3.5.63 \(\int x^5 \sqrt {-9+4 x^2} \, dx\) [463]

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

[Out]

27/64*(4*x^2-9)^(3/2)+9/160*(4*x^2-9)^(5/2)+1/448*(4*x^2-9)^(7/2)

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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}{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} \end {gather*}

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

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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.59 \begin {gather*} \frac {1}{280} \left (-9+4 x^2\right )^{3/2} \left (27+18 x^2+10 x^4\right ) \end {gather*}

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 = 0.08, size = 41, normalized size = 0.89

method	result	size
trager	\(\left (\frac {1}{7} x^{6}-\frac {9}{140} x^{4}-\frac {27}{140} x^{2}-\frac {243}{280}\right ) \sqrt {4 x^{2}-9}\)	\(28\)
risch	\(\frac {\left (40 x^{6}-18 x^{4}-54 x^{2}-243\right ) \sqrt {4 x^{2}-9}}{280}\)	\(29\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (10 x^{4}+18 x^{2}+27\right ) \sqrt {4 x^{2}-9}}{280}\)	\(34\)
default	\(\frac {x^{4} \left (4 x^{2}-9\right )^{\frac {3}{2}}}{28}+\frac {9 x^{2} \left (4 x^{2}-9\right )^{\frac {3}{2}}}{140}+\frac {27 \left (4 x^{2}-9\right )^{\frac {3}{2}}}{280}\)	\(41\)
meijerg	\(\frac {2187 \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (1-\frac {4 x^{2}}{9}\right )^{\frac {3}{2}} \left (\frac {80}{27} x^{4}+\frac {16}{3} x^{2}+8\right )}{105}\right )}{256 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 40, normalized size = 0.87 \begin {gather*} \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 {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]

1/28*(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 = 1.62, size = 28, normalized size = 0.61 \begin {gather*} \frac {1}{280} \, {\left (40 \, x^{6} - 18 \, x^{4} - 54 \, x^{2} - 243\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]

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

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Sympy [A]
time = 0.33, size = 61, normalized size = 1.33 \begin {gather*} \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 {gather*}

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)/28
0

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Giac [A]
time = 0.58, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{448} \, {\left (4 \, x^{2} - 9\right )}^{\frac {7}{2}} + \frac {9}{160} \, {\left (4 \, x^{2} - 9\right )}^{\frac {5}{2}} + \frac {27}{64} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} \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]

1/448*(4*x^2 - 9)^(7/2) + 9/160*(4*x^2 - 9)^(5/2) + 27/64*(4*x^2 - 9)^(3/2)

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Mupad [B]
time = 5.31, size = 28, normalized size = 0.61 \begin {gather*} -\sqrt {4\,x^2-9}\,\left (-\frac {x^6}{7}+\frac {9\,x^4}{140}+\frac {27\,x^2}{140}+\frac {243}{280}\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)*((27*x^2)/140 + (9*x^4)/140 - x^6/7 + 243/280)

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