy [A]  time = 1.16989, size = 49, normalized size = 1.07 \begin{align*} - \frac{x^{4} \sqrt{- 4 x^{2} - 9}}{20} + \frac{3 x^{2} \sqrt{- 4 x^{2} - 9}}{20} - \frac{27 \sqrt{- 4 x^{2} - 9}}{40} \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**4*sqrt(-4*x**2 - 9)/20 + 3*x**2*sqrt(-4*x**2 - 9)/20 - 27*sqrt(-4*x**2 - 9)/40

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Giac [C]  time = 2.09956, size = 46, normalized size = 1. \begin{align*} -\frac{1}{320} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} + \frac{3}{32} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - \frac{81}{64} i \, \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="giac")

[Out]

-1/320*I*(4*x^2 + 9)^(5/2) + 3/32*I*(4*x^2 + 9)^(3/2) - 81/64*I*sqrt(4*x^2 + 9)