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

Optimal. Leaf size=54 \[ \frac{1}{16} \sqrt{4 x^2-9} x^3+\frac{27}{128} \sqrt{4 x^2-9} x+\frac{243}{256} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0125138, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 217, 206} \[ \frac{1}{16} \sqrt{4 x^2-9} x^3+\frac{27}{128} \sqrt{4 x^2-9} x+\frac{243}{256} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]

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*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/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 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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

Mathematica [A]  time = 0.0129595, size = 43, normalized size = 0.8 \[ \frac{1}{256} \left (2 x \sqrt{4 x^2-9} \left (8 x^2+27\right )+243 \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )\right ) \]

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*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/256

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Maple [A]  time = 0.005, size = 49, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{16}\sqrt{4\,{x}^{2}-9}}+{\frac{27\,x}{128}\sqrt{4\,{x}^{2}-9}}+{\frac{243\,\sqrt{4}}{512}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x^3*(4*x^2-9)^(1/2)+27/128*x*(4*x^2-9)^(1/2)+243/512*ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

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Maxima [A]  time = 3.97886, size = 61, normalized size = 1.13 \begin{align*} \frac{1}{16} \, \sqrt{4 \, x^{2} - 9} x^{3} + \frac{27}{128} \, \sqrt{4 \, x^{2} - 9} x + \frac{243}{256} \, \log \left (8 \, x + 4 \, \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="maxima")

[Out]

1/16*sqrt(4*x^2 - 9)*x^3 + 27/128*sqrt(4*x^2 - 9)*x + 243/256*log(8*x + 4*sqrt(4*x^2 - 9))

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Fricas [A]  time = 1.31189, size = 103, normalized size = 1.91 \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 = 0.685653, size = 39, normalized size = 0.72 \begin{align*} \frac{x^{3} \sqrt{4 x^{2} - 9}}{16} + \frac{27 x \sqrt{4 x^{2} - 9}}{128} + \frac{243 \operatorname{acosh}{\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*acosh(2*x/3)/256

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Giac [A]  time = 1.90084, size = 50, normalized size = 0.93 \begin{align*} \frac{1}{128} \,{\left (8 \, x^{2} + 27\right )} \sqrt{4 \, x^{2} - 9} x - \frac{243}{256} \, \log \left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\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(abs(-2*x + sqrt(4*x^2 - 9)))