3.141 \(\int x^2 \sqrt{9-x^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{4} \sqrt{9-x^2} x^3-\frac{9}{8} \sqrt{9-x^2} x+\frac{81}{8} \sin ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

(-9*x*Sqrt[9 - x^2])/8 + (x^3*Sqrt[9 - x^2])/4 + (81*ArcSin[x/3])/8

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Rubi [A]  time = 0.0099616, 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, 216} \[ \frac{1}{4} \sqrt{9-x^2} x^3-\frac{9}{8} \sqrt{9-x^2} x+\frac{81}{8} \sin ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[9 - x^2],x]

[Out]

(-9*x*Sqrt[9 - x^2])/8 + (x^3*Sqrt[9 - x^2])/4 + (81*ArcSin[x/3])/8

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 216

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

Rubi steps

\begin{align*} \int x^2 \sqrt{9-x^2} \, dx &=\frac{1}{4} x^3 \sqrt{9-x^2}+\frac{9}{4} \int \frac{x^2}{\sqrt{9-x^2}} \, dx\\ &=-\frac{9}{8} x \sqrt{9-x^2}+\frac{1}{4} x^3 \sqrt{9-x^2}+\frac{81}{8} \int \frac{1}{\sqrt{9-x^2}} \, dx\\ &=-\frac{9}{8} x \sqrt{9-x^2}+\frac{1}{4} x^3 \sqrt{9-x^2}+\frac{81}{8} \sin ^{-1}\left (\frac{x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0150719, size = 33, normalized size = 0.73 \[ \frac{1}{8} \left (x \sqrt{9-x^2} \left (2 x^2-9\right )+81 \sin ^{-1}\left (\frac{x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[9 - x^2],x]

[Out]

(x*Sqrt[9 - x^2]*(-9 + 2*x^2) + 81*ArcSin[x/3])/8

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Maple [A]  time = 0.003, size = 32, normalized size = 0.7 \begin{align*} -{\frac{x}{4} \left ( -{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{8}\sqrt{-{x}^{2}+9}}+{\frac{81}{8}\arcsin \left ({\frac{x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x*(-x^2+9)^(3/2)+9/8*x*(-x^2+9)^(1/2)+81/8*arcsin(1/3*x)

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Maxima [A]  time = 1.40555, size = 42, normalized size = 0.93 \begin{align*} -\frac{1}{4} \,{\left (-x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{9}{8} \, \sqrt{-x^{2} + 9} x + \frac{81}{8} \, \arcsin \left (\frac{1}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.41846, size = 97, normalized size = 2.16 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{3} - 9 \, x\right )} \sqrt{-x^{2} + 9} - \frac{81}{4} \, \arctan \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.67154, size = 112, normalized size = 2.49 \begin{align*} \begin{cases} \frac{i x^{5}}{4 \sqrt{x^{2} - 9}} - \frac{27 i x^{3}}{8 \sqrt{x^{2} - 9}} + \frac{81 i x}{8 \sqrt{x^{2} - 9}} - \frac{81 i \operatorname{acosh}{\left (\frac{x}{3} \right )}}{8} & \text{for}\: \frac{\left |{x^{2}}\right |}{9} > 1 \\- \frac{x^{5}}{4 \sqrt{9 - x^{2}}} + \frac{27 x^{3}}{8 \sqrt{9 - x^{2}}} - \frac{81 x}{8 \sqrt{9 - x^{2}}} + \frac{81 \operatorname{asin}{\left (\frac{x}{3} \right )}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*x**5/(4*sqrt(x**2 - 9)) - 27*I*x**3/(8*sqrt(x**2 - 9)) + 81*I*x/(8*sqrt(x**2 - 9)) - 81*I*acosh(x
/3)/8, Abs(x**2)/9 > 1), (-x**5/(4*sqrt(9 - x**2)) + 27*x**3/(8*sqrt(9 - x**2)) - 81*x/(8*sqrt(9 - x**2)) + 81
*asin(x/3)/8, True))

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Giac [A]  time = 1.06193, size = 35, normalized size = 0.78 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{2} - 9\right )} \sqrt{-x^{2} + 9} x + \frac{81}{8} \, \arcsin \left (\frac{1}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*x^2 - 9)*sqrt(-x^2 + 9)*x + 81/8*arcsin(1/3*x)