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

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]

27/64*(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 = 0.02, 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 = {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 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])

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]]

Rubi steps

\begin {align*} \int x^5 \sqrt {-9+4 x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int 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} (-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 = 0.01, 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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fricas [A]  time = 0.92, size = 28, normalized size = 0.61 \[ \frac {1}{280} \, {\left (40 \, x^{6} - 18 \, x^{4} - 54 \, x^{2} - 243\right )} \sqrt {4 \, x^{2} - 9} \]

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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giac [A]  time = 0.97, size = 34, normalized size = 0.74 \[ \frac {1}{448} \, {\left (4 \, x^{2} - 9\right )}^{\frac {7}{2}} + \frac {9}{160} \, {\left (4 \, x^{2} - 9\right )}^{\frac {5}{2}} + \frac {27}{64} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} \]

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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maple [A]  time = 0.01, size = 34, normalized size = 0.74 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (10 x^{4}+18 x^{2}+27\right ) \sqrt {4 x^{2}-9}}{280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/280*(2*x-3)*(2*x+3)*(10*x^4+18*x^2+27)*(4*x^2-9)^(1/2)

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maxima [A]  time = 2.94, size = 40, normalized size = 0.87 \[ \frac {1}{28} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{4} + \frac {9}{140} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{2} + \frac {27}{280} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} \]

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

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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sympy [A]  time = 1.92, size = 61, normalized size = 1.33 \[ \frac {x^{6} \sqrt {4 x^{2} - 9}}{7} - \frac {9 x^{4} \sqrt {4 x^{2} - 9}}{140} - \frac {27 x^{2} \sqrt {4 x^{2} - 9}}{140} - \frac {243 \sqrt {4 x^{2} - 9}}{280} \]

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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