3.5.76 \(\int x^3 \sqrt {-9-4 x^2} \, dx\) [476]

Optimal. Leaf size=31 \[ \frac {3}{16} \left (-9-4 x^2\right )^{3/2}+\frac {1}{80} \left (-9-4 x^2\right )^{5/2} \]

[Out]

3/16*(-4*x^2-9)^(3/2)+1/80*(-4*x^2-9)^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 31, 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}{80} \left (-4 x^2-9\right )^{5/2}+\frac {3}{16} \left (-4 x^2-9\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[-9 - 4*x^2],x]

[Out]

(3*(-9 - 4*x^2)^(3/2))/16 + (-9 - 4*x^2)^(5/2)/80

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 x^3 \sqrt {-9-4 x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \sqrt {-9-4 x} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {9}{4} \sqrt {-9-4 x}-\frac {1}{4} (-9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{16} \left (-9-4 x^2\right )^{3/2}+\frac {1}{80} \left (-9-4 x^2\right )^{5/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {1}{40} \left (-9-4 x^2\right )^{3/2} \left (3-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[-9 - 4*x^2],x]

[Out]

((-9 - 4*x^2)^(3/2)*(3 - 2*x^2))/40

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 27, normalized size = 0.87

method	result	size
gosper	\(-\frac {\left (2 x^{2}-3\right ) \left (-4 x^{2}-9\right )^{\frac {3}{2}}}{40}\)	\(19\)
trager	\(\left (\frac {1}{5} x^{4}+\frac {3}{20} x^{2}-\frac {27}{40}\right ) \sqrt {-4 x^{2}-9}\)	\(23\)
default	\(-\frac {x^{2} \left (-4 x^{2}-9\right )^{\frac {3}{2}}}{20}+\frac {3 \left (-4 x^{2}-9\right )^{\frac {3}{2}}}{40}\)	\(27\)
risch	\(-\frac {\left (8 x^{4}+6 x^{2}-27\right ) \left (4 x^{2}+9\right )}{40 \sqrt {-4 x^{2}-9}}\)	\(31\)
meijerg	\(-\frac {243 i \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+\frac {4 x^{2}}{9}\right )^{\frac {3}{2}} \left (-\frac {4 x^{2}}{3}+2\right )}{15}\right )}{64 \sqrt {\pi }}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-4*x^2-9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 26, normalized size = 0.84 \begin {gather*} -\frac {1}{20} \, {\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{2} + \frac {3}{40} \, {\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.12, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{40} \, {\left (8 \, 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^3*(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/40*(8*x^4 + 6*x^2 - 27)*sqrt(-4*x^2 - 9)

________________________________________________________________________________________

Sympy [A]
time = 0.15, size = 49, normalized size = 1.58 \begin {gather*} \frac {x^{4} \sqrt {- 4 x^{2} - 9}}{5} + \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**3*(-4*x**2-9)**(1/2),x)

[Out]

x**4*sqrt(-4*x**2 - 9)/5 + 3*x**2*sqrt(-4*x**2 - 9)/20 - 27*sqrt(-4*x**2 - 9)/40

________________________________________________________________________________________

Giac [C] Result contains complex when optimal does not.
time = 0.57, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{80} i \, {\left (4 \, x^{2} + 9\right )}^{\frac {5}{2}} - \frac {3}{16} i \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/80*I*(4*x^2 + 9)^(5/2) - 3/16*I*(4*x^2 + 9)^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 5.15, size = 22, normalized size = 0.71 \begin {gather*} \sqrt {-4\,x^2-9}\,\left (\frac {x^4}{5}+\frac {3\,x^2}{20}-\frac {27}{40}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(- 4*x^2 - 9)^(1/2),x)

[Out]

(- 4*x^2 - 9)^(1/2)*((3*x^2)/20 + x^4/5 - 27/40)

________________________________________________________________________________________