3.6.65 \(\int \frac {x^5}{\sqrt {-9-4 x^2}} \, dx\) [565]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} -\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} \end {gather*}

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 45

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])

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {x^5}{\sqrt {-9-4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {-9-4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.59 \begin {gather*} \frac {1}{40} \sqrt {-9-4 x^2} \left (-27+6 x^2-2 x^4\right ) \end {gather*}

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 = 0.04, size = 41, normalized size = 0.89

method	result	size
trager	\(\left (-\frac {1}{20} x^{4}+\frac {3}{20} x^{2}-\frac {27}{40}\right ) \sqrt {-4 x^{2}-9}\)	\(23\)
gosper	\(-\frac {\left (2 x^{4}-6 x^{2}+27\right ) \sqrt {-4 x^{2}-9}}{40}\)	\(24\)
risch	\(\frac {\left (4 x^{2}+9\right ) \left (2 x^{4}-6 x^{2}+27\right )}{40 \sqrt {-4 x^{2}-9}}\)	\(31\)
meijerg	\(-\frac {243 i \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {32}{27} x^{4}-\frac {32}{9} x^{2}+16\right ) \sqrt {1+\frac {4 x^{2}}{9}}}{15}\right )}{128 \sqrt {\pi }}\)	\(39\)
default	\(-\frac {x^{4} \sqrt {-4 x^{2}-9}}{20}+\frac {3 x^{2} \sqrt {-4 x^{2}-9}}{20}-\frac {27 \sqrt {-4 x^{2}-9}}{40}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-4*x^2-9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 40, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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 = 1.00, size = 23, normalized size = 0.50 \begin {gather*} -\frac {1}{40} \, {\left (2 \, x^{4} - 6 \, x^{2} + 27\right )} \sqrt {-4 \, x^{2} - 9} \end {gather*}

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 = 0.24, size = 49, normalized size = 1.07 \begin {gather*} - \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 {gather*}

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] Result contains complex when optimal does not.
time = 1.11, size = 34, normalized size = 0.74 \begin {gather*} -\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} \, \sqrt {-4 \, x^{2} - 9} \end {gather*}

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*sqrt(-4*x^2 - 9)

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Mupad [B]
time = 5.29, size = 23, normalized size = 0.50 \begin {gather*} -\sqrt {-4\,x^2-9}\,\left (\frac {x^4}{20}-\frac {3\,x^2}{20}+\frac {27}{40}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(- 4*x^2 - 9)^(1/2),x)

[Out]

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

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