3.5.41 \(\int x^5 \sqrt {9+4 x^2} \, dx\) [441]

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 = 0.06, size = 41, normalized size = 0.89

method	result	size
gosper	\(\frac {\left (4 x^{2}+9\right )^{\frac {3}{2}} \left (10 x^{4}-18 x^{2}+27\right )}{280}\)	\(24\)
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	\(\frac {\left (40 x^{6}+18 x^{4}-54 x^{2}+243\right ) \sqrt {4 x^{2}+9}}{280}\)	\(29\)
meijerg	\(-\frac {2187 \left (\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (1+\frac {4 x^{2}}{9}\right )^{\frac {3}{2}} \left (\frac {80}{27} x^{4}-\frac {16}{3} x^{2}+8\right )}{105}\right )}{256 \sqrt {\pi }}\)	\(38\)
default	\(\frac {x^{4} \left (4 x^{2}+9\right )^{\frac {3}{2}}}{28}-\frac {9 x^{2} \left (4 x^{2}+9\right )^{\frac {3}{2}}}{140}+\frac {27 \left (4 x^{2}+9\right )^{\frac {3}{2}}}{280}\)	\(41\)

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 = 0.49, 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 = 1.18, 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 = 0.61, 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 = 4.55, size = 25, normalized size = 0.54 \begin {gather*} \sqrt {x^2+\frac {9}{4}}\,\left (\frac {2\,x^6}{7}+\frac {9\,x^4}{70}-\frac {27\,x^2}{70}+\frac {243}{140}\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]

(x^2 + 9/4)^(1/2)*((9*x^4)/70 - (27*x^2)/70 + (2*x^6)/7 + 243/140)

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