3.2.22 \(\int \frac {x^3}{(9+4 x^2)^{3/2}} \, dx\) [122]

Optimal. Leaf size=31 \[ \frac {9}{16 \sqrt {9+4 x^2}}+\frac {1}{16} \sqrt {9+4 x^2} \]

[Out]

9/16/(4*x^2+9)^(1/2)+1/16*(4*x^2+9)^(1/2)

________________________________________________________________________________________

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}{16} \sqrt {4 x^2+9}+\frac {9}{16 \sqrt {4 x^2+9}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(9 + 4*x^2)^(3/2),x]

[Out]

9/(16*Sqrt[9 + 4*x^2]) + Sqrt[9 + 4*x^2]/16

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {9+2 x^2}{8 \sqrt {9+4 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(9 + 4*x^2)^(3/2),x]

[Out]

(9 + 2*x^2)/(8*Sqrt[9 + 4*x^2])

________________________________________________________________________________________

Mathics [A]
time = 1.96, size = 18, normalized size = 0.58 \begin {gather*} \frac {9+2 x^2}{8 \sqrt {9+4 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^3/(9 + 4*x^2)^(3/2),x]')

[Out]

(9 + 2 x ^ 2) / (8 Sqrt[9 + 4 x ^ 2])

________________________________________________________________________________________

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

method	result	size
gosper	\(\frac {2 x^{2}+9}{8 \sqrt {4 x^{2}+9}}\)	\(19\)
trager	\(\frac {2 x^{2}+9}{8 \sqrt {4 x^{2}+9}}\)	\(19\)
risch	\(\frac {2 x^{2}+9}{8 \sqrt {4 x^{2}+9}}\)	\(19\)
default	\(\frac {x^{2}}{4 \sqrt {4 x^{2}+9}}+\frac {9}{8 \sqrt {4 x^{2}+9}}\)	\(27\)
meijerg	\(\frac {-\frac {3 \sqrt {\pi }}{8}+\frac {3 \sqrt {\pi }\, \left (\frac {16 x^{2}}{9}+8\right )}{64 \sqrt {1+\frac {4 x^{2}}{9}}}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2/(4*x^2+9)^(1/2)+9/8/(4*x^2+9)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.35, size = 26, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{4 \, \sqrt {4 \, x^{2} + 9}} + \frac {9}{8 \, \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)^(3/2),x, algorithm="maxima")

[Out]

1/4*x^2/sqrt(4*x^2 + 9) + 9/8/sqrt(4*x^2 + 9)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 18, normalized size = 0.58 \begin {gather*} \frac {2 \, x^{2} + 9}{8 \, \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)^(3/2),x, algorithm="fricas")

[Out]

1/8*(2*x^2 + 9)/sqrt(4*x^2 + 9)

________________________________________________________________________________________

Sympy [A]
time = 0.21, size = 27, normalized size = 0.87 \begin {gather*} \frac {x^{2}}{4 \sqrt {4 x^{2} + 9}} + \frac {9}{8 \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)**(3/2),x)

[Out]

x**2/(4*sqrt(4*x**2 + 9)) + 9/(8*sqrt(4*x**2 + 9))

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 34, normalized size = 1.10 \begin {gather*} \frac {\frac {\sqrt {4 x^{2}+9}}{4}+\frac {9}{4 \sqrt {4 x^{2}+9}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(4*x^2+9)^(3/2),x)

[Out]

1/16*sqrt(4*x^2 + 9) + 9/16/sqrt(4*x^2 + 9)

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 24, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x^2+\frac {9}{4}}\,\left (2\,x^2+9\right )}{4\,\left (4\,x^2+9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(4*x^2 + 9)^(3/2),x)

[Out]

((x^2 + 9/4)^(1/2)*(2*x^2 + 9))/(4*(4*x^2 + 9))

________________________________________________________________________________________