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

Optimal. Leaf size=27 \[ \frac {1}{8} x \sqrt {4 x^2+9}-\frac {9}{16} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]

[Out]

-9/16*arcsinh(2/3*x)+1/8*x*(4*x^2+9)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {321, 215} \[ \frac {1}{8} x \sqrt {4 x^2+9}-\frac {9}{16} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[9 + 4*x^2],x]

[Out]

(x*Sqrt[9 + 4*x^2])/8 - (9*ArcSinh[(2*x)/3])/16

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]

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]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {9+4 x^2}} \, dx &=\frac {1}{8} x \sqrt {9+4 x^2}-\frac {9}{8} \int \frac {1}{\sqrt {9+4 x^2}} \, dx\\ &=\frac {1}{8} x \sqrt {9+4 x^2}-\frac {9}{16} \sinh ^{-1}\left (\frac {2 x}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {1}{8} x \sqrt {4 x^2+9}-\frac {9}{16} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[9 + 4*x^2],x]

[Out]

(x*Sqrt[9 + 4*x^2])/8 - (9*ArcSinh[(2*x)/3])/16

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fricas [A]  time = 0.76, size = 29, normalized size = 1.07 \[ \frac {1}{8} \, \sqrt {4 \, x^{2} + 9} x + \frac {9}{16} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 + 9)*x + 9/16*log(-2*x + sqrt(4*x^2 + 9))

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giac [A]  time = 1.07, size = 29, normalized size = 1.07 \[ \frac {1}{8} \, \sqrt {4 \, x^{2} + 9} x + \frac {9}{16} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 + 9)*x + 9/16*log(-2*x + sqrt(4*x^2 + 9))

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maple [A]  time = 0.01, size = 20, normalized size = 0.74 \[ \frac {\sqrt {4 x^{2}+9}\, x}{8}-\frac {9 \arcsinh \left (\frac {2 x}{3}\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/16*arcsinh(2/3*x)+1/8*(4*x^2+9)^(1/2)*x

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maxima [A]  time = 2.92, size = 19, normalized size = 0.70 \[ \frac {1}{8} \, \sqrt {4 \, x^{2} + 9} x - \frac {9}{16} \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(4*x^2 + 9)*x - 9/16*arcsinh(2/3*x)

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mupad [B]  time = 0.03, size = 17, normalized size = 0.63 \[ \frac {x\,\sqrt {x^2+\frac {9}{4}}}{4}-\frac {9\,\mathrm {asinh}\left (\frac {2\,x}{3}\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(4*x^2 + 9)^(1/2),x)

[Out]

(x*(x^2 + 9/4)^(1/2))/4 - (9*asinh((2*x)/3))/16

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sympy [A]  time = 0.23, size = 22, normalized size = 0.81 \[ \frac {x \sqrt {4 x^{2} + 9}}{8} - \frac {9 \operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(4*x**2 + 9)/8 - 9*asinh(2*x/3)/16

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