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

Optimal. Leaf size=45 \[ \frac{1}{4} \sqrt{4 x^2+9} x^3+\frac{9}{32} \sqrt{4 x^2+9} x-\frac{81}{64} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(9*x*Sqrt[9 + 4*x^2])/32 + (x^3*Sqrt[9 + 4*x^2])/4 - (81*ArcSinh[(2*x)/3])/64

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Rubi [A]  time = 0.0095313, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 215} \[ \frac{1}{4} \sqrt{4 x^2+9} x^3+\frac{9}{32} \sqrt{4 x^2+9} x-\frac{81}{64} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[9 + 4*x^2],x]

[Out]

(9*x*Sqrt[9 + 4*x^2])/32 + (x^3*Sqrt[9 + 4*x^2])/4 - (81*ArcSinh[(2*x)/3])/64

Rule 279

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

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 x^2 \sqrt{9+4 x^2} \, dx &=\frac{1}{4} x^3 \sqrt{9+4 x^2}+\frac{9}{4} \int \frac{x^2}{\sqrt{9+4 x^2}} \, dx\\ &=\frac{9}{32} x \sqrt{9+4 x^2}+\frac{1}{4} x^3 \sqrt{9+4 x^2}-\frac{81}{32} \int \frac{1}{\sqrt{9+4 x^2}} \, dx\\ &=\frac{9}{32} x \sqrt{9+4 x^2}+\frac{1}{4} x^3 \sqrt{9+4 x^2}-\frac{81}{64} \sinh ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0143258, size = 36, normalized size = 0.8 \[ \sqrt{4 x^2+9} \left (\frac{x^3}{4}+\frac{9 x}{32}\right )-\frac{81}{64} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[9 + 4*x^2],x]

[Out]

Sqrt[9 + 4*x^2]*((9*x)/32 + x^3/4) - (81*ArcSinh[(2*x)/3])/64

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Maple [A]  time = 0.004, size = 32, normalized size = 0.7 \begin{align*}{\frac{x}{16} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{32}\sqrt{4\,{x}^{2}+9}}-{\frac{81}{64}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x*(4*x^2+9)^(3/2)-9/32*x*(4*x^2+9)^(1/2)-81/64*arcsinh(2/3*x)

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Maxima [A]  time = 2.7876, size = 42, normalized size = 0.93 \begin{align*} \frac{1}{16} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} x - \frac{9}{32} \, \sqrt{4 \, x^{2} + 9} x - \frac{81}{64} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*x^2 + 9)^(3/2)*x - 9/32*sqrt(4*x^2 + 9)*x - 81/64*arcsinh(2/3*x)

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Fricas [A]  time = 1.44884, size = 97, normalized size = 2.16 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{4 \, x^{2} + 9} + \frac{81}{64} \, \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^2*(4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

1/32*(8*x^3 + 9*x)*sqrt(4*x^2 + 9) + 81/64*log(-2*x + sqrt(4*x^2 + 9))

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Sympy [A]  time = 2.66339, size = 54, normalized size = 1.2 \begin{align*} \frac{x^{5}}{\sqrt{4 x^{2} + 9}} + \frac{27 x^{3}}{8 \sqrt{4 x^{2} + 9}} + \frac{81 x}{32 \sqrt{4 x^{2} + 9}} - \frac{81 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5/sqrt(4*x**2 + 9) + 27*x**3/(8*sqrt(4*x**2 + 9)) + 81*x/(32*sqrt(4*x**2 + 9)) - 81*asinh(2*x/3)/64

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Giac [A]  time = 2.48487, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{2} + 9\right )} \sqrt{4 \, x^{2} + 9} x + \frac{81}{64} \, \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^2*(4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/32*(8*x^2 + 9)*sqrt(4*x^2 + 9)*x + 81/64*log(-2*x + sqrt(4*x^2 + 9))