3.122 \(\int \frac{x^3}{(9+4 x^2)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0146203, antiderivative size = 31, normalized size of antiderivative = 1., 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} \[ \frac{1}{16} \sqrt{4 x^2+9}+\frac{9}{16 \sqrt{4 x^2+9}} \]

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 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]]

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])

Rubi steps

\begin{align*} \int \frac{x^3}{\left (9+4 x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(9+4 x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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 = 0.0076352, size = 22, normalized size = 0.71 \[ \frac{2 x^2+9}{8 \sqrt{4 x^2+9}} \]

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])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{2}+9}{8}{\frac{1}{\sqrt{4\,{x}^{2}+9}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41136, size = 35, normalized size = 1.13 \begin{align*} \frac{x^{2}}{4 \, \sqrt{4 \, x^{2} + 9}} + \frac{9}{8 \, \sqrt{4 \, x^{2} + 9}} \end{align*}

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)

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Fricas [A]  time = 1.82282, size = 45, normalized size = 1.45 \begin{align*} \frac{2 \, x^{2} + 9}{8 \, \sqrt{4 \, x^{2} + 9}} \end{align*}

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)

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Sympy [A]  time = 0.546797, size = 27, normalized size = 0.87 \begin{align*} \frac{x^{2}}{4 \sqrt{4 x^{2} + 9}} + \frac{9}{8 \sqrt{4 x^{2} + 9}} \end{align*}

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))

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Giac [A]  time = 1.04861, size = 31, normalized size = 1. \begin{align*} \frac{1}{16} \, \sqrt{4 \, x^{2} + 9} + \frac{9}{16 \, \sqrt{4 \, x^{2} + 9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(4*x^2+9)^(3/2),x, algorithm="giac")

[Out]

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