3.6.32 \(\int \frac {x^5}{\sqrt {9+4 x^2}} \, dx\) [532]

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 = 0.05, size = 41, normalized size = 0.89

method	result	size
trager	\(\sqrt {4 x^{2}+9}\, \left (\frac {1}{20} x^{4}-\frac {3}{20} x^{2}+\frac {27}{40}\right )\)	\(23\)
gosper	\(\frac {\sqrt {4 x^{2}+9}\, \left (2 x^{4}-6 x^{2}+27\right )}{40}\)	\(24\)
risch	\(\frac {\sqrt {4 x^{2}+9}\, \left (2 x^{4}-6 x^{2}+27\right )}{40}\)	\(24\)
meijerg	\(\frac {-\frac {81 \sqrt {\pi }}{40}+\frac {81 \sqrt {\pi }\, \left (\frac {32}{27} x^{4}-\frac {32}{9} x^{2}+16\right ) \sqrt {1+\frac {4 x^{2}}{9}}}{640}}{\sqrt {\pi }}\)	\(38\)
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 = 0.50, 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 = 1.05, 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 = 0.23, size = 44, normalized size = 0.96 \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 [A]
time = 0.57, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{320} \, {\left (4 \, x^{2} + 9\right )}^{\frac {5}{2}} - \frac {3}{32} \, {\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*(4*x^2 + 9)^(5/2) - 3/32*(4*x^2 + 9)^(3/2) + 81/64*sqrt(4*x^2 + 9)

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Mupad [B]
time = 0.02, size = 21, normalized size = 0.46 \begin {gather*} \frac {\sqrt {x^2+\frac {9}{4}}\,\left (\frac {x^4}{5}-\frac {3\,x^2}{5}+\frac {27}{10}\right )}{2} \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)*(x^4/5 - (3*x^2)/5 + 27/10))/2

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