3.2.41 \(\int x^2 \sqrt {9-x^2} \, dx\) [141]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {285, 327, 222} \begin {gather*} -\frac {9}{8} \sqrt {9-x^2} x+\frac {1}{4} \sqrt {9-x^2} x^3+\frac {81}{8} \sin ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

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 222

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]

Rule 285

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 327

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 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*}

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Mathematica [A]
time = 0.06, size = 46, normalized size = 1.02 \begin {gather*} \frac {1}{8} x \sqrt {9-x^2} \left (-9+2 x^2\right )-\frac {81}{4} \tan ^{-1}\left (\frac {\sqrt {9-x^2}}{3+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[9 - x^2],x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.70, size = 92, normalized size = 2.04 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (81 x-27 x^3+2 x^5-81 \text {ArcCosh}\left [\frac {x}{3}\right ] \sqrt {-9+x^2}\right )}{8 \sqrt {-9+x^2}},\text {Abs}\left [x^2\right ]>9\right \}\right \},\frac {-81 x}{8 \sqrt {9-x^2}}+\frac {27 x^3}{8 \sqrt {9-x^2}}-\frac {x^5}{4 \sqrt {9-x^2}}+\frac {81 \text {ArcSin}\left [\frac {x}{3}\right ]}{8}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^2*Sqrt[9 - x^2],x]')

[Out]

Piecewise[{{I / 8 (81 x - 27 x ^ 3 + 2 x ^ 5 - 81 ArcCosh[x / 3] Sqrt[-9 + x ^ 2]) / Sqrt[-9 + x ^ 2], Abs[x ^
2] > 9}}, -81 x / (8 Sqrt[9 - x ^ 2]) + 27 x ^ 3 / (8 Sqrt[9 - x ^ 2]) - x ^ 5 / (4 Sqrt[9 - x ^ 2]) + 81 Arc
Sin[x / 3] / 8]

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Maple [A]
time = 0.11, size = 32, normalized size = 0.71

method	result	size
default	\(-\frac {x \left (-x^{2}+9\right )^{\frac {3}{2}}}{4}+\frac {9 x \sqrt {-x^{2}+9}}{8}+\frac {81 \arcsin \left (\frac {x}{3}\right )}{8}\)	\(32\)
risch	\(-\frac {x \left (2 x^{2}-9\right ) \left (x^{2}-9\right )}{8 \sqrt {-x^{2}+9}}+\frac {81 \arcsin \left (\frac {x}{3}\right )}{8}\)	\(32\)
meijerg	\(-\frac {81 i \left (-\frac {i \sqrt {\pi }\, x \left (-\frac {2 x^{2}}{3}+3\right ) \sqrt {-\frac {x^{2}}{9}+1}}{18}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {x}{3}\right )}{2}\right )}{4 \sqrt {\pi }}\)	\(41\)
trager	\(\frac {x \left (2 x^{2}-9\right ) \sqrt {-x^{2}+9}}{8}+\frac {81 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+9}+x \right )}{8}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.36, size = 31, normalized size = 0.69 \begin {gather*} -\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 {gather*}

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 = 0.31, size = 39, normalized size = 0.87 \begin {gather*} \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 {gather*}

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 = 1.68, size = 110, normalized size = 2.44 \begin {gather*} \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}\: \left |{x^{2}}\right | > 9 \\- \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 {gather*}

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), (-x**5/(4*sqrt(9 - x**2)) + 27*x**3/(8*sqrt(9 - x**2)) - 81*x/(8*sqrt(9 - x**2)) + 81*a
sin(x/3)/8, True))

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Giac [A]
time = 0.00, size = 33, normalized size = 0.73 \begin {gather*} 2 \left (\frac {1}{8} x x-\frac {9}{16}\right ) x \sqrt {-x^{2}+9}+\frac {81}{8} \arcsin \left (\frac {x}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 27, normalized size = 0.60 \begin {gather*} \frac {81\,\mathrm {asin}\left (\frac {x}{3}\right )}{8}-\sqrt {9-x^2}\,\left (\frac {9\,x}{8}-\frac {x^3}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(9 - x^2)^(1/2),x)

[Out]

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

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