3.533 \(\int \frac{x^4}{\sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{16} \sqrt{4 x^2+9} x^3-\frac{27}{128} \sqrt{4 x^2+9} x+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(-27*x*Sqrt[9 + 4*x^2])/128 + (x^3*Sqrt[9 + 4*x^2])/16 + (243*ArcSinh[(2*x)/3])/256

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Rubi [A]  time = 0.0093216, antiderivative size = 45, 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 = {321, 215} \[ \frac{1}{16} \sqrt{4 x^2+9} x^3-\frac{27}{128} \sqrt{4 x^2+9} x+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[9 + 4*x^2],x]

[Out]

(-27*x*Sqrt[9 + 4*x^2])/128 + (x^3*Sqrt[9 + 4*x^2])/16 + (243*ArcSinh[(2*x)/3])/256

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^4}{\sqrt{9+4 x^2}} \, dx &=\frac{1}{16} x^3 \sqrt{9+4 x^2}-\frac{27}{16} \int \frac{x^2}{\sqrt{9+4 x^2}} \, dx\\ &=-\frac{27}{128} x \sqrt{9+4 x^2}+\frac{1}{16} x^3 \sqrt{9+4 x^2}+\frac{243}{128} \int \frac{1}{\sqrt{9+4 x^2}} \, dx\\ &=-\frac{27}{128} x \sqrt{9+4 x^2}+\frac{1}{16} x^3 \sqrt{9+4 x^2}+\frac{243}{256} \sinh ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0106941, size = 34, normalized size = 0.76 \[ \frac{1}{256} \left (2 x \sqrt{4 x^2+9} \left (8 x^2-27\right )+243 \sinh ^{-1}\left (\frac{2 x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[9 + 4*x^2],x]

[Out]

(2*x*Sqrt[9 + 4*x^2]*(-27 + 8*x^2) + 243*ArcSinh[(2*x)/3])/256

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \begin{align*}{\frac{243}{256}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) }-{\frac{27\,x}{128}\sqrt{4\,{x}^{2}+9}}+{\frac{{x}^{3}}{16}\sqrt{4\,{x}^{2}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/256*arcsinh(2/3*x)-27/128*x*(4*x^2+9)^(1/2)+1/16*x^3*(4*x^2+9)^(1/2)

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Maxima [A]  time = 3.57066, size = 45, normalized size = 1. \begin{align*} \frac{1}{16} \, \sqrt{4 \, x^{2} + 9} x^{3} - \frac{27}{128} \, \sqrt{4 \, x^{2} + 9} x + \frac{243}{256} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

1/16*sqrt(4*x^2 + 9)*x^3 - 27/128*sqrt(4*x^2 + 9)*x + 243/256*arcsinh(2/3*x)

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Fricas [A]  time = 1.44861, size = 103, normalized size = 2.29 \begin{align*} \frac{1}{128} \,{\left (8 \, x^{3} - 27 \, x\right )} \sqrt{4 \, x^{2} + 9} - \frac{243}{256} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/128*(8*x^3 - 27*x)*sqrt(4*x^2 + 9) - 243/256*log(-2*x + sqrt(4*x^2 + 9))

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Sympy [A]  time = 0.671648, size = 39, normalized size = 0.87 \begin{align*} \frac{x^{3} \sqrt{4 x^{2} + 9}}{16} - \frac{27 x \sqrt{4 x^{2} + 9}}{128} + \frac{243 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{256} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(4*x**2+9)**(1/2),x)

[Out]

x**3*sqrt(4*x**2 + 9)/16 - 27*x*sqrt(4*x**2 + 9)/128 + 243*asinh(2*x/3)/256

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Giac [A]  time = 3.02374, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{128} \,{\left (8 \, x^{2} - 27\right )} \sqrt{4 \, x^{2} + 9} x - \frac{243}{256} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/128*(8*x^2 - 27)*sqrt(4*x^2 + 9)*x - 243/256*log(-2*x + sqrt(4*x^2 + 9))